Opposites

📁 Source: Mathlib/CategoryTheory/Opposites.lean

Statistics

Metric	Count
Definitions opposite, opposite, leftOp, op, rightOp, unop, leftOp, op, rightOp, leftOp, leftOpComp, leftOpId, leftOpRightOpEquiv, leftOpRightOpIso, op, opComp, opHom, opId, opInv, opUnopEquiv, opUnopIso, rightOp, rightOpComp, rightOpId, rightOpLeftOpIso, unop, unopComp, unopId, unopOpIso, op, unop, op, removeOp, unop, leftOp, op, removeLeftOp, removeOp, removeRightOp, removeUnop, rightOp, unop, decidableEqOfUnop, isoOpEquiv, opEquiv, opOp, opOpEquivalence, unopUnop	48
Theorems leftOp_counitIso_hom_app, leftOp_counitIso_inv_app, leftOp_functor_map, leftOp_functor_obj, leftOp_inverse_map, leftOp_inverse_obj, leftOp_unitIso_hom_app, leftOp_unitIso_inv_app, op_counitIso, op_functor, op_inverse, op_unitIso, rightOp_counitIso_hom_app, rightOp_counitIso_inv_app, rightOp_functor_map, rightOp_functor_obj, rightOp_inverse_map, rightOp_inverse_obj, rightOp_unitIso_hom_app, rightOp_unitIso_inv_app, unop_counitIso, unop_functor, unop_inverse, unop_unitIso, instEssSurjOppositeLeftOp, instEssSurjOppositeOp, instEssSurjOppositeRightOp, instFaithfulOppositeOp, instFullOppositeOp, instIsEquivalenceOppositeLeftOp, instIsEquivalenceOppositeOp, instIsEquivalenceOppositeRightOp, leftOpComp_hom_app, leftOpComp_inv_app, leftOpId_hom_app, leftOpId_inv_app, leftOpRightOpEquiv_counitIso_hom_app_app, leftOpRightOpEquiv_counitIso_inv_app_app, leftOpRightOpEquiv_functor_map_app, leftOpRightOpEquiv_functor_obj_map, leftOpRightOpEquiv_functor_obj_obj, leftOpRightOpEquiv_inverse_map, leftOpRightOpEquiv_inverse_obj, leftOpRightOpEquiv_unitIso_hom_app, leftOpRightOpEquiv_unitIso_inv_app, leftOpRightOpIso_hom_app, leftOpRightOpIso_inv_app, leftOp_faithful, leftOp_full, leftOp_map, leftOp_obj, opComp_hom_app, opComp_inv_app, opHom_map_app, opHom_obj, opId_hom_app, opId_inv_app, opInv_map, opInv_obj, opUnopEquiv_counitIso, opUnopEquiv_functor, opUnopEquiv_inverse, opUnopEquiv_unitIso, opUnopIso_hom_app, opUnopIso_inv_app, op_map, op_obj, rightOpComp_hom_app, rightOpComp_inv_app, rightOpId_hom_app, rightOpId_inv_app, rightOpLeftOpIso_hom_app, rightOpLeftOpIso_inv_app, rightOp_faithful, rightOp_full, rightOp_leftOp_eq, rightOp_map, rightOp_map_unop, rightOp_obj, unopComp_hom_app, unopComp_inv_app, unopId_hom_app, unopId_inv_app, unopOpIso_hom_app, unopOpIso_inv_app, unop_map, unop_obj, op_hom, op_inv, op_refl, op_symm, op_trans, op_unop, unop_hom, unop_hom_inv_id_app, unop_hom_inv_id_app_assoc, unop_inv, unop_inv_hom_id_app, unop_inv_hom_id_app_assoc, unop_op, unop_refl, unop_symm, unop_trans, op_associator, op_hom, op_inv, op_isoWhiskerLeft, op_isoWhiskerRight, op_leftUnitor, op_refl, op_rightUnitor, op_symm, op_trans, removeOp_hom, removeOp_inv, unop_associator, unop_hom, unop_inv, unop_leftUnitor, unop_refl, unop_rightUnitor, unop_symm, unop_trans, unop_whiskerLeft, unop_whiskerRight, leftOpWhiskerRight, leftOpWhiskerRight_assoc, leftOp_app, leftOp_comp, leftOp_id, op_app, op_comp, op_comp_assoc, op_id, op_whiskerLeft, op_whiskerLeft_assoc, op_whiskerRight, op_whiskerRight_assoc, removeLeftOp_app, removeLeftOp_id, removeOp_app, removeOp_id, removeRightOp_app, removeRightOp_id, removeUnop_app, removeUnop_id, rightOpWhiskerRight, rightOpWhiskerRight_assoc, rightOp_app, rightOp_comp, rightOp_id, unop_app, unop_comp, unop_comp_assoc, unop_id, unop_whiskerLeft, unop_whiskerLeft_assoc, unop_whiskerRight, unop_whiskerRight_assoc, instIsEquivalenceOppositeOpOp, instIsEquivalenceOppositeUnopUnop, isIso_of_op, isIso_op, isIso_op_iff, isIso_unop, isIso_unop_iff, isoOpEquiv_apply, isoOpEquiv_symm_apply, opEquiv_apply, opEquiv_symm_apply, opOpEquivalence_counitIso, opOpEquivalence_functor, opOpEquivalence_inverse, opOpEquivalence_unitIso, opOp_map, opOp_obj, op_comp, op_comp_assoc, op_comp_unop, op_id, op_id_unop, op_inv, subsingleton_of_unop, unopUnop_map, unopUnop_obj, unop_comp, unop_comp_assoc, unop_id, unop_id_op, unop_inv, op_inj, op_unop, unop_inj, unop_mk, unop_op, unop_op'	196
Total	244

CategoryTheory

Definitions

Name	Category	Theorems
decidableEqOfUnop 📖	CompOp	—
isoOpEquiv 📖	CompOp	2 math math: isoOpEquiv_symm_apply, isoOpEquiv_apply
opEquiv 📖	CompOp	4 math math: Coyoneda.objOpOp_inv_app, Coyoneda.objOpOp_hom_app, opEquiv_apply, opEquiv_symm_apply
opOp 📖	CompOp	22 math math: opOp_obj, opOp_map, Pretriangulated.Opposite.OpOpCommShift.iso_inv_app, monoidalOpOp_δ, Pretriangulated.commShiftIso_opOp_inv_app, Cat.opEquivalence_counitIso, Pretriangulated.commShiftIso_opOp_hom_app_assoc, Pretriangulated.instIsTriangulatedOppositeOpOp, Functor.rightOpId_inv_app, opOpEquivalence_inverse, Cat.opEquivalence_unitIso, instIsEquivalenceOppositeOpOp, Pretriangulated.Opposite.OpOpCommShift.iso_hom_app_assoc, monoidalOpOp_μ, monoidalOpOp_η, Pretriangulated.Opposite.OpOpCommShift.iso_inv_app_assoc, monoidalOpOp_ε, opOpEquivalence_counitIso, Pretriangulated.commShiftIso_opOp_hom_app, Pretriangulated.commShiftIso_opOp_inv_app_assoc, Functor.rightOpId_hom_app, Pretriangulated.Opposite.OpOpCommShift.iso_hom_app
opOpEquivalence 📖	CompOp	17 math math: Equivalence.leftOp_unitIso_hom_app, Equivalence.rightOp_counitIso_inv_app, Equivalence.leftOp_unitIso_inv_app, Pretriangulated.instIsTriangulatedOppositeOpOpEquivalence, Equivalence.leftOp_counitIso_inv_app, Equivalence.rightOp_unitIso_hom_app, opOpEquivalence_inverse, Adjunction.rightOp_eq, Pretriangulated.instCommShiftOppositeOpOpEquivalenceInt, opOpEquivalence_unitIso, Adjunction.leftOp_eq, Equivalence.rightOp_counitIso_hom_app, instIsMonoidalOppositeOpOpEquivalence, opOpEquivalence_counitIso, Equivalence.rightOp_unitIso_inv_app, Equivalence.leftOp_counitIso_hom_app, opOpEquivalence_functor
unopUnop 📖	CompOp	22 math math: instIsEquivalenceOppositeUnopUnop, Pretriangulated.Opposite.UnopUnopCommShift.iso_inv_app, unopUnop_obj, Pretriangulated.Opposite.UnopUnopCommShift.iso_inv_app_assoc, Cat.opEquivalence_counitIso, Functor.leftOpId_hom_app, Cat.opEquivalence_unitIso, monoidalUnopUnop_ε, Pretriangulated.Opposite.UnopUnopCommShift.iso_hom_app_assoc, monoidalUnopUnop_δ, Pretriangulated.commShiftIso_unopUnop_inv_app, Functor.leftOpId_inv_app, opOpEquivalence_counitIso, unopUnop_map, Pretriangulated.commShiftIso_unopUnop_inv_app_assoc, monoidalUnopUnop_μ, Pretriangulated.Opposite.UnopUnopCommShift.iso_hom_app, Pretriangulated.commShiftIso_unopUnop_hom_app, Pretriangulated.commShiftIso_unopUnop_hom_app_assoc, monoidalUnopUnop_η, Pretriangulated.instIsTriangulatedOppositeUnopUnop, opOpEquivalence_functor

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
instIsEquivalenceOppositeOpOp 📖	mathematical	—	Functor.IsEquivalence Opposite Category.opposite opOp	—	Equivalence.isEquivalence_inverse
instIsEquivalenceOppositeUnopUnop 📖	mathematical	—	Functor.IsEquivalence Opposite Category.opposite unopUnop	—	Equivalence.isEquivalence_functor
isIso_of_op 📖	mathematical	—	IsIso	—	IsIso.inv_hom_id IsIso.hom_inv_id
isIso_op 📖	mathematical	—	IsIso Opposite Category.opposite Opposite.op Quiver.Hom.op CategoryStruct.toQuiver Category.toCategoryStruct	—	IsIso.inv_hom_id IsIso.hom_inv_id
isIso_op_iff 📖	mathematical	—	IsIso Opposite Category.opposite Opposite.op Quiver.Hom.op CategoryStruct.toQuiver Category.toCategoryStruct	—	isIso_of_op isIso_op
isIso_unop 📖	mathematical	—	IsIso Opposite.unop Quiver.Hom.unop CategoryStruct.toQuiver Category.toCategoryStruct	—	isIso_unop_iff
isIso_unop_iff 📖	mathematical	—	IsIso Opposite.unop Quiver.Hom.unop CategoryStruct.toQuiver Category.toCategoryStruct Opposite Category.opposite	—	isIso_op_iff Quiver.Hom.op_unop
isoOpEquiv_apply 📖	mathematical	—	DFunLike.coe Equiv Iso Opposite Category.opposite Opposite.unop EquivLike.toFunLike Equiv.instEquivLike isoOpEquiv Iso.unop	—	—
isoOpEquiv_symm_apply 📖	mathematical	—	DFunLike.coe Equiv Iso Opposite.unop Opposite Category.opposite EquivLike.toFunLike Equiv.instEquivLike Equiv.symm isoOpEquiv Iso.op	—	—
opEquiv_apply 📖	mathematical	—	DFunLike.coe Equiv Quiver.Hom Opposite Quiver.opposite CategoryStruct.toQuiver Category.toCategoryStruct Opposite.unop EquivLike.toFunLike Equiv.instEquivLike opEquiv Quiver.Hom.unop	—	—
opEquiv_symm_apply 📖	mathematical	—	DFunLike.coe Equiv Quiver.Hom CategoryStruct.toQuiver Category.toCategoryStruct Opposite.unop Opposite Quiver.opposite EquivLike.toFunLike Equiv.instEquivLike Equiv.symm opEquiv Quiver.Hom.op	—	—
opOpEquivalence_counitIso 📖	mathematical	—	Equivalence.counitIso Opposite Category.opposite opOpEquivalence Iso.refl Functor Functor.category Functor.comp opOp unopUnop	—	—
opOpEquivalence_functor 📖	mathematical	—	Equivalence.functor Opposite Category.opposite opOpEquivalence unopUnop	—	—
opOpEquivalence_inverse 📖	mathematical	—	Equivalence.inverse Opposite Category.opposite opOpEquivalence opOp	—	—
opOpEquivalence_unitIso 📖	mathematical	—	Equivalence.unitIso Opposite Category.opposite opOpEquivalence Iso.refl Functor Functor.category Functor.id	—	—
opOp_map 📖	mathematical	—	Functor.map Opposite Category.opposite opOp Quiver.Hom.op Quiver.opposite CategoryStruct.toQuiver Category.toCategoryStruct Opposite.op	—	—
opOp_obj 📖	mathematical	—	Functor.obj Opposite Category.opposite opOp Opposite.op	—	—
op_comp 📖	mathematical	—	Quiver.Hom.op CategoryStruct.toQuiver CategoryStruct.comp Opposite CategoryStruct.opposite Opposite.op	—	—
op_comp_assoc 📖	mathematical	—	CategoryStruct.comp Opposite CategoryStruct.opposite Category.toCategoryStruct Opposite.op Quiver.Hom.op CategoryStruct.toQuiver	—	Category.assoc
op_comp_unop 📖	mathematical	—	Quiver.Hom.op CategoryStruct.toQuiver Opposite.unop CategoryStruct.comp Quiver.Hom.unop Opposite CategoryStruct.opposite	—	—
op_id 📖	mathematical	—	Quiver.Hom.op CategoryStruct.toQuiver CategoryStruct.id Opposite CategoryStruct.opposite Opposite.op	—	—
op_id_unop 📖	mathematical	—	Quiver.Hom.op CategoryStruct.toQuiver Opposite.unop CategoryStruct.id Opposite CategoryStruct.opposite	—	—
op_inv 📖	mathematical	—	Quiver.Hom.op CategoryStruct.toQuiver Category.toCategoryStruct inv Opposite Category.opposite Opposite.op isIso_op	—	IsIso.eq_inv_of_hom_inv_id isIso_op op_comp IsIso.inv_hom_id op_id
subsingleton_of_unop 📖	mathematical	—	Quiver.Hom Opposite Quiver.opposite CategoryStruct.toQuiver Category.toCategoryStruct	—	Equiv.subsingleton
unopUnop_map 📖	mathematical	—	Functor.map Opposite Category.opposite unopUnop Quiver.Hom.unop CategoryStruct.toQuiver Category.toCategoryStruct Opposite.unop	—	—
unopUnop_obj 📖	mathematical	—	Functor.obj Opposite Category.opposite unopUnop Opposite.unop	—	—
unop_comp 📖	mathematical	—	Quiver.Hom.unop CategoryStruct.toQuiver CategoryStruct.comp Opposite CategoryStruct.opposite Opposite.unop	—	—
unop_comp_assoc 📖	mathematical	—	CategoryStruct.comp Category.toCategoryStruct Opposite.unop Quiver.Hom.unop CategoryStruct.toQuiver Opposite CategoryStruct.opposite	—	Category.assoc
unop_id 📖	mathematical	—	Quiver.Hom.unop CategoryStruct.toQuiver CategoryStruct.id Opposite CategoryStruct.opposite Opposite.unop	—	—
unop_id_op 📖	mathematical	—	Quiver.Hom.unop CategoryStruct.toQuiver Opposite.op CategoryStruct.id Opposite CategoryStruct.opposite	—	—
unop_inv 📖	mathematical	—	Quiver.Hom.unop CategoryStruct.toQuiver Category.toCategoryStruct inv Opposite Category.opposite Opposite.unop isIso_unop	—	IsIso.eq_inv_of_hom_inv_id isIso_unop unop_comp IsIso.inv_hom_id unop_id

CategoryTheory.Category

Definitions

Name	Category	Theorems
opposite 📖	CompOp	6428 math math: CategoryTheory.Functor.IsDenseSubsite.mapPreimage_map_of_fac, AlgebraicGeometry.continuousMapPresheaf_obj, CategoryTheory.Sheaf.cartesianMonoidalCategoryLift_val, CategoryTheory.GrothendieckTopology.coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation_pt, SSet.OneTruncation₂.nerveEquiv_apply, HomotopicalAlgebra.fibration_op_iff, CategoryTheory.Presheaf.instIsCardinalPresentableFunctorOppositeFreeYonedaOfHasColimitsOfSize, CategoryTheory.Comon.tensorObj_comul', AlgebraicTopology.DoldKan.natTransPInfty_app, Condensed.finYoneda_obj, AlgebraicGeometry.Scheme.toSpecΓ_apply, CategoryTheory.LocalizerMorphism.LeftResolution.opFunctor_map_f, PresheafOfModules.Monoidal.tensorObj_obj, CategoryTheory.GrothendieckTopology.isoSheafify_hom, SSet.op_δ, AlgebraicGeometry.Γ_map_morphismRestrict, CategoryTheory.Limits.pushoutIsoOpPullback_inr_hom_assoc, CategoryTheory.CommSq.LiftStruct.op_l, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_inv_app_eq, CategoryTheory.Limits.colimitHomIsoLimitYoneda'_hom_comp_π, AlgebraicGeometry.Scheme.Modules.pushforward_obj_presheaf_map, CochainComplex.acyclic_op, CategoryTheory.Functor.functorHomEquiv_apply_app, CategoryTheory.Pseudofunctor.DescentData.isEquivalence_toDescentData_of_sieve_le, CategoryTheory.SimplicialObject.id_left_app, CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence.functor_obj_hom_right, CategoryTheory.Functor.FullyFaithful.homNatIsoMaxRight_inv_app, CategoryTheory.GrothendieckTopology.W_sheafToPresheaf_map_iff_isIso, SSet.stdSimplex.objMk_bijective, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_skyscraperPresheafHomEquiv_symm, SSet.Subcomplex.lift_ι, PresheafOfModules.instIsRightAdjointPushforwardCompFunctorOppositeRingCatWhiskerLeftOp, CategoryTheory.Adjunction.compUliftCoyonedaIso_hom_app_app_down, CategoryTheory.sheafBotEquivalence_functor, smoothSheafCommRing.ι_forgetStalk_inv, CategoryTheory.GrothendieckTopology.plusFunctorWhiskerLeftIso_inv_app, SSet.Truncated.HomotopyCategory.BinaryProduct.iso_inv_toFunctor, CategoryTheory.GrothendieckTopology.Plus.toPlus_mk, CategoryTheory.Limits.walkingParallelPairOpEquiv_unitIso_hom_app_zero, CategoryTheory.uliftCoyonedaEquiv_apply, CategoryTheory.Adjunction.leftOp_unit, CategoryTheory.GrothendieckTopology.Point.Hom.presheafFiber_app, CategoryTheory.PreOneHypercover.forkOfIsColimit_ι_map_inj_assoc, CategoryTheory.CosimplicialObject.Augmented.leftOpRightOpIso_hom_left, SheafOfModules.pushforward_assoc, CategoryTheory.instSmallOppositeObjFunctorTypeYoneda, CategoryTheory.PresheafOfGroups.OneCochain.one_ev, CategoryTheory.ShortComplex.HomologyMapData.op_left, AlgebraicGeometry.Scheme.ΓSpecIso_inv_naturality_assoc, SSet.RelativeMorphism.image_le, CategoryTheory.Limits.IndizationClosedUnderFilteredColimitsAux.exists_nonempty_limit_obj_of_isColimit, CategoryTheory.ShortComplex.LeftHomologyData.op_g', PresheafOfModules.Sheafify.app_eq_of_isLocallyInjective, AddCommGrpCat.coyoneda_obj_obj_coe, AlgebraicGeometry.Proj.awayMap_awayToSection_assoc, CategoryTheory.Equivalence.leftOp_unitIso_hom_app, CategoryTheory.linearCoyoneda_obj_additive, CategoryTheory.SimplicialObject.whiskering_obj_map_app, CategoryTheory.Functor.IsDenseSubsite.instIsIsoSheafAppCounitSheafAdjunctionCocontinuous, CategoryTheory.instIsEquivalenceOppositeUnopUnop, CategoryTheory.ComposableArrows.opEquivalence_counitIso_inv_app_app, CategoryTheory.Equivalence.preregular_isSheaf_iff, CategoryTheory.Pseudofunctor.DescentData.ofObj_hom, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_fst_assoc, AlgebraicGeometry.Proj.fromOfGlobalSections_resLE, HomologicalComplex.evalCompCoyonedaCorepresentableByDoubleId_homEquiv_apply, CategoryTheory.Functor.natTransEquiv_apply_app, AlgebraicGeometry.Scheme.map_PrimeSpectrum_basicOpen_of_affine, CategoryTheory.uliftCoyonedaIsoCoyoneda_hom_app_app, CategoryTheory.Pseudofunctor.DescentData.subtypeCompatibleHomEquiv_toCompatible_presheafHomObjHomEquiv, SheafOfModules.instAdditivePresheafOfModulesObjFunctorOppositeRingCatIsSheafForget, CategoryTheory.coyonedaEquiv_symm_app_apply, CategoryTheory.Presheaf.instIsLocallySurjectiveHomWhiskerRightOppositeForget, AlgebraicGeometry.Scheme.Modules.pseudofunctor_map_adj, CategoryTheory.Functor.leibnizPullback_obj_map, CategoryTheory.MonoidalCategory.MonoidalRightAction.rightActionOfOppositeRightAction_actionRight_unop, CategoryTheory.Sheaf.ΓHomEquiv_naturality_left_symm, CategoryTheory.isIso_sheafificationAdjunction_counit, SSet.Subcomplex.preimage_eq_top_iff, AlgebraicGeometry.StructureSheaf.instIsScalarTowerCarrierStalkCommRingCatStructurePresheafInCommRingCatCarrierAbPresheafOpensCarrierTopModuleStructurePresheaf, CategoryTheory.kernelUnopOp_inv, CategoryTheory.Sheaf.isLocallySurjective_sheafToPresheaf_map_iff, CategoryTheory.Limits.reflectsLimitsOfShape_leftOp, CategoryTheory.ParametrizedAdjunction.homEquiv_symm_naturality_three_assoc, CategoryTheory.Functor.sheafPushforwardContinuousIso_inv, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.shortComplex_X₃, CategoryTheory.preservesFiniteLimits_iff_lan_preservesFiniteLimits, CategoryTheory.GrothendieckTopology.Cover.index_left, CategoryTheory.Pretriangulated.shiftFunctorCompIsoId_op_hom_app, CategoryTheory.Localization.isLocalization_op, CategoryTheory.GrothendieckTopology.toPlus_plusLift_assoc, CategoryTheory.Pretriangulated.shift_opShiftFunctorEquivalence_counitIso_inv_app, CategoryTheory.Limits.preservesColimitsOfShape_rightOp, AlgebraicGeometry.Spec.toPresheafedSpace_obj, CategoryTheory.GrothendieckTopology.liftToDiagramLimitObjAux_fac, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_appTop, LightProfinite.proj_comp_transitionMap, CategoryTheory.GrothendieckTopology.preservesColimitsOfShape_yoneda_of_ofArrows_inj_mem, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo, CategoryTheory.ULiftYoneda.instFullFunctorOppositeTypeUliftYoneda, CategoryTheory.yoneda_preservesLimit, CategoryTheory.Pseudofunctor.DescentData'.comm, CategoryTheory.ObjectProperty.instIsClosedUnderLimitsOfShapeUnopOppositeOfIsClosedUnderColimitsOfShape, CategoryTheory.opHom_apply, LightProfinite.Extend.functorOp_obj, AlgebraicGeometry.Scheme.zeroLocus_iInf, CategoryTheory.Limits.colimitHomIsoLimitYoneda_hom_comp_π, AlgebraicGeometry.Scheme.Modules.pushforwardId_inv_app_app, AddCommMonCat.coyoneda_obj_obj_coe, CategoryTheory.IsPullback.op_iff, AlgebraicGeometry.IsAffineOpen.map_fromSpec_assoc, LightCondensed.lanPresheafIso_hom, CategoryTheory.additive_yonedaObj, CategoryTheory.Equivalence.symmEquivFunctor_obj, AlgebraicGeometry.IsAffineOpen.isLocalization_of_eq_basicOpen, CategoryTheory.DinatTrans.dinaturality_assoc, CategoryTheory.Presieve.shrinkFunctorHomEquiv_apply_coe, SSet.stdSimplex.mem_nonDegenerate_iff_strictMono, SSet.stdSimplex.coe_triangle_down_toOrderHom, AlgebraicGeometry.Etale.etale_appLE, CategoryTheory.SimplicialObject.δ_comp_δ_self_assoc, CategoryTheory.Presheaf.isSheaf_of_isTerminal, CategoryTheory.Limits.isIndObject_limit_of_discrete, CategoryTheory.ShortComplex.hasRightHomology_iff_op, CategoryTheory.regularTopology.EqualizerCondition.bijective_mapToEqualizer_pullback', CategoryTheory.Limits.limitUnopIsoUnopColimit_hom_comp_ι, CategoryTheory.ObjectProperty.colimitsOfShape_op, CategoryTheory.Limits.reflectsColimitsOfShape_of_rightOp, AlgebraicGeometry.Scheme.Hom.germ_stalkMap_assoc, AlgebraicGeometry.LocallyQuasiFinite.quasiFinite_appLE, CategoryTheory.GrothendieckTopology.Point.skyscraperPresheafAdjunction_homEquiv_symm_apply, TopCat.presheafToType_map, CategoryTheory.Functor.leftOpRightOpEquiv_functor_obj_map, TopCat.Sheaf.interUnionPullbackCone_snd, CategoryTheory.linearCoyoneda_map_app, CategoryTheory.linearCoyoneda_obj_obj_carrier, LightCondensed.isoFinYonedaComponents_hom_apply, CategoryTheory.ShortComplex.Splitting.op_r, SSet.RelativeMorphism.botEquiv_symm_apply_map, CategoryTheory.ShortComplex.SnakeInput.op_δ, CategoryTheory.Limits.isLimitConeRightOpOfCocone_lift, CategoryTheory.ObjectProperty.isClosedUnderColimitsOfShape_iff_op, AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToSection_apply, CategoryTheory.Limits.preservesFiniteCoproducts_rightOp, CategoryTheory.NatTrans.ofOpSequence_app, SSet.Truncated.tensor_map_apply_snd, SSet.Subcomplex.prod_top_le_unionProd, CategoryTheory.ShortComplex.cyclesOpIso_inv_op_iCycles_assoc, CategoryTheory.Pseudofunctor.isEquivalence_toDescentDataAsCoalgebra_iff_isEquivalence_comonadComparison, CategoryTheory.MorphismProperty.instHasOfPrecompPropertyOppositeOpOfHasOfPostcompProperty, PresheafOfModules.ofPresheaf_obj_isModule, CategoryTheory.Limits.pullbackIsoUnopPushout_inv_snd, SSet.oneTruncation₂_obj, AlgebraicGeometry.Scheme.Modules.instFaithfulPresheafOfModulesToPresheafOfModules, AlgebraicGeometry.PresheafedSpace.stalkMap_germ, CategoryTheory.Presieve.FamilyOfElements.isCompatible_map_smul, CategoryTheory.CategoryOfElements.costructuredArrowYonedaEquivalenceInverseπ_hom_app, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_add_unitIso_hom_app_eq, CategoryTheory.op_hom_leftUnitor, SSet.OneTruncation₂.ofNerve₂.natIso_hom_app_map, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit_app, CategoryTheory.shrinkYoneda_map, CategoryTheory.isSeparator_iff_faithful_preadditiveCoyoneda, CategoryTheory.GrothendieckTopology.Point.skyscraperSheafFunctor_obj_obj, CategoryTheory.regularTopology.equalizerCondition_precomp_of_preservesPullback, CategoryTheory.ParametrizedAdjunction.homEquiv_naturality_two, AlgebraicGeometry.LocallyRingedSpace.Γ_def, CategoryTheory.GrothendieckTopology.yonedaOpCompCoyoneda_hom_app_app_down, AlgebraicGeometry.Scheme.IsQuasiAffine.toIsImmersion, CategoryTheory.GrothendieckTopology.pseudofunctorOver_toPrelaxFunctor_toPrelaxFunctorStruct_map₂, CategoryTheory.ShortComplex.homologyOpIso_hom_naturality, AlgebraicGeometry.StructureSheaf.globalSectionsIso_inv, CategoryTheory.Limits.coneOpEquiv_unitIso, PresheafOfModules.pullback_id_comp, AlgebraicGeometry.Scheme.ofRestrict_appIso, SSet.PtSimplex.RelStruct.δ_map_of_lt, CategoryTheory.isSeparating_unop_iff, SSet.Subcomplex.toRange_ι, CategoryTheory.NatTrans.unop_whiskerLeft, ContinuousMap.yonedaPresheaf'_obj, AlgebraicGeometry.RingedSpace.basicOpen_res, SSet.Homotopy.congr_homologyMap_singularChainComplexFunctor, CategoryTheory.Comma.opFunctor_obj, AlgebraicGeometry.Scheme.Hom.app_invApp'_assoc, AlgebraicGeometry.LocallyRingedSpace.restrict_presheaf_obj, CategoryTheory.ShortComplex.LeftHomologyMapData.unop_φQ, CategoryTheory.Limits.multicospanIndexEnd_fst, AlgebraicTopology.DoldKan.σ_comp_PInfty_assoc, CategoryTheory.Functor.IsDenseSubsite.mapPreimage_id, CategoryTheory.Equalizer.Sieve.equalizer_sheaf_condition, CategoryTheory.SimplicialObject.δ_comp_δ''_assoc, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv, CategoryTheory.Limits.preservesFiniteLimits_of_leftOp, CategoryTheory.Pretriangulated.Opposite.mem_distinguishedTriangles_iff, LightCondensed.ihomPoints_apply, Condensed.instPreservesFiniteProductsOppositeStoneanObjFunctorIsSheafCoherentTopology, AlgebraicGeometry.Scheme.app_eq, CategoryTheory.shrinkYonedaEquiv_comp, CategoryTheory.Functor.sheafAdjunctionCocontinuous_homEquiv_apply_hom, CategoryTheory.ShortComplex.LeftHomologyData.unop_p, CategoryTheory.eval_app, CategoryTheory.GrothendieckTopology.toSheafify_naturality_assoc, CategoryTheory.OverPresheafAux.costructuredArrowPresheafToOver_map, CategoryTheory.Subfunctor.Subpresheaf.range_eq_ofSection', HomotopicalAlgebra.instWeakEquivalenceOppositeOp, CategoryTheory.Square.unop_X₂, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_presheafFiberDesc, CategoryTheory.Localization.isoOfHom_unop, SSet.Subcomplex.prodIso_hom, CategoryTheory.Groupoid.invEquivalence_inverse_map, AlgebraicTopology.NormalizedMooreComplex.obj_d, CategoryTheory.SimplicialObject.δ_comp_σ_of_gt, CategoryTheory.Functor.IsDenseSubsite.instIsEquivalenceSheafSheafPushforwardContinuous, CategoryTheory.ObjectProperty.essentiallySmall_unop_iff, CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan.uliftYonedaEquiv_ι_presheafHom, CategoryTheory.Pseudofunctor.DescentData.pullFunctorCompIso_inv_app_hom, CategoryTheory.Limits.Cone.unop_ι, CategoryTheory.nerve.instFullCatTruncatedOfNatNatNerveFunctor₂, CategoryTheory.Pseudofunctor.isEquivalence_toDescentData, CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinsetLimitCone_isLimit_lift, CategoryTheory.Pseudofunctor.DescentData'.comp_pullHom', CategoryTheory.Equivalence.sheafCongr_functor, CategoryTheory.GrothendieckTopology.Cover.index_fst, CategoryTheory.GrothendieckTopology.uliftYonedaIsoYoneda_inv_app_hom_app_down, SSet.horn₃₁.desc.multicofork_π_two, SSet.stdSimplex.δ_zero_eq_const, AlgebraicGeometry.LocallyOfFiniteType.finiteType_of_affine_subset, AlgebraicGeometry.Scheme.zeroLocus_empty_eq_univ, CategoryTheory.CategoryOfElements.costructuredArrowYonedaEquivalence_functor, CategoryTheory.Limits.walkingSpanOpEquiv_inverse_map, AlgebraicGeometry.mono_pushoutSection_of_iSup_eq, CategoryTheory.Limits.widePushoutShapeUnop_obj, CompHausLike.LocallyConstant.functor_obj_obj, CategoryTheory.Subfunctor.isSeparated, CategoryTheory.Limits.reflectsFiniteColimits_op, CategoryTheory.Presheaf.coherentExtensiveEquivalence_inverse_map_hom, AlgebraicGeometry.StructureSheaf.comap_id, CategoryTheory.OverPresheafAux.unitAux_hom, Profinite.Extend.cocone_pt, CategoryTheory.isCofiltered_op_of_isFiltered, CategoryTheory.cocones_map_app_app, CategoryTheory.CategoryOfElements.costructuredArrowULiftYonedaEquivalence_unitIso, CategoryTheory.IndParallelPairPresentation.hf, CategoryTheory.Sieve.functor_obj, RingHom.IsStableUnderBaseChange.pullback_fst_appTop, PresheafOfModules.instIsLocallySurjectiveToSheafify, PresheafOfModules.add_app, CategoryTheory.Functor.mapTriangleOpCompTriangleOpEquivalenceFunctorApp_inv_hom₁, SSet.Truncated.sk.full, CategoryTheory.Equivalence.sheafCongr.inverse_obj_obj_map, CategoryTheory.Subfunctor.Subpresheaf.range_eq_ofSection, SSet.ι₀_snd_assoc, CategoryTheory.Limits.widePushoutShapeOpEquiv_functor, CategoryTheory.Limits.coconeLeftOpOfConeEquiv_functor_map_hom, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverse_obj_π_app, SSet.Truncated.HomotopyCategory.descOfTruncation_comp, condensedSetToTopCat_obj_carrier, CategoryTheory.Limits.FormalCoproduct.cechIsoCechNerve_hom_app, TopCat.Presheaf.germ_eq_of_isBasis, CategoryTheory.Functor.rightOp_map, SSet.Subcomplex.mem_ofSimplex_obj_iff, AlgebraicTopology.DoldKan.N₁_map_f, AlgebraicGeometry.instIsIsoSchemeCoprodComparisonOppositeCommRingCatSpec, SSet.Edge.CompStruct.exists_of_simplex, CategoryTheory.MorphismProperty.IsStableUnderCobaseChange.op, CategoryTheory.Limits.preservesLimits_of_leftOp, CondensedMod.IsSolid.isIso_solidification_map, CategoryTheory.Limits.coneRightOpOfCoconeEquiv_functor_obj, CategoryTheory.instFullMonFunctorOppositeMonCatYonedaMon, AlgebraicGeometry.IsAffineOpen.map_fromSpec, CategoryTheory.SimplicialThickening.SimplicialCategory.comp_id, IsOpenMap.pullbackIso_hom_app_app, AlgebraicGeometry.StructureSheaf.const_mul_rev, CategoryTheory.NonemptyParallelPairPresentationAux.hf, AlgebraicGeometry.ΓSpec.isIso_adjunction_counit, AlgebraicGeometry.Spec_Γ_naturality, SSet.degenerate_eq_top_of_hasDimensionLT, CategoryTheory.FunctorToTypes.functorHomEquiv_symm_apply_app_app, PresheafOfModules.toPresheaf_preservesFiniteLimits, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo_Spec_fromSpecStalk_assoc, TopCat.Presheaf.Pushforward.id_inv_app, LightCondensed.instPreservesFiniteProductsOppositeLightProfiniteObjFunctorIsSheafCoherentTopology, CategoryTheory.Equivalence.sheafCongrPrecoherent_unitIso_inv_app_hom_app, CategoryTheory.MorphismProperty.LeftFraction.op_map, CategoryTheory.GrothendieckTopology.Point.toPresheafFiberOfIsCofiltered_w_assoc, CategoryTheory.WithInitial.opEquiv_unitIso_inv_app, AlgebraicGeometry.AffineScheme.Spec_faithful, CategoryTheory.Limits.pushoutIsoOpPullback_inl_hom, SSet.Subcomplex.mem_unionProd_iff, CategoryTheory.MorphismProperty.isIso_fst'_self, CategoryTheory.GrothendieckTopology.Point.skyscraperPresheafFunctor_map_app, AlgebraicTopology.DoldKan.HigherFacesVanish.inclusionOfMooreComplexMap, AlgebraicGeometry.IsLocallyArtinian.isArtinianRing_presheaf_obj, SSet.Truncated.mapHomotopyCategory_homMk, AlgebraicGeometry.Scheme.Opens.toSpecΓ_naturality, CategoryTheory.Limits.cospanUnop_hom_app, TopCat.Presheaf.germ_stalkPullbackHom, CategoryTheory.orderDualEquivalence_unitIso, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_presheaf_map_top, skyscraperPresheaf_obj, AlgebraicGeometry.LocallyRingedSpace.notMem_prime_iff_unit_in_stalk, CategoryTheory.Equivalence.sheafCongr.unitIso_inv_app_hom_app, SSet.modelCategoryQuillen.I_le_monomorphisms, CategoryTheory.Pseudofunctor.DescentData'.pullHom'_eq_pullHom_assoc, CategoryTheory.opOp_obj, AlgebraicGeometry.Scheme.restrictFunctorΓ_inv_app, CategoryTheory.Sheaf.instIsLocallySurjectiveAppArrowILocallySurjectiveLocallyInjectiveFunctorialLocallySurjectiveInjectiveFactorization, CategoryTheory.op_inv_associator, Condensed.hom_ext_iff, CategoryTheory.SimplicialObject.σ_naturality_assoc, CategoryTheory.Limits.walkingParallelPairOp_left, CategoryTheory.Arrow.augmentedCechNerve_hom_app, imageToKernel_unop, CategoryTheory.ran_isSheaf_of_isCocontinuous, SSet.instIsDiscreteHomotopyCategoryObjTruncatedOfNatNatTruncationSimplexCategoryStdSimplexMk, CategoryTheory.shrinkYonedaEquiv_symm_map_assoc, SSet.oneTruncation₂_map, AlgebraicGeometry.Scheme.Modules.restrictAdjunction_counit_app_app, CategoryTheory.ShortComplex.op_g, CategoryTheory.Limits.ι_comp_colimitOpIsoOpLimit_hom_assoc, AlgebraicGeometry.preservesColimit_Γ, CategoryTheory.GrothendieckTopology.diagram_obj, AlgebraicGeometry.instIsDomainCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTopOfIsIntegral, AlgebraicGeometry.instPreservesLimitSchemeOppositeCommRingCatRightOpΓOfIsAffineHomMapOfCompactSpaceOfQuasiSeparatedSpaceCarrierCarrierObj, CategoryTheory.GrothendieckTopology.yonedaULiftEquiv_apply, CategoryTheory.Pseudofunctor.DescentData'.instIsIsoαCategoryObjLocallyDiscreteOppositeCatMkOpPullbackHom, SSet.horn.faceSingletonComplIso_inv_ι_assoc, CategoryTheory.AddMonObj.ofRepresentableBy_add, commBialgCatEquivComonCommAlgCat_unitIso_inv_app, CategoryTheory.lan_preservesFiniteLimits_of_flat, AlgebraicTopology.DoldKan.MorphComponents.preComp_a, AlgebraicGeometry.StructureSheaf.comap_comp, CategoryTheory.GrothendieckTopology.yonedaEquiv_symm_app_apply, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_hom_eq_germ, CategoryTheory.Limits.parallelPairOpIso_inv_app_zero, CategoryTheory.SimplicialObject.whiskering_obj_obj_σ, SheafOfModules.pushforwardNatIso_inv, CategoryTheory.ObjectProperty.instIsClosedUnderColimitsOfShapeOppositeOpOfIsClosedUnderLimitsOfShape, CategoryTheory.Arrow.mapCechNerve_app, CategoryTheory.Functor.CorepresentableBy.uniqueUpToIso_inv, CategoryTheory.PreOneHypercover.forkOfIsColimit_pt, SSet.degenerate_iff_of_mono, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_eq_iff', CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence.functor_obj_left_hom, CategoryTheory.ShortComplex.unop_X₂, AlgebraicGeometry.Proj.fromOfGlobalSections_toSpecZero_assoc, CategoryTheory.GrothendieckTopology.liftToDiagramLimitObjAux_fac_assoc, CategoryTheory.GrothendieckTopology.Point.skyscraperPresheafHomEquiv_naturality_left_symm, CategoryTheory.ShortComplex.ShortExact.op, CategoryTheory.Functor.hom_map, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_symm_apply, PresheafOfModules.Sheafify.add_smul, CategoryTheory.Functor.uliftCoyonedaCoreprXIso_hom_app, CategoryTheory.NatTrans.removeLeftOp_id, CategoryTheory.SimplicialObject.comp_right, CategoryTheory.eval_map, CategoryTheory.isCoseparator_iff_faithful_preadditiveYoneda, CategoryTheory.Functor.IsCoverDense.Types.sheafIso_inv_hom, CategoryTheory.SimplicialObject.Augmented.point_map, Profinite.Extend.functorOp_map, AlgebraicGeometry.functionField_isFractionRing_of_isAffineOpen, CategoryTheory.GrothendieckTopology.W_eq_inverseImage_isomorphisms, AlgebraicGeometry.Scheme.IdealSheafData.ideal_sSup, AlgebraicGeometry.Scheme.fromSpecStalk_toSpecΓ_assoc, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.counit, SSet.stdSimplex.faceSingletonIso_zero_hom_comp_ι_eq_δ, SSet.stdSimplex.ι₁_whiskerLeft_toSSetObjI_μ, AlgebraicGeometry.Scheme.Modules.pushforwardCongr_hom_app_app, AlgebraicGeometry.AffineSpace.SpecIso_hom_appTop, CategoryTheory.Presheaf.IsSheaf.amalgamate_map, PresheafOfModules.restrictScalars_map_app, CategoryTheory.MonoidalCategory.MonoidalLeftAction.oppositeLeftAction_actionHomRight, CategoryTheory.MorphismProperty.LeftFraction.op_X', CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.coalgebraEquivalence_inverse_obj_hom, AlgebraicGeometry.LocallyRingedSpace.basicOpen_eq_bot_iff_forall_evaluation_eq_zero, CategoryTheory.Limits.coneRightOpOfCoconeEquiv_functor_map_hom, CategoryTheory.NatIso.op_rightUnitor, CategoryTheory.op_tensorHom, SSet.σ_mem_degenerate, AlgebraicGeometry.Scheme.Modules.pushforward_obj_obj, CategoryTheory.Limits.coneUnopOfCoconeEquiv_counitIso, SSet.ofSimplex_le_skeleton, CategoryTheory.Comon.MonOpOpToComon_obj, SSet.Subcomplex.fromPreimage_ι_assoc, SSet.Subcomplex.toSSetFunctor_map, CategoryTheory.Pretriangulated.Opposite.UnopUnopCommShift.iso_inv_app, AlgebraicGeometry.Spec.locallyRingedSpaceObj_presheaf_map, CategoryTheory.MonoidalCategory.MonoidalLeftAction.oppositeLeftAction_actionAssocIso, SSet.Subcomplex.unionProd.isPushout, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_inverse, SSet.iSup_subcomplexOfSimplex_prod_eq_top, CategoryTheory.Equivalence.rightOp_counitIso_inv_app, CategoryTheory.Sheaf.toImage_ι_assoc, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_inv_eq_germ_assoc, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.app_invApp, CategoryTheory.Functor.PullbackObjObj.mapArrowRight_right, HomologicalComplex.op_d, CategoryTheory.Functor.unopOpIso_inv_app, CategoryTheory.GrothendieckTopology.overMapPullback_comp_id, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.inv_naturality, PresheafOfModules.epi_iff_surjective, SSet.Truncated.Path.mk₂_arrow, AlgebraicGeometry.StructureSheaf.const_self, CategoryTheory.Limits.widePushoutShapeOpEquiv_inverse, AlgebraicGeometry.Scheme.mem_basicOpen', LightCondensed.ihomPoints_symm_comp, SSet.Truncated.HomotopyCategory.BinaryProduct.inverse_map_mkHom_id_homMk, CategoryTheory.Functor.sheafPullbackConstruction.instPreservesFiniteLimitsSheafSheafPullback, CategoryTheory.Limits.preservesFiniteLimits_op, CategoryTheory.Functor.PullbackObjObj.mapArrowLeft_id, CategoryTheory.Retract.op_i, CategoryTheory.RanIsSheafOfIsCocontinuous.liftAux_map', CategoryTheory.yoneda_map_app, CategoryTheory.Functor.WellOrderInductionData.map_succ, HomologicalComplex.opSymm_d, CategoryTheory.unop_tensorHom, CategoryTheory.Pseudofunctor.CoGrothendieck.categoryStruct_id_fiber, AlgebraicTopology.DoldKan.N₂Γ₂ToKaroubiIso_inv_app, CondensedSet.topCatAdjunctionUnit_hom_app_apply, AlgebraicGeometry.Scheme.zeroLocus_eq_univ_iff_subset_nilradical, CategoryTheory.Limits.IsZero.op, AlgebraicGeometry.Scheme.Opens.topIso_inv, CategoryTheory.OverPresheafAux.restrictedYoneda_map, SSet.stdSimplex.obj₀Equiv_symm_apply, CategoryTheory.whiskering_linearCoyoneda, CategoryTheory.Discrete.opposite_functor_obj_as, CategoryTheory.Equivalence.leftOp_unitIso_inv_app, CategoryTheory.Limits.Cone.equiv_inv_pt, CategoryTheory.ShortComplex.instCategoryWithHomologyOpposite, CategoryTheory.unopUnop_obj, PresheafOfModules.pullback_comp_id, AlgebraicGeometry.Proj.pow_apply, CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence_inverse, CategoryTheory.yonedaGrp_obj, CategoryTheory.Functor.Elements.shrinkYoneda_map_app_coconeπOpCompShrinkYonedaObj_ι_app_assoc, SSet.OneTruncation₂.nerveHomEquiv_id, CategoryTheory.MonoidalCategory.MonoidalRightAction.oppositeRightAction_actionObj, CategoryTheory.Functor.Monoidal.RepresentableBy.tensorObj_homEquiv, CategoryTheory.GrothendieckTopology.Point.skyscraperPresheafHomEquiv_naturality_left_assoc, CategoryTheory.op_mono_of_epi, PresheafOfModules.evaluation_preservesColimitsOfShape, CategoryTheory.Comon.ComonToMonOpOp_obj, SSet.finite_iSup_iff, CategoryTheory.yonedaCommGrpGrpObj_map, TopCat.Presheaf.SheafConditionEqualizerProducts.piOpens.hom_ext_iff, CategoryTheory.Functor.WellOrderInductionData.Extension.ofLE_val, AlgebraicTopology.DoldKan.identity_N₂, CategoryTheory.MonoidalCategory.MonoidalLeftAction.leftActionOfOppositeLeftAction_actionHomRight, PresheafOfModules.comp_app, CategoryTheory.Limits.FormalCoproduct.cosimplicialObjectFunctor_obj_obj, CategoryTheory.Square.unop_X₁, CategoryTheory.ShortComplex.RightHomologyMapData.op_φH, AlgebraicGeometry.Spec_presheaf, CategoryTheory.nerve.functorOfNerveMap_map, SSet.opFunctorCompOpFunctorIso_inv_app_app, CategoryTheory.Pretriangulated.shiftFunctorZero_op_inv_app, CategoryTheory.yonedaAddMonObjIsoOfRepresentableBy_hom_app_hom_apply, CategoryTheory.Sheaf.isPullback_χ_truth, HomologicalComplex.unopFunctor_obj, CategoryTheory.HasLiftingProperty.op, CategoryTheory.Functor.sheafPushforwardContinuous_map_hom_app, LightCondSet.continuous_coinducingCoprod, CategoryTheory.Sieve.W_shrinkFunctor_ι_of_mem, CategoryTheory.Join.inclRightCompOpEquivInverse_inv_app_op, CategoryTheory.Injective.instProjectiveOppositeOp, CategoryTheory.Equivalence.leftOp_functor_obj, CategoryTheory.Limits.reflectsLimits_rightOp, CategoryTheory.Presieve.isSheaf_iff_preservesFiniteProducts, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.invApp_app_assoc, SSet.Truncated.StrictSegal.spine_spineToSimplex, CategoryTheory.GrothendieckTopology.plusFunctorWhiskerRightIso_hom_app, CategoryTheory.Presheaf.coconeOfRepresentable_pt, CategoryTheory.Pseudofunctor.CoGrothendieck.map_comp_eq, CategoryTheory.Limits.end_.map_π, CategoryTheory.SimplicialObject.Truncated.whiskering_obj_obj_obj, CategoryTheory.cosimplicialSimplicialEquiv_counitIso_hom_app_app, CategoryTheory.preservesFiniteColimits_liftToFinset, LightCondSet.epi_iff_locallySurjective_on_lightProfinite, CategoryTheory.GrothendieckTopology.sheafifyMap_sheafifyLift_assoc, CategoryTheory.WithTerminal.opEquiv_inverse_obj, CategoryTheory.Sheaf.classifier_χ, CategoryTheory.Limits.opProdIsoCoprod_hom_snd_assoc, SSet.modelCategoryQuillen.mono_of_cofibration, SSet.horn₃₁.exists_desc, AlgebraicTopology.DoldKan.Γ₀.Obj.map_epi_on_summand_id_assoc, CategoryTheory.Presieve.isSeparatedFor_singleton, CategoryTheory.Functor.IsLeftAdjoint.op, SSet.Subcomplex.range_eq_ofSimplex, AlgebraicGeometry.StructureSheaf.toOpen_comp_comap_apply, CategoryTheory.GrothendieckTopology.plusCompIso_whiskerRight, CategoryTheory.Limits.preservesColimits_rightOp, CategoryTheory.shrinkYonedaMonObjObjEquiv_symm_comp, CategoryTheory.Functor.FullyFaithful.homNatIsoMaxRight_hom_app_down, CategoryTheory.Functor.leftOpRightOpEquiv_counitIso_inv_app_app, CategoryTheory.Adjunction.compCoyonedaIso_inv_app_app, CategoryTheory.Pretriangulated.Opposite.mem_distinguishedTriangles_iff', CategoryTheory.isIso_unop_iff, CategoryTheory.Limits.IsCofiltered.sequentialFunctor_initial, CategoryTheory.CosimplicialObject.Augmented.leftOp_right, CochainComplex.exactAt_op, SSet.spine_map_subinterval, CategoryTheory.Limits.preservesColimits_of_unop, Opens.mayerVietorisSquare_X₃, CategoryTheory.sheafToPresheafCompYonedaCompWhiskeringLeftSheafToPresheaf_inv_app_app, CategoryTheory.Limits.IsIndObject.isFiltered, CategoryTheory.Equivalence.rightOp_functor_map, SSet.PtSimplex.MulStruct.δ_map_of_gt, CategoryTheory.ParametrizedAdjunction.whiskerLeft_map_counit, CategoryTheory.GrothendieckTopology.preservesLimitsOfShape_plusFunctor, CategoryTheory.OverPresheafAux.restrictedYoneda_obj, LightCondensed.finYoneda_obj, AlgebraicGeometry.Scheme.inv_app, CategoryTheory.GrothendieckTopology.ι_plusCompIso_hom_assoc, CategoryTheory.Abelian.Ext.preadditiveYoneda_homologySequenceδ_singleTriangle_apply, LightCondensed.isoFinYoneda_inv_app, CategoryTheory.ObjectProperty.isDetecting_op_iff, CategoryTheory.ObjectProperty.instContainsZeroOppositeOp, CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheaf_obj, AlgebraicGeometry.Scheme.Hom.id_appTop, CategoryTheory.Equivalence.sheafCongr.functor_obj_obj_map, CategoryTheory.Functor.RepresentableBy.homEquiv_eq, SSet.nonDegenerate_iff_of_mono, CategoryTheory.Presheaf.imageSieve_app, CategoryTheory.Presheaf.isSheaf_iff_isLimit_coverage, CategoryTheory.Limits.walkingParallelPairOpEquiv_counitIso_hom_app_op_one, CondensedMod.isDiscrete_tfae, CategoryTheory.SimplicialObject.Homotopy.h_last_comp_δ_last_assoc, AlgebraicGeometry.Proj.basicOpenIsoAway_hom, SSet.instHasDimensionLETensorUnitOfNatNat, CategoryTheory.linearYoneda_obj_map, CategoryTheory.Limits.coend.map_id, smoothSheafCommRing.ι_evalHom_apply, CategoryTheory.PreOneHypercover.multicospanIndex_fst, CategoryTheory.NatIso.unop_rightUnitor, CategoryTheory.Pretriangulated.instIsTriangulatedOppositeOpOpEquivalence, SheafOfModules.evaluationPreservesLimit, CategoryTheory.NatTrans.unop_comp_assoc, TopCat.Presheaf.germ_exist_of_isBasis, PresheafOfModules.homEquivOfIsLocallyBijective_symm_apply, CategoryTheory.Limits.widePullbackShapeUnop_map, CategoryTheory.prodOpEquiv_inverse_map, CategoryTheory.instFullSheafSheafComposeOfFaithful, CategoryTheory.ShortComplex.opMap_τ₁, SimplexCategory.toTopHomeo_symm_naturality, CategoryTheory.SimplicialObject.Homotopy.h_succ_comp_δ_castSucc_of_lt, CategoryTheory.Functor.commShiftOp_iso_eq, CategoryTheory.ParametrizedAdjunction.inr_arrowHomEquiv_symm_apply_left, CategoryTheory.simplicialCosimplicialEquiv_counitIso_inv_app_app, CategoryTheory.Limits.PushoutCocone.unop_π_app, CategoryTheory.NatTrans.Equifibered.rightOp, CategoryTheory.ShiftedHom.opEquiv_symm_add, CategoryTheory.Equivalence.precoherent_isSheaf_iff_of_essentiallySmall, CategoryTheory.GrothendieckTopology.plusMap_id, CategoryTheory.PresheafOfGroups.Cochain₀.inv_apply, AlgebraicGeometry.Scheme.zeroLocus_mul, CategoryTheory.Pretriangulated.shiftFunctorCompIsoId_op_inv_app, CategoryTheory.Presheaf.functorToRepresentables_map, CategoryTheory.op_whiskerRight, SSet.tensorHom_app_apply, CategoryTheory.ShortComplex.Homotopy.unop_h₂, CategoryTheory.isConnected_op_iff_isConnected, SheafOfModules.forgetToSheafModuleCat_map_hom, CategoryTheory.simplicialCosimplicialEquiv_inverse_map, CategoryTheory.yonedaMonObjIsoOfRepresentableBy_hom_app_hom_apply, AlgebraicGeometry.Proj.mul_apply, AlgebraicGeometry.StructureSheaf.comap_id', CategoryTheory.SimplicialObject.Augmented.toArrow_map_right, CategoryTheory.Limits.IndObjectPresentation.extend_ι_app_app, CategoryTheory.unop_inv_associator, CategoryTheory.Pretriangulated.opShiftFunctorEquivalenceSymmHomEquiv_left_inv, SimplicialObject.opFunctor_obj_σ, CategoryTheory.Limits.Fork.unop_ι_app_zero, CategoryTheory.Presieve.isSheaf_comp_uliftFunctor, CategoryTheory.Limits.preservesLimitsOfShape_leftOp, CategoryTheory.ParametrizedAdjunction.inl_arrowHomEquiv_symm_apply_left, SSet.Edge.map_id, CategoryTheory.Functor.OneHypercoverDenseData.isSheaf_iff, AlgebraicGeometry.Scheme.Modules.germ_restrictStalkNatIso_inv_app, CategoryTheory.Limits.isClosedUnderLimitsOfShape_isIndObject_walkingParallelPair, CategoryTheory.Limits.reflectsColimitsOfShape_leftOp, CategoryTheory.Limits.SequentialProduct.cone_π_app_comp_Pi_π_pos_assoc, CategoryTheory.IsPullback.op, CategoryTheory.Limits.opCompYonedaSectionsEquiv_symm_apply_coe, CategoryTheory.Yoneda.naturality, CategoryTheory.Limits.reflectsLimit_of_leftOp, CondensedSet.hom_naturality_apply, CategoryTheory.Limits.yonedaCompLimIsoCocones_inv_app, SSet.Truncated.sk_coreflective, CategoryTheory.nerve.homEquiv_edgeMk_map_nerveMap, CategoryTheory.nerveFunctor.faithful, CategoryTheory.Comon.Comon_EquivMon_OpOp_counitIso, CompHausLike.LocallyConstantModule.functorToPresheaves_obj_obj_isModule, CategoryTheory.Functor.WellOrderInductionData.Extension.map_zero, SSet.S.equivElements_symm_apply_dim, CategoryTheory.Pseudofunctor.DescentData.pullFunctor_map_hom, SSet.hoFunctor.preservesTerminal', CompHausLike.LocallyConstant.adjunction_counit, CategoryTheory.Limits.hasLimit_leftOp_of_hasColimit, CategoryTheory.MorphismProperty.IsMultiplicative.of_unop, CategoryTheory.typeEquiv_functor_map_hom_app, AlgebraicGeometry.Scheme.ideal_ker_le_ker_ΓSpecIso_inv_comp, CategoryTheory.Preadditive.DoldKan.equivalence_unitIso, SSet.prodStdSimplex.instHasDimensionLETensorObjObjSimplexCategoryStdSimplexMkHAddNat, SSet.Subcomplex.toImage_ι, CategoryTheory.sheafificationNatIso_hom_app_hom, SSet.Truncated.Edge.map_fst, CategoryTheory.instFaithfulGrpFunctorOppositeGrpCatYonedaGrp, CategoryTheory.Functor.IsCoverDense.restrictHomEquivHom_naturality_left_symm_assoc, HomologicalComplex.opcyclesOpIso_inv_naturality_assoc, smoothSheafCommRing.eval_germ, CategoryTheory.SimplicialObject.augment_left, CategoryTheory.Limits.Cocone.unop_π, AlgebraicTopology.DoldKan.instIsIsoFunctorSimplicialObjectKaroubiNatTrans, CompHausLike.LocallyConstantModule.functor_obj_obj_map_hom_apply_apply, PresheafOfModules.pushforward_map_app_apply, SheafOfModules.inl_freeSumIso_hom, CategoryTheory.GrothendieckTopology.whiskerRight_toSheafify_sheafifyCompIso_hom_assoc, PresheafOfModules.limitPresheafOfModules_map, CategoryTheory.preadditiveYonedaObj_obj_carrier, LightCondensed.finYoneda_map, SSet.stdSimplex.objMk₁_bijective, CategoryTheory.Presheaf.isSeparating, SimplicialObject.Splitting.cofan_inj_epi_naturality_assoc, CompHausLike.LocallyConstantModule.functor_obj_obj_obj_carrier, CategoryTheory.Limits.colimitCoyonedaHomIsoLimit'_π_apply, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_map, AlgebraicGeometry.Scheme.isoSpec_inv_toSpecΓ, CategoryTheory.GrothendieckTopology.instFaithfulSheafTypeUliftYoneda, SSet.Subcomplex.image_obj, CategoryTheory.yonedaMonObjIsoOfRepresentableBy_inv_app_hom_apply, CategoryTheory.Functor.cones_map_app, CategoryTheory.Limits.pullbackIsoOpPushout_hom_inr_assoc, CategoryTheory.ShortComplex.LeftHomologyData.unop_ι, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.instIsIsoInvApp, CategoryTheory.isCodetector_unop_iff, CategoryTheory.NatTrans.leftOp_app, CommRingCat.moduleCatRestrictScalarsPseudofunctor_mapComp, SSet.horn.yonedaEquiv_ι, CategoryTheory.Functor.closedSieves_obj, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.invApp_app_apply, CategoryTheory.MonoidalCategory.DayConvolution.corepresentableBy₂_homEquiv, CategoryTheory.Functor.op_comp_isSheaf_of_isSheaf, CategoryTheory.Equivalence.leftOp_counitIso_inv_app, CategoryTheory.NatTrans.unop_whiskerLeft_assoc, AlgebraicGeometry.StructureSheaf.const_add, SSet.Subcomplex.range_eq_top, CategoryTheory.Limits.coyonedaCompLimIsoCones_inv_app, AlgebraicGeometry.PresheafedSpace.GlueData.ιIsOpenImmersion, PresheafOfModules.sections_property, CategoryTheory.SimplicialObject.Homotopy.h_comp_σ_castSucc_of_le_assoc, CategoryTheory.Subfunctor.ofSection_eq_range, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.shortComplex_X₁, HomologicalComplex.instHasHomologyOppositeOp, CompHausLike.LocallyConstantModule.functor_obj_obj_obj_isAddCommGroup_sub_apply, CategoryTheory.shrinkYonedaGrp_obj, CategoryTheory.Limits.walkingParallelPairOpEquiv_counitIso_inv_app_op_zero, CategoryTheory.Presheaf.instIsLeftKanExtensionFunctorOppositeTypeLanOpHomCompULiftYonedaIsoULiftYonedaCompLan, AlgebraicTopology.AlternatingFaceMapComplex.ε_app_f_succ, AlgebraicGeometry.isIso_ΓSpec_adjunction_unit_app_basicOpen, AlgebraicGeometry.isNoetherian_iff_of_finite_affine_openCover, CategoryTheory.Limits.reflectsLimitsOfSize_of_unop, CategoryTheory.Limits.multispanIndexCoend_right, CategoryTheory.OverPresheafAux.YonedaCollection.map₂_yonedaEquivFst, CondensedMod.LocallyConstant.instFullModuleCatSheafCompHausCoherentTopologyConstantSheaf, CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan.natTrans_app_uliftYoneda_obj, CategoryTheory.ParametrizedAdjunction.homEquiv_symm_naturality_one, CategoryTheory.SimplicialObject.Truncated.whiskering_map_app_app, Condensed.discrete_map, PresheafOfModules.toSheafify_app_apply', CategoryTheory.isConnected_op, SSet.Finite.instIsFinitelyPresentableObjSimplexCategoryStdSimplex, CategoryTheory.opOp_map, CategoryTheory.Limits.limitOpIsoOpColimit_hom_comp_ι_assoc, PresheafOfModules.instPreservesLimitsOfShapeModuleCatCarrierObjOppositeRingCatEvaluation, CategoryTheory.Functor.IsCoverDense.Types.sheafIso_hom_hom, CategoryTheory.Limits.pushoutIsoUnopPullback_inr_hom, PresheafOfModules.toFreeYonedaCoproduct_fromFreeYonedaCoproduct, HomotopicalAlgebra.instFibrationOppositeOpOfCofibration, CategoryTheory.Equalizer.firstObjEqFamily_inv, CategoryTheory.instIsRegularEpiOppositeOpOfIsRegularMono, AlgebraicGeometry.SmoothOfRelativeDimension.exists_isStandardSmoothOfRelativeDimension, CategoryTheory.unop_mono_iff, CategoryTheory.yonedaEquiv_apply, AlgebraicGeometry.exists_pow_mul_eq_zero_of_res_basicOpen_eq_zero_of_isCompact, CategoryTheory.GrothendieckTopology.overMapPullbackId_inv_app_hom_app, AlgebraicGeometry.Scheme.IdealSheafData.ideal_le_comap_ideal, CategoryTheory.MonoidalCategory.MonoidalLeftAction.leftActionOfOppositeLeftAction_actionAssocIso, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.s_comp_δ₀_assoc, CategoryTheory.Limits.IndObjectPresentation.instFinalICostructuredArrowFunctorOppositeTypeYonedaToCostructuredArrow, SSet.Truncated.HomotopyCategory.BinaryProduct.functorCompInverseIso_inv_app, CategoryTheory.Functor.opInv_obj, CategoryTheory.Functor.PullbackObjObj.π_fst, CategoryTheory.Functor.WellOrderInductionData.surjective, CategoryTheory.sheafToPresheafCompCoyonedaCompWhiskeringLeftSheafToPresheaf_inv_app_app, instIsLeftKanExtensionSimplexCategoryTopCatSSetToTopInvFunctorToTopSimplex, CategoryTheory.GrothendieckTopology.Point.instPreservesFiniteLimitsSheafSheafFiberOfLocallySmallOfHasFiniteLimitsOfAB5OfSize, CategoryTheory.IsFiltered.iff_nonempty_limit, CategoryTheory.Comon.monoidal_rightUnitor_inv_hom, CategoryTheory.Comma.opEquiv_counitIso, CategoryTheory.SimplicialObject.Truncated.cosk.full, SSet.horn₃₂.desc.multicofork_π_one, PresheafOfModules.Derivation.d_map, CategoryTheory.SimplicialObject.Homotopy.h_comp_σ_castSucc_of_le, IsOpenMap.pullbackObjIso_hom_app, CategoryTheory.Pseudofunctor.CoGrothendieck.instIsEquivalenceαCategoryObjLocallyDiscreteOppositeCatMkOpFiberForgetInducedFunctor, SSet.yonedaEquiv_symm_zero, CategoryTheory.ShortComplex.hasLeftHomology_iff_op, CategoryTheory.ShortComplex.shortExact_iff_op, AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine_apply, CategoryTheory.Functor.Elements.coconeπOpCompShrinkYonedaObj_pt, AlgebraicGeometry.structureSheafInType.mul_apply, TopCat.Presheaf.germ_stalkSpecializes_apply, SSet.Subcomplex.topIso_inv_app_coe, SSet.prodStdSimplex.orderHomOfSimplex_coe, SSet.horn₃₂.ι₁_desc, HomologicalComplex.extend_op_d_assoc, CategoryTheory.constantCommuteCompose_hom_app_val, CategoryTheory.Square.op_f₁₂, CategoryTheory.cokernelUnopUnop_inv, CategoryTheory.Limits.op_pushoutMap, AlgebraicGeometry.StructureSheaf.exists_const, CategoryTheory.Limits.desc_op_comp_opCoproductIsoProduct'_hom, skyscraperPresheafCocone_pt, CategoryTheory.GrothendieckTopology.pseudofunctorOver_mapComp_inv_toNatTrans_app_hom_app, CategoryTheory.forgetEnrichmentOppositeEquivalence_functor, AlgebraicGeometry.Scheme.Modules.Hom.zero_app, CategoryTheory.WithInitial.opEquiv_functor_map, TopCat.Sheaf.pushforward_map, CategoryTheory.Functor.PullbackObjObj.π_iso_of_iso_left_inv, CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence.functor_obj_right_as, CategoryTheory.isFilteredOrEmpty_op_of_isCofilteredOrEmpty, AlgebraicGeometry.AffineScheme.instIsEquivalenceOppositeCommRingCatOpRightOpΓ, CategoryTheory.Pseudofunctor.DescentData.isoMk_inv_hom, CategoryTheory.Sheaf.isLocallySurjective_comp, HomotopicalAlgebra.weakEquivalences_op, CategoryTheory.GrothendieckTopology.preservesLimitsOfShape_diagramFunctor, CategoryTheory.SimplicialObject.Homotopy.h_last_comp_δ_last, AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine_symm_apply, AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex_inv, CategoryTheory.Iso.op_inv, AlgebraicGeometry.Scheme.congr_app, CategoryTheory.unop_tensor_unop, SimplicialObject.opFunctor_obj_map, SimplicialObject.Split.forget_obj, AlgebraicGeometry.Scheme.IdealSheafData.ideal_mul, CategoryTheory.Arrow.mapAugmentedCechNerve_left, CategoryTheory.kernelOpUnop_hom, SSet.prodStdSimplex.nonDegenerateEquiv₁_snd, CategoryTheory.Functor.IsCoverDense.isoOver_hom_app, AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.structurePresheafInCommRing_obj_carrier, AlgebraicGeometry.Scheme.SpecΓIdentity_hom_app, SSet.Subcomplex.homOfLE_refl, CategoryTheory.Pretriangulated.triangleOpEquivalence_unitIso, CategoryTheory.Functor.FullyFaithful.compUliftCoyonedaCompWhiskeringLeft_hom_app_app_down, CategoryTheory.MonoidalCategory.MonoidalRightAction.oppositeRightAction_actionHomRight, AlgebraicGeometry.locallyQuasiFinite_iff, CategoryTheory.Preadditive.DoldKan.equivalence_functor, CategoryTheory.ShortComplex.Homotopy.op_h₀, CategoryTheory.Coyoneda.objOpOp_inv_app, SheafOfModules.inr_freeSumIso_hom, CategoryTheory.Functor.opComp_inv_app, smoothSheaf.ι_evalHom_apply, HomologicalComplex.isStrictlySupportedOutside_op_iff, CategoryTheory.Sheaf.adjunction_unit_app_hom, Condensed.instPreservesFiniteProductsOppositeProfiniteObjFunctorIsSheafCoherentTopology, TopCat.Presheaf.SheafConditionEqualizerProducts.fork_π_app_walkingParallelPair_one, SSet.Truncated.Edge.CompStruct.d₂, CategoryTheory.ObjectProperty.IsConservativeFamilyOfPoints.isMonoidal_W, CategoryTheory.Limits.isColimitCoconeLeftOpOfCone_desc, SSet.opFunctorCompOpFunctorIso_hom_app_app, CategoryTheory.sheafToPresheafCompYonedaCompWhiskeringLeftSheafToPresheaf_app_app, CategoryTheory.instIsClosedUnderLimitsOfShapeFunctorOppositeTypeIsIndObjectDiscreteOfHasLimitsOfShape, CategoryTheory.Pseudofunctor.CoGrothendieck.map_map_base, CategoryTheory.Limits.pushoutIsoUnopPullback_inl_hom_assoc, AlgebraicTopology.DoldKan.HigherFacesVanish.comp_σ, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_inv_app', AlgebraicGeometry.StructureSheaf.toOpen_comp_comap, SSet.Truncated.liftOfStrictSegal.hδ'₂, CategoryTheory.Equivalence.sheafCongrPrecoherent_functor_obj_obj_obj, CategoryTheory.Equivalence.instPreservesFiniteLimitsFunctorOppositeSheafTransportAndSheafify, CategoryTheory.Equivalence.sheafCongrPreregular_functor_obj_obj_obj, CategoryTheory.Presieve.shrinkFunctorHomEquiv_symm_apply_app, CategoryTheory.SimplicialObject.Augmented.w₀, IsOpenMap.pullbackObjIso_hom_naturality, SSet.Truncated.HomotopyCategory.BinaryProduct.functorCompInverseIso_hom_app, AlgebraicGeometry.AffineScheme.instIsEquivalenceOppositeCommRingCatRightOpΓ, AlgebraicTopology.AlternatingCofaceMapComplex.d_eq_unop_d, TopCat.Presheaf.pushforwardToOfIso_app, SSet.Subcomplex.toRange_app_val, CategoryTheory.Pseudofunctor.CoGrothendieck.map_map_fiber, AlgebraicGeometry.affineLocally_iff_forall_isAffineOpen, CategoryTheory.Pseudofunctor.DescentData'.pullHom'_self', HomologicalComplex.unopInverse_map, CategoryTheory.Presheaf.freeYonedaHomEquiv_comp, PresheafOfModules.free_map_app, CategoryTheory.SimplicialObject.σ_naturality, CategoryTheory.OverPresheafAux.yonedaCollectionPresheaf_map, CategoryTheory.ShortComplex.RightHomologyData.unop_i, AlgebraicGeometry.Scheme.Hom.formallyUnramified_appLE, CategoryTheory.SimplicialObject.instIsRightKanExtensionOppositeTruncatedSimplexCategoryObjCoskAppTruncatedCounitCoskAdjTruncation, CategoryTheory.Presheaf.comp_χ_eq, CategoryTheory.WithInitial.opEquiv_counitIso_hom_app, CategoryTheory.Pretriangulated.shiftFunctorAdd'_op_hom_app, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.ofRestrict_invApp, SSet.hasDimensionLT_prod, CategoryTheory.uliftCoyonedaIsoCoyoneda_inv_app_app_down, CategoryTheory.Presheaf.isSheaf_iff_preservesFiniteProducts, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivCounitIso_hom_app_hom, CategoryTheory.GrothendieckTopology.sheafificationWhiskerRightIso_hom_app, CategoryTheory.Pseudofunctor.toDescentDataAsCoalgebraCompCoalgebraEquivalenceFunctorIso_inv_app_f, SheafOfModules.instIsRightAdjointPushforwardCompSheafRingCatMapSheafPushforwardContinuous, SimplexCategory.toTopHomeo_naturality_apply, CategoryTheory.CostructuredArrow.unop_left_comp_ofMkLEMk_unop, CategoryTheory.Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor, CategoryTheory.Limits.multicospanIndexEnd_snd, CategoryTheory.GrothendieckTopology.diagramCompIso_hom_ι, AlgebraicGeometry.Scheme.isoSpec_inv_preimage_zeroLocus, CategoryTheory.unop_tensorUnit, PresheafOfModules.congr_map_apply, CategoryTheory.ShortComplex.SnakeInput.op_L₂, AlgebraicGeometry.Scheme.LocalRepresentability.isRepresentable, CategoryTheory.PresheafOfGroups.OneCocycle.ev_trans, SSet.instIsStableUnderFilteredColimitsMonomorphisms, CategoryTheory.Adjunction.compYonedaIso_hom_app_app, CategoryTheory.Functor.WellOrderInductionData.sectionsMk_val_op_bot, SSet.StrictSegalCore.map_mkOfSucc_zero_concat, PresheafOfModules.freeYonedaEquiv_symm_app, SSet.Subcomplex.toRange_ι_assoc, CommRingCat.coyoneda_map_app, HomologicalComplex.unopEquivalence_functor, CategoryTheory.ShortComplex.HomologyMapData.unop_right, CategoryTheory.Limits.hasCoequalizers_opposite, CategoryTheory.Sheaf.isLocallySurjective_iff_epi, AlgebraicGeometry.ProjectiveSpectrum.Proj.toOpen_toSpec_val_c_app_assoc, TopCat.Presheaf.germ_res', AlgebraicGeometry.PresheafedSpace.Γ_obj_op, CondensedMod.LocallyConstant.instFaithfulModuleCatCondensedDiscrete, AlgebraicGeometry.Scheme.Modules.pseudofunctor_map_l, CategoryTheory.ShortComplex.unopFunctor_map, CategoryTheory.CostructuredArrow.overEquivPresheafCostructuredArrow_inverse_map_toOverCompCoyoneda, CategoryTheory.GrothendieckTopology.Point.Hom.presheafFiber_comp, SSet.RelativeMorphism.Homotopy.h₀_assoc, PresheafOfModules.restrictScalarsObj_map, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.SheafCondition.map_f₂₄_op_glue, CategoryTheory.Limits.coconeUnopOfConeEquiv_unitIso, AlgebraicGeometry.Spec.locallyRingedSpaceObj_presheaf', CategoryTheory.Over.opEquivOpUnder_unitIso, CategoryTheory.Limits.coconeOfConeLeftOp_pt, PresheafOfModules.forgetToPresheafModuleCatObj_map, SSet.Subcomplex.unionProd.image_β_inv, CategoryTheory.Limits.pushoutIsoUnopPullback_inr_hom_assoc, Profinite.NobelingProof.spanFunctorIsoIndexFunctor_hom_app_hom_hom_apply_coe, HomotopicalAlgebra.cofibration_op_iff, CategoryTheory.Functor.LeibnizAdjunction.adj_unit_app_left, CategoryTheory.ShortComplex.Homotopy.op_h₂, TopCat.Sheaf.interUnionPullbackCone_fst, CategoryTheory.ObjectProperty.instIsClosedUnderColimitsOfShapeUnopOppositeOfIsClosedUnderLimitsOfShape, CategoryTheory.Pretriangulated.Opposite.OpOpCommShift.iso_inv_app, AugmentedSimplexCategory.equivAugmentedSimplicialObject_inverse_map_app, AlgebraicGeometry.functionField_isScalarTower, CategoryTheory.GrothendieckTopology.OneHypercover.multiforkLift_map, CategoryTheory.ShortComplex.SnakeInput.op_v₁₂, CategoryTheory.GrothendieckTopology.Point.toPresheafFiberMap_presheafFiberMapObjIso_hom_assoc, CategoryTheory.SimplicialObject.Augmented.toArrow_map_left, CategoryTheory.uliftYonedaEquiv_symm_comp, AlgebraicTopology.alternatingFaceMapComplex_obj_d, instIsLeftAdjointSSetTopCatToTop, imageToKernel_op, AlgebraicGeometry.Scheme.Modules.toOpen_fromTildeΓ_app, TopCat.Presheaf.stalkPushforward_germ_apply, InfiniteGalois.toAlgEquivAux_eq_proj_of_mem, CategoryTheory.Functor.sheafPushforwardContinuousNatTrans_app_hom, CategoryTheory.op_epi_iff, CategoryTheory.Pseudofunctor.DescentData.comp_hom, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.s'_comp_ε, StalkSkyscraperPresheafAdjunctionAuxs.germ_fromStalk, SSet.prodStdSimplex.objEquiv_apply_fst, SSet.stdSimplex.map_id, SSet.Subcomplex.lift_ι_assoc, CategoryTheory.Presieve.FamilyOfElements.compatible_singleton_iff, SheafOfModules.evaluationPreservesLimitsOfShape, LightCondensed.discrete_map, CategoryTheory.monoidalOpOp_δ, SimplicialObject.Splitting.cofan_inj_eq, CategoryTheory.Limits.instHasProductOppositeOp, SSet.Homotopy.h₁_assoc, CategoryTheory.Limits.reflectsColimits_leftOp, CategoryTheory.Presieve.isSheafFor_comp_uliftFunctor_iff, CategoryTheory.MonObj.ofRepresentableBy_one, AlgebraicGeometry.Scheme.LocalRepresentability.instIsLocallySurjectiveHomYonedaGluedToSheafOfIsLocallySurjectiveZariskiTopologyDescFunctorOppositeType, AlgebraicGeometry.Scheme.exists_germ_injective, SSet.Truncated.Path.map_arrow, CategoryTheory.Limits.hasColimit_of_hasLimit_leftOp, CategoryTheory.Sheaf.cartesianMonoidalCategoryLift_hom, CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence.functor_obj_left_right, CategoryTheory.SimplicialObject.Augmented.toArrow_obj_left, CategoryTheory.Limits.pullbackIsoUnopPushout_hom_inr_assoc, CategoryTheory.Sheaf.isPullback_square_op_map_yoneda_presheafToSheaf_yoneda_iff, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_naturality_assoc, CategoryTheory.NatIso.op_trans, SSet.exists_nonDegenerate, CategoryTheory.Limits.PushoutCocone.op_snd, CategoryTheory.Limits.isColimitOfConeRightOpOfCocone_desc, CategoryTheory.Limits.coend.condition, CategoryTheory.Limits.FormalCoproduct.cechIsoAugmentedCechNerve_hom_left, CategoryTheory.Limits.reflectsColimit_of_rightOp, CategoryTheory.sheafToPresheaf_μ, CategoryTheory.SimplicialObject.Truncated.cosk.faithful, AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_app_eq, CategoryTheory.Presheaf.isIso_of_isLeftKanExtension, CategoryTheory.LocalizerMorphism.hasLeftResolutions_iff_op, CategoryTheory.GrothendieckTopology.toSheafify_sheafifyLift_assoc, LightProfinite.proj_comp_transitionMapLE', CategoryTheory.Presheaf.isSheaf_iff_preservesFiniteProducts_and_equalizerCondition, TopCat.Presheaf.germ_res_apply, CategoryTheory.GrothendieckTopology.isIso_toSheafify, CategoryTheory.OverPresheafAux.OverArrows.val_mk, CategoryTheory.Limits.PullbackCone.unop_inl, CategoryTheory.NatTrans.rightOp_comp, CategoryTheory.ShortComplex.op_X₁, CategoryTheory.uliftCoyonedaEquiv_uliftCoyoneda_map, CochainComplex.homotopyUnop_hom_eq, CategoryTheory.Functor.RepresentableBy.coyoneda_homEquiv, SSet.Subcomplex.lift_app_coe, CategoryTheory.Functor.PullbackObjObj.mapArrowLeft_comp, CategoryTheory.coyonedaEquiv_comp, CategoryTheory.uliftYonedaEquiv_symm_apply_app, CategoryTheory.Pretriangulated.TriangleOpEquivalence.functor_map_hom₂, CategoryTheory.ParametrizedAdjunction.arrowHomEquiv_apply_left, CategoryTheory.Limits.inr_opProdIsoCoprod_inv_assoc, CategoryTheory.Limits.isLimitOfCoconeOfConeRightOp_lift, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_snd_assoc, SimplicialObject.Split.forget_map, AlgebraicGeometry.Scheme.Hom.toNormalization_app_preimage, CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheafMap_π_assoc, CategoryTheory.Join.opEquiv_functor_obj_op_right, CategoryTheory.Limits.colimitYonedaHomIsoLimit'_π_apply, CategoryTheory.Limits.π_comp_colimitLeftOpIsoUnopLimit_inv, AlgebraicGeometry.Proj.one_apply, AlgebraicGeometry.AffineScheme.mk_obj, Condensed.epi_iff_surjective_on_stonean, HomologicalComplex.opcyclesOpIso_inv_naturality, AlgebraicGeometry.Scheme.Hom.toImage_app_injective, AlgebraicGeometry.Scheme.isoSpec_hom, SSet.modelCategoryQuillen.fibrations_eq, CategoryTheory.Presheaf.isLocallyInjective_presheafToSheaf_map_iff, CategoryTheory.Functor.sheafPushforwardContinuousId'_hom_app_hom_app, CategoryTheory.Groupoid.invFunctor_map, CategoryTheory.GrothendieckTopology.uliftYonedaEquiv_naturality, AlgebraicGeometry.StructureSheaf.to_basicOpen_epi, CategoryTheory.Functor.IsRepresentedBy.of_natIso, CategoryTheory.yonedaAddMon_obj, AlgebraicGeometry.IsAffineOpen.opensRange_fromSpec, SSet.Subcomplex.unionProd.symmIso_inv, HomologicalComplex.extend.XOpIso_hom_d_op, CategoryTheory.Limits.parallelPairOpIso_inv_app_one, CategoryTheory.Functor.instFullOppositeTypeRestrictedULiftYonedaOfIsDense, CategoryTheory.instEpiSheafTypeToImage, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo_Spec_fromSpecStalk, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.app_inv_app', CommMonCat.coyonedaType_obj_map, AlgebraicGeometry.Scheme.IdealSheafData.ideal_le_ker_glueDataObjι, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.isoRestrict_hom_c_app, TopCat.Presheaf.germ_stalkPullbackInv_assoc, LightProfinite.Extend.cocone_ι_app, CategoryTheory.Functor.sheafAdjunctionCocontinuous_counit_app_val, CategoryTheory.Limits.PreservesFiniteLimitsOfIsFilteredCostructuredArrowYonedaAux.functorToInterchangeIso_inv_app_app, AlgebraicGeometry.Scheme.eqToHom_c_app, TopCat.Sheaf.exact_iff_stalkFunctor_map_exact, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.biprodAddEquiv_symm_biprodIsoProd_hom_toBiprod_apply, AlgebraicGeometry.Scheme.Hom.resLE_app_top, CategoryTheory.Limits.colimitHomIsoLimitYoneda'_hom_comp_π_assoc, AlgebraicGeometry.IsAffineOpen.self_le_iSup_basicOpen_iff, CategoryTheory.Functor.LeibnizAdjunction.adj_counit_app_left, AlgebraicGeometry.Scheme.zeroLocus_biInf, AlgebraicGeometry.PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π, AlgebraicGeometry.PresheafedSpace.componentwiseDiagram_map, AlgebraicGeometry.PresheafedSpace.stalkMap_germ_assoc, HomologicalComplex.opEquivalence_unitIso, CategoryTheory.toSheafify_sheafifyLift_assoc, AlgebraicGeometry.Proj.basicOpenToSpec_app_top, SimpleGraph.componentComplFunctor_obj, CategoryTheory.Limits.compYonedaSectionsEquiv_apply_app, CategoryTheory.cokernelOpOp_hom, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.ofRestrict_invApp, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.inv_naturality, CategoryTheory.ShortComplex.op_pOpcycles_opcyclesOpIso_hom, CategoryTheory.SimplicialObject.δ_comp_σ_succ'_assoc, CategoryTheory.SimplicialObject.δ_def, SSet.modelCategoryQuillen.cofibrations_eq, Condensed.lanPresheafExt_inv, AlgebraicGeometry.Scheme.AffineZariskiSite.generate_presieveOfSections, CategoryTheory.NatIso.op_isoWhiskerLeft, CategoryTheory.ObjectProperty.unop_isoClosure, PresheafOfModules.pushforward₀_obj_obj_carrier, commBialgCatEquivComonCommAlgCat_unitIso_hom_app, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.invApp_app, CategoryTheory.Presieve.isSheafFor_iff_preservesProduct, CategoryTheory.Limits.Cofork.unop_op_π, CategoryTheory.Functor.imageSieve_eq_imageSieve, CategoryTheory.WithTerminal.opEquiv_counitIso_inv_app, CategoryTheory.GrothendieckTopology.instFullSheafTypeYoneda, CategoryTheory.HasLiftingProperty.transfiniteComposition.wellOrderInductionData.liftHom_fac_assoc, CategoryTheory.GrothendieckTopology.preserveFiniteLimits_plusFunctor, PresheafOfModules.id_app, SimplicialObject.Split.comp_F, CategoryTheory.Pretriangulated.commShiftIso_opOp_inv_app, CategoryTheory.ShortComplex.RightHomologyMapData.op_φK, SSet.opEquivalence_inverse, CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan.presheafHom_naturality, SSet.Truncated.StrictSegal.spineToSimplex_spine, CategoryTheory.ShortComplex.opMap_id, CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheafObjObjIso.inv_π, CategoryTheory.ShortComplex.LeftHomologyMapData.op_φH, CategoryTheory.Functor.IsDenseSubsite.mapPreimage_map, SimplexCategory.II_σ, SSet.stdSimplex.objMk₁_surjective, CategoryTheory.Presheaf.classifier_Ω, CategoryTheory.Limits.multicospanIndexEnd_right, CategoryTheory.Limits.PullbackCone.op_pt, AlgebraicGeometry.Scheme.toSpecΓ_naturality_assoc, CategoryTheory.PresheafOfGroups.Cochain₀.one_apply, CategoryTheory.PreOneHypercover.multicospanIndex_right, SSet.Subcomplex.degenerate_eq_top_iff, AlgebraicGeometry.Scheme.Modules.pushforwardComp_inv_app_app, CategoryTheory.Sheaf.mem_essImage_of_isConstant, CategoryTheory.GrothendieckTopology.preservesLimitsOfShape_sheafification, AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex_f, CategoryTheory.Presheaf.isLocallySurjective_whisker, CategoryTheory.Presheaf.truth_app, CategoryTheory.Presheaf.restrictedULiftYoneda_obj_map, CategoryTheory.leftDualFunctor_obj, CommRingCat.coyoneda_obj_obj_carrier, CategoryTheory.Limits.pushoutIsoUnopPullback_inv_snd, CategoryTheory.LocalizerMorphism.RightResolution.op_X₁, AlgebraicGeometry.SurjectiveOnStalks.iff_of_isAffine, CategoryTheory.Functor.rightOpComp_hom_app, CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinsetObj_obj, CategoryTheory.Functor.rightAdjointObjIsDefined_iff, AlgebraicTopology.DoldKan.Γ₀.Obj.map_on_summand_assoc, SSet.instFiniteTensorUnit, CategoryTheory.MorphismProperty.instHasLeftCalculusOfFractionsOppositeOpOfHasRightCalculusOfFractions, SSet.Edge.ofTruncated_edge, CategoryTheory.Limits.reflectsFiniteCoproducts_leftOp, AlgebraicGeometry.SpecMap_ΓSpecIso_hom, HomologicalComplex.extend.XOpIso_hom_d_op_assoc, CategoryTheory.ObjectProperty.isCoseparating_op_iff, CategoryTheory.Comma.opFunctor_map, CategoryTheory.image_ι_op_comp_imageUnopOp_hom, CategoryTheory.GrothendieckTopology.overMapPullbackComp_hom_app_hom_app, CategoryTheory.Presheaf.comp_isLocallySurjective_iff, CategoryTheory.OverPresheafAux.unitAuxAuxAux_hom, SSet.Truncated.HomotopyCategory.BinaryProduct.inverse_comp_mapHomotopyCategory_fst, CategoryTheory.Functor.instIsCorepresentableObjOppositeTypeUliftCoyoneda, AlgebraicGeometry.IsAffineOpen.isLocalization_basicOpen, CategoryTheory.hasLimitsEssentiallySmallSite, CategoryTheory.Functor.leftOp_obj, CategoryTheory.Limits.coneUnopOfCocone_pt, CategoryTheory.Equivalence.inverseFunctor_obj, AlgebraicTopology.DoldKan.PInfty_f_naturality_assoc, SSet.horn₂₂.sq, CategoryTheory.Presheaf.preservesColimitsOfSize_of_isLeftKanExtension, AlgebraicTopology.DoldKan.instIsIsoFunctorKaroubiSimplicialObjectNatTrans, CategoryTheory.GrothendieckTopology.overMapPullbackId_hom_app_hom_app, CategoryTheory.Functor.IsCoverDense.Types.appHom_valid_glue, CategoryTheory.Limits.Cofork.unop_π_app_one, AlgebraicGeometry.Scheme.zeroLocus_iUnion, SSet.iSup_skeleton, SSet.OneTruncation₂.ofNerve₂.natIso_inv_app_obj_obj, PresheafOfModules.toPresheaf_map_toSheafify, CategoryTheory.Functor.rightOpLeftOpIso_hom_app, CategoryTheory.Limits.hasCoproducts_opposite, CategoryTheory.Limits.reflectsFiniteColimits_leftOp, PresheafOfModules.restrictScalarsObj_obj, AlgebraicGeometry.IsReduced.component_reduced, CategoryTheory.Equalizer.firstObjEqFamily_hom, CategoryTheory.Adjunction.Triple.op_adj₁, CategoryTheory.Subfunctor.Subpresheaf.ofSection_eq_range', CategoryTheory.Iso.op_symm, CategoryTheory.Limits.isLimitOfCoconeOfConeLeftOp_lift, CategoryTheory.Iso.unop_refl, CategoryTheory.Equivalence.sheafCongr.functor_obj_obj_obj, AlgebraicGeometry.SheafedSpace.comp_hom_c_app, CategoryTheory.ParametrizedAdjunction.arrowHomEquiv_apply_right_snd, CategoryTheory.Limits.inl_opProdIsoCoprod_inv_assoc, CategoryTheory.ObjectProperty.isClosedUnderLimitsOfShape_op_iff_unop, SSet.nonDegenerateEquivOfIso_symm_apply_coe, CategoryTheory.Limits.spanUnop_inv_app, CommRingCat.coyoneda_obj_map, CategoryTheory.yonedaFunctor_reflectsLimits, CategoryTheory.Limits.piConst_map_app, AlgebraicGeometry.IsIntegralHom.isIntegral_app, AlgebraicGeometry.Scheme.ker_ideal_of_isPullback_of_isOpenImmersion, AlgebraicGeometry.RingedSpace.exists_res_eq_zero_of_germ_eq_zero, CategoryTheory.isIso_toSheafify, CategoryTheory.ParametrizedAdjunction.arrowHomEquiv_symm_apply_right, CategoryTheory.Adjunction.sheafPushforwardContinuous_unit_app_hom_app, AlgebraicTopology.DoldKan.N₁_obj_p, CategoryTheory.Functor.instFinalOppositeRightOpOfInitial, CategoryTheory.Under.mapFunctor_obj, SSet.Subcomplex.image_id, CategoryTheory.Presieve.isSheafFor_singleton, PresheafOfModules.Derivation.postcomp_d_apply, TopCat.Sheaf.id_app, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_inv_naturality_assoc, SimplicialObject.opFunctor_map_app, CategoryTheory.Limits.FormalCoproduct.cochainComplexFunctor_map_f, CategoryTheory.Preadditive.DoldKan.equivalence_counitIso, CategoryTheory.Equivalence.sheafCongrPreregular_counitIso_hom_app_hom_app, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.isPushout, AlgebraicGeometry.LocallyRingedSpace.Γ_obj_op, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_appTop_coord, SSet.Truncated.HomotopyCategory₂.mk_surjective, PresheafOfModules.Derivation.d_one, CategoryTheory.Functor.mapTriangleOpCompTriangleOpEquivalenceFunctorApp_inv_hom₃, SSet.spine_vertex, CategoryTheory.Pretriangulated.Opposite.contractibleTriangleIso_hom_hom₂, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.isIso_of_subset, PresheafOfModules.sectionsMap_coe, SimpleGraph.infinite_iff_in_eventualRange, CategoryTheory.SimplicialObject.δ_comp_δ_assoc, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.ofRestrict_invApp_apply, CategoryTheory.shrinkYonedaMon_map, CategoryTheory.Pretriangulated.Opposite.UnopUnopCommShift.iso_inv_app_assoc, CategoryTheory.Presieve.FamilyOfElements.singletonEquiv_symm_apply_self, HomologicalComplex.opFunctor_obj, CategoryTheory.Limits.walkingParallelPairOpEquiv_counitIso_zero, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.const_s', AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_over_assoc, AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToΓ_ΓToStalk, CategoryTheory.Functor.FullyFaithful.homNatIso_inv_app_down, CategoryTheory.Cat.opEquivalence_counitIso, AlgebraicTopology.DoldKan.comp_P_eq_self_iff, CategoryTheory.IsCofiltered.iff_nonempty_limit, CategoryTheory.ShortComplex.HomologyData.op_right, AlgebraicGeometry.StructureSheaf.comap_const, CategoryTheory.Limits.pushoutIsoUnopPullback_inv_fst, AlgebraicGeometry.Scheme.Hom.appLE_map', SSet.Subcomplex.preimage_min, SSet.Subcomplex.eq_top_iff_of_hasDimensionLT, CategoryTheory.instIsSplitMonoOppositeOpOfIsSplitEpi, AlgebraicGeometry.Scheme.isoSpec_hom_naturality, CategoryTheory.LocalizerMorphism.LeftResolution.unop_X₁, topCatOpToFrm_map, AlgebraicGeometry.Scheme.zeroLocus_singleton, SSet.stdSimplex.ext_iff, HomologicalComplex.instQuasiIsoAtMapOppositeSymmUnopFunctorOp, SSet.δ_naturality_apply, CategoryTheory.MorphismProperty.LeftFraction.unop_X', Condensed.id_hom, AlgebraicGeometry.PresheafedSpace.comp_c_app, CategoryTheory.MorphismProperty.op_isomorphisms, CategoryTheory.GrothendieckTopology.uliftYonedaEquiv_naturality', CategoryTheory.Limits.opCoproductIsoProduct'_comp_self, CategoryTheory.Subfunctor.sheafify_le, PresheafOfModules.Sheafify.map_smul_eq, PresheafOfModules.map_comp_apply, AlgebraicGeometry.Scheme.toSpecΓ_isoSpec_inv_assoc, AlgebraicGeometry.LocallyRingedSpace.component_nontrivial, CategoryTheory.Presheaf.χ_app, CategoryTheory.uliftCoyonedaEquiv_symm_apply_app, CategoryTheory.ShortComplex.leftHomologyMap_op, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.sheafCondition_of_sheaf, CategoryTheory.Sheaf.instHasColimitsOfShape, AlgebraicGeometry.Scheme.IdealSheafData.ideal_map, CategoryTheory.Functor.representableByUliftFunctorEquiv_symm_apply_homEquiv, CategoryTheory.Functor.opId_hom_app, CategoryTheory.LocalizerMorphism.RightResolution.unop_X₁, CategoryTheory.GrothendieckTopology.Point.presheafFiberOfIsCofilteredCocone_ι_app, PresheafOfModules.pushforward_map_app_apply', AlgebraicGeometry.Proj.sub_apply, AlgebraicGeometry.isLocallyArtinian_iff_of_isOpenCover, CategoryTheory.Functor.sheafPullbackConstruction.instIsRightAdjointSheafSheafPushforwardContinuousOfHasWeakSheafify, CategoryTheory.Pseudofunctor.toDescentDataAsCoalgebra_obj_obj, CategoryTheory.Pseudofunctor.isStackFor_ofArrows_iff, CategoryTheory.Limits.opCoproductIsoProduct'_hom_comp_proj, CategoryTheory.Coyoneda.ULiftCoyoneda.instFullOppositeFunctorTypeUliftCoyoneda, CategoryTheory.instEssentiallySmallOpposite, CategoryTheory.Functor.shift_map_op_assoc, PresheafOfModules.instFaithfulFunctorOppositeAbToPresheaf, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_jointly_surjective, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverseObj_π_app, CategoryTheory.Functor.leftOp_faithful, CategoryTheory.LocalizerMorphism.LeftResolution.opEquivalence_unitIso, CategoryTheory.GrothendieckTopology.Point.tensorHom_comp_toPresheafFiber_μ_assoc, CategoryTheory.Functor.IsDenseSubsite.mapPreimage_comp_assoc, CategoryTheory.SimplicialObject.Truncated.trunc_obj_obj, SSet.mem_degenerate_iff_notMem_nonDegenerate, CategoryTheory.GrothendieckTopology.diagramCompIso_hom_ι_assoc, CategoryTheory.GrothendieckTopology.sheafifyCompIso_inv_eq_sheafifyLift, AlgebraicGeometry.Scheme.Hom.comp_app, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_hom_naturality, CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily_def, CategoryTheory.LocalizerMorphism.instHasLeftResolutionsOppositeOpOpOfHasRightResolutions, SSet.opFunctor_map, CategoryTheory.Adjunction.Quadruple.op_rightTriple, CategoryTheory.ShortComplex.homologyOpIso_inv_naturality, CategoryTheory.GrothendieckTopology.sheafifyMap_id, CategoryTheory.countableCategoryOpposite, PresheafOfModules.Derivation.d_mul, AddCommMonCat.coyonedaType_obj_map, CategoryTheory.MorphismProperty.RespectsIso.op, CategoryTheory.OverPresheafAux.counitForward_naturality₁, HomologicalComplex.cyclesOpIso_inv_naturality_assoc, SSet.Subcomplex.toImage_ι_assoc, CategoryTheory.Limits.piFunctor_obj_obj, Condensed.isSheafStonean, CategoryTheory.linearYoneda_obj_obj_carrier, CategoryTheory.Limits.walkingCospanOpEquiv_counitIso_hom_app, AlgebraicGeometry.continuousMapPresheaf_map, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.coalgebraEquivalence_unitIso_inv_app_hom, PresheafOfModules.instFullSheafOfModulesCompObjFunctorOppositeRingCatIsSheafForgetRestrictScalars, CategoryTheory.op_tensorUnit, CategoryTheory.Pretriangulated.commShiftIso_opOp_hom_app_assoc, CategoryTheory.ObjectProperty.limitsOfShape_op, CategoryTheory.Functor.instEssSurjOppositeOp, CategoryTheory.Functor.OneHypercoverDenseData.essSurj, SSet.Subcomplex.topIso_inv_ι, AlgebraicGeometry.Scheme.toSpecΓ_naturality, AlgebraicGeometry.PresheafedSpace.GlueData.snd_invApp_t_app_assoc, SSet.prodStdSimplex.strictMono_orderHomOfSimplex_iff, CategoryTheory.SimplicialObject.Truncated.rightExtensionInclusion_hom_app, lightDiagramToLightProfinite_obj, CategoryTheory.Coyoneda.objOpOp_hom_app, HomologicalComplex.Acyclic.op, CategoryTheory.ShortComplex.Homotopy.unop_h₃, CategoryTheory.Functor.unopId_inv_app, CategoryTheory.Presheaf.instIsIsoFunctorOfIsLeftKanExtensionOppositeType, sSetTopAdj_homEquiv_stdSimplex_zero, SSet.horn₃₁.ι₂_desc_assoc, CategoryTheory.SubobjectRepresentableBy.homEquiv_eq, CategoryTheory.GrothendieckTopology.uliftYonedaEquiv_symm_map, CategoryTheory.Limits.coconeLeftOpOfConeEquiv_inverse_obj, SSet.prod_δ_snd, CategoryTheory.Limits.π_comp_colimitUnopIsoOpLimit_inv, SSet.Subcomplex.ofSimplexProd_eq_range, smoothSheaf.ι_evalHom_assoc, CategoryTheory.ShortComplex.exact_op_iff, CategoryTheory.MorphismProperty.RightFraction.unop_s, CategoryTheory.sheafComposeIso_hom_fac, CategoryTheory.Comon.Comon_EquivMon_OpOp_functor, SSet.Truncated.Edge.CompStruct.tensor_simplex_snd, CategoryTheory.Limits.fst_opProdIsoCoprod_hom_assoc, CategoryTheory.CategoryOfElements.instHasInitialElementsOppositeOfIsRepresentable, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_inv_naturality, Profinite.isIso_indexCone_lift, SSet.stdSimplex.face_eq_ofSimplex, CategoryTheory.Limits.coneOfCoconeRightOp_π, CategoryTheory.extensiveTopology.isSheaf_yoneda_obj, CategoryTheory.GrothendieckTopology.sheafifyMap_sheafifyLift, CategoryTheory.Pretriangulated.shiftFunctorAdd'_op_inv_app, AlgebraicGeometry.Scheme.Hom.ι_toNormalization, CompHausLike.LocallyConstant.counitApp_app, CategoryTheory.GrothendieckTopology.Plus.toPlus_eq_mk, CategoryTheory.Equalizer.Sieve.w, CategoryTheory.Functor.const.opObjUnop_hom_app, AlgebraicGeometry.LocallyRingedSpace.Γ_Spec_left_triangle, CategoryTheory.Functor.IsCoverDense.sheafHom_eq, CategoryTheory.op_tensorObj, AlgebraicGeometry.StructureSheaf.comap_apply, CommMonCat.coyoneda_obj_map, CategoryTheory.GrothendieckTopology.Plus.res_mk_eq_mk_pullback, SSet.PtSimplex.RelStruct.δ_castSucc_map, CategoryTheory.eHomFunctor_obj_obj, Condensed.isoFinYonedaComponents_hom_apply, SSet.prod_σ_fst, AlgebraicGeometry.Scheme.Hom.smooth_appLE, SSet.stdSimplex.obj₀Equiv_apply, SSet.horn₃₁.desc.multicofork_pt, AlgebraicTopology.AlternatingFaceMapComplex.map_f, CategoryTheory.Pretriangulated.instIsTriangulatedOppositeOpOp, AlgebraicGeometry.Scheme.IdealSheafData.ideal_pow, CategoryTheory.ShortComplex.cyclesOpIso_hom_naturality, AlgebraicGeometry.RingedSpace.mem_top_basicOpen, AlgebraicGeometry.Spec.faithful, CompHausLike.LocallyConstantModule.functorToPresheaves_map_app, AlgebraicGeometry.Scheme.Opens.germ_stalkIso_inv, CategoryTheory.Iso.op_hom, CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheafObj_mapPreimage_condition, CategoryTheory.isSeparating_op_iff, CategoryTheory.StructuredArrow.functor_map, CategoryTheory.Functor.RepresentableBy.comp_homEquiv_symm, CategoryTheory.Pretriangulated.Opposite.contractibleTriangleIso_hom_hom₁, CategoryTheory.Pseudofunctor.DescentData.iso_hom, CategoryTheory.Limits.preservesColimit_rightOp, SSet.Truncated.HomotopyCategory.BinaryProduct.inverse_comp_mapHomotopyCategory_snd, CategoryTheory.Functor.PullbackObjObj.π_iso_of_iso_right_hom, SSet.skeleton_le_skeletonOfMono, CategoryTheory.Limits.PullbackCone.unop_inr, CategoryTheory.Sheaf.instIsIsoAppCounitConstantSheafAdjOfFaithfulOfFullConstantSheafOfIsConstant, SSet.prodStdSimplex.objEquiv_δ_apply, CategoryTheory.Enriched.FunctorCategory.enrichedId_π_assoc, AlgebraicGeometry.isSheaf_propQCTopology_iff, Subobject.presheaf_obj, CategoryTheory.Functor.opUnopEquiv_counitIso, CategoryTheory.shrinkYonedaGrp_map, CategoryTheory.Adjunction.Quadruple.op_leftTriple, AddCommMonCat.coyonedaType_map_app, CategoryTheory.NatTrans.op_id, CategoryTheory.instFullMonFunctorOppositeMonCatyonedaAddMon, CategoryTheory.Limits.pullbackIsoUnopPushout_hom_inl, CategoryTheory.Presheaf.IsSheaf.amalgamateOfArrows_map, CategoryTheory.Limits.reflectsLimits_op, CategoryTheory.Limits.reflectsColimitsOfShape_of_leftOp, CategoryTheory.Limits.preservesColimitsOfShape_of_leftOp, SheafOfModules.evaluationPreservesLimitsOfSize, CategoryTheory.GrothendieckTopology.Point.instPreservesColimitsOfSizeFunctorOppositePresheafFiber, CategoryTheory.Comon.MonOpOpToComon_map_hom, CategoryTheory.Functor.instFaithfulOppositeOp, CategoryTheory.SimplicialObject.Augmented.rightOpLeftOpIso_inv_right, CategoryTheory.Functor.sheafAdjunctionCocontinuous_unit_app_hom, CategoryTheory.Limits.spanOp_hom_app, AlgebraicGeometry.IsAffineOpen.ideal_le_iff, CategoryTheory.Limits.coconeOfConeUnop_pt, CategoryTheory.Presheaf.isLimit_iff_isSheafFor_presieve, AlgebraicGeometry.Scheme.comp_appLE, CategoryTheory.faithful_linearYoneda, TopCat.Presheaf.IsSheaf.isSheafUniqueGluing_types, AlgebraicTopology.karoubi_alternatingFaceMapComplex_d, CategoryTheory.Abelian.Ext.preadditiveCoyoneda_homologySequenceδ_singleTriangle_apply, CategoryTheory.Limits.coyonedaOpColimitIsoLimitCoyoneda_hom_comp_π_assoc, AlgebraicGeometry.isIso_fromTildeΓ_of_presentation, CategoryTheory.Join.opEquiv_inverse_map_inclRight_op, Profinite.Extend.functorOp_obj, CategoryTheory.GrothendieckTopology.Point.W_isInvertedBy_presheafFiber', AlgebraicGeometry.Scheme.Hom.congr_app, CategoryTheory.HasClassifier.reflectsIsomorphismsOp, AlgebraicGeometry.instFullOppositeCommRingCatLocallyRingedSpaceToLocallyRingedSpace, CategoryTheory.shrinkYonedaEquiv_symm_app_shrinkYonedaObjObjEquiv_symm, CategoryTheory.Functor.PullbackObjObj.mapArrowRight_comp_assoc, CategoryTheory.Functor.IsCoverDense.sheafIso_hom_hom, CategoryTheory.Limits.preservesColimitsOfSize_of_rightOp, CategoryTheory.Limits.hasLimitsOfShape_op_of_hasColimitsOfShape, CategoryTheory.Presheaf.isLocallySurjective_toSheafify, CategoryTheory.Limits.coconeRightOpOfConeEquiv_functor_map_hom, CategoryTheory.Over.opEquivOpUnder_inverse_obj, CategoryTheory.Limits.limitCompCoyonedaIsoCone_hom_app, CategoryTheory.Functor.IsCoverDense.restrictHomEquivHom_naturality_right_symm, CategoryTheory.Limits.widePullbackShapeOpEquiv_inverse, SSet.op_σ, CategoryTheory.Limits.reflectsFiniteProducts_op, CategoryTheory.uliftYonedaEquiv_apply, CategoryTheory.Limits.ι_comp_colimitRightOpIsoUnopLimit_hom_assoc, AlgebraicGeometry.Scheme.Opens.toSpecΓ_naturality_assoc, AlgebraicGeometry.Scheme.ofRestrict_appLE, AlgebraicGeometry.Scheme.Hom.QuasiFiniteAt.quasiFiniteAt, CategoryTheory.Sheaf.terminal_obj, CategoryTheory.SimplicialObject.hom_ext_iff, CategoryTheory.Sheaf.imageι_hom, CategoryTheory.Limits.FormalCoproduct.powerBifunctor_map_app, CategoryTheory.Limits.preservesLimitsOfSize_op, AlgebraicGeometry.IsAffineOpen.isoSpec_inv, CategoryTheory.HasSheafify.isLeftExact, AlgebraicGeometry.Scheme.Hom.id_app, TopCat.Presheaf.comp_app, CategoryTheory.LocalizerMorphism.nonempty_rightResolution_iff_op, CategoryTheory.ShortComplex.hasLeftHomology_iff_unop, AugmentedSimplexCategory.equivAugmentedSimplicialObjectFunctorCompToArrowIso_inv_app_left, CategoryTheory.Presheaf.isSheaf_yoneda', AlgebraicGeometry.Scheme.Γ_obj, LightCondensed.comp_hom, PresheafOfModules.unit_map_one, CategoryTheory.ShortComplex.exact_unop_iff, SSet.horn.faceSingletonComplIso_inv_ι, CategoryTheory.Functor.instInitialOppositeLeftOpOfFinal, SSet.Subcomplex.toSSetFunctor_obj, CategoryTheory.Limits.reflectsColimits_of_leftOp, CategoryTheory.GrothendieckTopology.pseudofunctorOver_toPrelaxFunctor_toPrelaxFunctorStruct_toPrefunctor_map_toFunctor_obj_obj_obj, PresheafOfModules.zsmul_app, CategoryTheory.GrothendieckTopology.Point.Hom.sheafFiber_comp_assoc, AlgebraicGeometry.isLocallyNoetherian_iff_of_affine_openCover, AlgebraicGeometry.Scheme.kerAdjunction_unit_app, Condensed.finYoneda_map, PresheafOfModules.toPresheaf_preservesLimit, AlgebraicTopology.alternatingFaceMapComplex_map_f, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.s_comp_δ, SSet.hasDimensionLT_subcomplex_top_iff, PresheafOfModules.toPresheaf_obj_coe, SSet.stdSimplex.δ_zero_toSSetObjI, SSet.Truncated.Edge.id_tensor_id, SSet.horn₃₁.ι₃_desc_assoc, CategoryTheory.op_whiskerLeft, SSet.Subcomplex.instSubsingletonHomToSSetBot, CategoryTheory.ShortComplex.LeftHomologyData.op_H, CategoryTheory.Functor.mapCoconeOp_inv_hom, AlgebraicGeometry.Scheme.IdealSheafData.mem_support_iff_of_mem, HomotopicalAlgebra.instHasFactorizationOppositeCofibrationsTrivialFibrationsOfTrivialCofibrationsFibrations, CategoryTheory.MorphismProperty.IsStableUnderComposition.op, CategoryTheory.IsCommMonObj.ofRepresentableBy, TopCat.Presheaf.germ_stalkSpecializes_assoc, CategoryTheory.Limits.Cocone.op_pt, CategoryTheory.GrothendieckTopology.Point.skyscraperPresheafHomEquiv_apply_app, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.coassoc_assoc, CategoryTheory.GrothendieckTopology.whiskerRight_toPlus_comp_plusCompIso_hom, CategoryTheory.Presieve.FamilyOfElements.singletonEquiv_apply, CategoryTheory.prodOpEquiv_unitIso_hom_app, CategoryTheory.Limits.coneRightOpOfCoconeEquiv_inverse_obj, CategoryTheory.NatTrans.op_whiskerRight_assoc, CategoryTheory.preadditiveYoneda_obj, CategoryTheory.Injective.injective_iff_projective_op, CategoryTheory.regularTopology.equalizerConditionMap_iff_nonempty_isLimit, LightProfinite.lightToProfinite_map_proj_eq, SSet.instBalanced, CategoryTheory.Limits.reflectsColimits_of_rightOp, CategoryTheory.GrothendieckTopology.yonedaULiftEquiv_naturality', CategoryTheory.preservesColimitsOfShape_of_isCardinalPresentable_of_essentiallySmall, CategoryTheory.Pseudofunctor.presheafHom_obj, CategoryTheory.isFinitelyPresentable_iff_preservesFilteredColimitsOfSize, AlgebraicGeometry.IsLocallyArtinian.isArtinianRing_of_isAffine, CategoryTheory.linearCoyoneda_obj_obj_isAddCommGroup, CategoryTheory.Functor.FullyFaithful.compUliftYonedaCompWhiskeringLeft_inv_app_app_down, AlgebraicGeometry.StructureSheaf.toPushforwardStalkAlgHom_apply, AlgebraicGeometry.Scheme.Hom.germ_stalkMap_apply, CategoryTheory.GrothendieckTopology.preservesSheafification_iff_of_adjunctions, SSet.Subcomplex.unionProd.image_β_hom, CompHausLike.LocallyConstant.incl_of_counitAppApp, CategoryTheory.Functor.leftOpId_hom_app, AlgebraicGeometry.Scheme.preimage_basicOpen_top, CategoryTheory.Subfunctor.isGeneratedBy_iff, CategoryTheory.WithTerminal.opEquiv_unitIso_hom_app, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_naturality_apply, AlgebraicGeometry.Scheme.ΓSpecIso_inv, AlgebraicGeometry.Scheme.Hom.app_appIso_inv_assoc, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv_counitIso, CategoryTheory.Limits.preservesColimit_leftOp, CompHausLike.sigmaComparison_eq_comp_isos, CategoryTheory.Limits.preservesLimitsOfShape_of_rightOp, CategoryTheory.Adjunction.compPreadditiveYonedaIso_inv_app_app_apply, CategoryTheory.HasLiftingProperty.transfiniteComposition.wellOrderInductionData.liftHom_fac, CategoryTheory.Functor.leftOpRightOpEquiv_unitIso_inv_app, CategoryTheory.Functor.map_opShiftFunctorEquivalence_unitIso_inv_app_unop, CategoryTheory.Subfunctor.Subpresheaf.isGeneratedBy_iff, AlgebraicGeometry.IsAffineOpen.basicOpen_fromSpec_app, PresheafOfModules.isoMk_hom_app, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_presheaf_map_assoc, CategoryTheory.WithTerminal.opEquiv_counitIso_hom_app, TopCat.Presheaf.stalkPushforward_germ_assoc, SSet.prodStdSimplex.objEquiv_naturality, CategoryTheory.ObjectProperty.instIsClosedUnderIsomorphismsOppositeOp, CategoryTheory.Functor.IsCoverDense.faithful_sheafPushforwardContinuous, TopCat.toSSetObj₀Equiv_symm_apply, CategoryTheory.BraidedCategory.unop_tensorμ, CategoryTheory.Sheaf.tensorProd_isSheaf, CategoryTheory.ComposableArrows.opEquivalence_unitIso_inv_app, TopCat.Presheaf.pushforward_map_app, AlgebraicGeometry.Scheme.Hom.coequifibered_normalizationDiagramMap, AlgebraicGeometry.ΓSpec.adjunction_counit_app', CategoryTheory.SimplicialObject.Augmented.const_obj_hom, TopologicalSpace.OpenNhds.op_map_id_obj, CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheafMap_restriction_assoc, CategoryTheory.PreOneHypercover.multifork_ι, AugmentedSimplexCategory.equivAugmentedSimplicialObjectFunctorCompPointIso_inv_app, CategoryTheory.GrothendieckTopology.Point.toPresheafFiberOfIsCofiltered_naturality, AlgebraicGeometry.Scheme.isoSpec_hom_naturality_assoc, AugmentedSimplexCategory.equivAugmentedSimplicialObjectFunctorCompPointIso_hom_app, CategoryTheory.ParametrizedAdjunction.hasLiftingProperty_iff, CategoryTheory.Pseudofunctor.DescentData'.comp_hom, SheafOfModules.isSheaf, AlgebraicGeometry.Scheme.isoSpec_inv_image_zeroLocus, CategoryTheory.Pretriangulated.TriangleOpEquivalence.functor_map_hom₃, AlgebraicGeometry.Scheme.IdealSheafData.range_glueDataObjι_ι, CommGrpCat.coyonedaType_map_app, CategoryTheory.instIsIsoFunctorOppositeSheafSheafComposeNatTrans, CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheafObjObjIso.inv_π_assoc, AlgebraicGeometry.Scheme.germToFunctionField_injective, CategoryTheory.NatTrans.removeLeftOp_app, CommRingCat.moduleCatRestrictScalarsPseudofunctor_mapId, CategoryTheory.instIsMonoidalFunctorOppositeWOfHasSheafComposeForgetOfHasEnoughPoints, CategoryTheory.Subfunctor.IsGeneratedBy.ofSection_le, CategoryTheory.Sheaf.toImage_ι, CategoryTheory.op_tensor_op, CategoryTheory.CommSq.HasLift.iff_op, SSet.modelCategoryQuillen.instHasLiftingPropertyιHornHAddNatOfNatOfFibration, CategoryTheory.Functor.IsCoverDense.restrictHomEquivHom_naturality_left, CategoryTheory.Idempotents.DoldKan.hε, CategoryTheory.uliftYonedaEquiv_symm_map, CategoryTheory.GrothendieckTopology.W.transport_isMonoidal, AlgebraicGeometry.Scheme.Modules.germ_restrictStalkNatIso_hom_app, HomologicalComplex.cyclesOpNatIso_inv_app, CategoryTheory.Limits.PreservesFiniteLimitsOfIsFilteredCostructuredArrowYonedaAux.iso_hom, CategoryTheory.Functor.cones_obj, CategoryTheory.preadditiveYonedaMap_app, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit_app', CategoryTheory.Pseudofunctor.DescentData'.isoMk_inv_hom, CategoryTheory.homOfLE_op_comp_eqToHom_assoc, SSet.stdSimplex.spineId_arrow_apply_zero, CategoryTheory.Square.unop_f₁₂, CategoryTheory.shrinkYoneda_obj, CategoryTheory.sheafBotEquivalence_inverse_map_hom, SSet.prod_δ_fst, CategoryTheory.Iso.unop_trans, SSet.comp_app_assoc, CondensedMod.isDiscrete_iff_isDiscrete_forget, AddCommGrpCat.coyonedaForget_inv_app_app, SSet.nonDegenerate_eq_bot_of_hasDimensionLT, CategoryTheory.instPreservesFiniteColimitsSheafExtensiveTopologyFunctorOppositeSheafToPresheafOfPreadditiveOfHasFiniteColimits, HomotopicalAlgebra.weakEquivalences_op_iff, CategoryTheory.Limits.FormalCoproduct.cechIsoAugmentedCechNerve_inv_left, PresheafOfModules.map_smul, CategoryTheory.Pseudofunctor.DescentData.iso_inv, CategoryTheory.Abelian.subobjectIsoSubobjectOp_apply, AlgebraicGeometry.isIso_morphismRestrict_iff_isIso_app, AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex_homotopyHomInvId, CategoryTheory.Limits.preservesLimit_leftOp, CategoryTheory.kernelUnopUnop_inv, CategoryTheory.Limits.coyonedaOpColimitIsoLimitCoyoneda'_inv_comp_π, SimplicialObject.opEquivalence_counitIso, CategoryTheory.MorphismProperty.fst'_self_eq_snd, CategoryTheory.Presieve.isSheafFor_iff_bijective_shrinkFunctor_ι_comp, CategoryTheory.NatTrans.leftOp_id, LightCondensed.ofSheafLightProfinite_obj, CategoryTheory.Enriched.FunctorCategory.diagram_map_app, CategoryTheory.SimplicialObject.Augmented.whiskering_map_app_right, AlgebraicGeometry.Scheme.Opens.ι_image_basicOpen_topIso_inv, PresheafOfModules.Monoidal.tensorObj_map_tmul, AlgebraicGeometry.Scheme.zeroLocus_setMul, AlgebraicGeometry.SpecMap_ΓSpecIso_inv_toSpecΓ, CategoryTheory.RetractArrow.op_r_right, CategoryTheory.Adjunction.compUliftCoyonedaIso_inv_app_app_down, CategoryTheory.LocalizerMorphism.LeftResolution.opEquivalence_inverse, CategoryTheory.presheafHom_obj, Alexandrov.principals_map, CategoryTheory.MorphismProperty.instHasFactorizationOppositeOp, AlgebraicTopology.DoldKan.MorphComponents.preComp_b, CategoryTheory.ObjectProperty.InheritedFromTarget.op, AlgebraicGeometry.Scheme.toSpecΓ_preimage_zeroLocus, CategoryTheory.Limits.PullbackCone.op_inr, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctorObj_π_app, AlgebraicTopology.DoldKan.Γ₂_obj_p_app, AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open, CategoryTheory.NatIso.unop_trans, CategoryTheory.Functor.whiskerLeft_obj_map_bijective_of_isCoverDense, CategoryTheory.Limits.reflectsLimit_of_rightOp, CategoryTheory.RelCat.instIsEquivalenceOppositeUnopFunctor, CategoryTheory.Limits.parallelPairOpIso_hom_app_one, CategoryTheory.Limits.walkingCospanOpEquiv_counitIso_inv_app, CategoryTheory.Subpresheaf.sheafify_le, CategoryTheory.ObjectProperty.IsConservativeFamilyOfPoints.jointlyReflectMonomorphisms, CategoryTheory.Functor.IsCoverDense.sheafHom_restrict_eq, TopCat.Presheaf.stalk_open_algebraMap, CategoryTheory.ShortComplex.opMap_τ₃, CategoryTheory.PreOneHypercover.multicospanIndex_snd, SSet.Subcomplex.eqToIso_hom, AlgebraicGeometry.Scheme.Hom.app_surjective, CategoryTheory.Pseudofunctor.CoGrothendieck.instFullαCategoryObjLocallyDiscreteOppositeCatMkOpFiberForgetInducedFunctor, CategoryTheory.Limits.PullbackCone.op_ι_app, CategoryTheory.ShortComplex.opcyclesOpIso_hom_naturality_assoc, SSet.yonedaEquiv_symm_comp, CategoryTheory.Pretriangulated.TriangleOpEquivalence.unitIso_hom_app, SSet.Truncated.spine_map_subinterval, TopCat.Presheaf.presheafEquivOfIso_inverse_obj_map, TopCat.Presheaf.pushforwardEq_hom_app, CategoryTheory.typeEquiv_inverse_obj, SSet.Subcomplex.eq_top_iff_contains_nonDegenerate, SSet.stdSimplex.range_δ, SSet.PtSimplex.RelStruct.δ_castSucc_map_assoc, CategoryTheory.Equivalence.symmEquivInverse_obj_counitIso_hom, AlgebraicGeometry.affineLocally_iff_affineOpens_le, AlgebraicTopology.DoldKan.HigherFacesVanish.comp_δ_eq_zero_assoc, AlgebraicGeometry.Scheme.fromSpecStalk_appTop, AlgebraicTopology.AlternatingFaceMapComplex.d_squared, CategoryTheory.Square.op_f₂₄, CondensedMod.LocallyConstant.instFullModuleCatCondensedDiscrete, CategoryTheory.essImage_yonedaMon, AlgebraicGeometry.RingedSpace.basicOpen_pow, CategoryTheory.Presheaf.isLocallySurjective_comp_iff, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.inv_naturality_assoc, CategoryTheory.Idempotents.DoldKan.N₂_map_isoΓ₀_hom_app_f, CategoryTheory.Pseudofunctor.DescentData.nonempty_fullyFaithful_toDescentData_iff_of_sieve_eq, CategoryTheory.Presieve.IsSheaf.comp_of_W_map_of_adjunction, CategoryTheory.Limits.opCoproductIsoProduct_hom_comp_π, SSet.PtSimplex.MulStruct.δ_succ_succ_map_assoc, CategoryTheory.Equivalence.inverseFunctor_map, TopCat.presheafToTop_obj, CategoryTheory.Limits.preservesFiniteColimits_rightOp, AlgebraicGeometry.HasRingHomProperty.iff_of_iSup_eq_top, CategoryTheory.Presheaf.coherentExtensiveEquivalence_unitIso_hom_app_hom_app, CategoryTheory.Sheaf.epi_of_isLocallySurjective', CategoryTheory.ShortComplex.unop_f, CategoryTheory.Adjunction.Triple.op_rightToLeft, AlgebraicGeometry.Scheme.Opens.toScheme_presheaf_obj, PresheafOfModules.instPreservesLimitsOfSizeModuleCatCarrierObjOppositeRingCatEvaluation, CategoryTheory.GrothendieckTopology.W_iff_isIso_map_of_adjunction, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.shortComplex_shortExact, AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal_ideal, CategoryTheory.Limits.reflectsLimit_rightOp, AlgebraicGeometry.PresheafedSpace.Γ_map_op, CategoryTheory.simplicialToCosimplicialAugmented_map_left, AddCommGrpCat.coyonedaForget_hom_app_app_hom, CategoryTheory.Limits.preservesLimit_of_unop, CategoryTheory.GrothendieckTopology.W_isInvertedBy_whiskeringRight_presheafToSheaf, HomotopicalAlgebra.cofibrations_op, CategoryTheory.OverPresheafAux.YonedaCollection.map₂_snd, CategoryTheory.Limits.hasLimit_of_hasColimit_rightOp, TopCat.Presheaf.pullbackObjObjOfImageOpen_hom_naturality, CategoryTheory.Limits.reflectsColimitsOfSize_op, CategoryTheory.Functor.isDense_iff_fullyFaithful_restrictedULiftYoneda, CategoryTheory.Sheaf.ΓHomEquiv_naturality_left, CategoryTheory.Sheaf.classifier_truth, CategoryTheory.SimplicialObject.Truncated.rightExtensionInclusion_left, SSet.stdSimplex.objMk₁_apply, CategoryTheory.SubobjectRepresentableBy.pullback_homEquiv_symm_obj_Ω₀, CategoryTheory.kernelOpUnop_inv, CategoryTheory.Limits.Fork.unop_ι_app_one, CategoryTheory.Limits.coyonedaOpColimitIsoLimitCoyoneda_inv_comp_π_assoc, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.invApp_app_apply, AlgebraicGeometry.Scheme.id_appTop, CategoryTheory.GrothendieckTopology.isoToPlus_hom, CategoryTheory.Arrow.cechNerve_obj, CategoryTheory.Limits.reflectsColimitsOfSize_of_rightOp, CategoryTheory.yonedaMon_naturality_assoc, SSet.instFiniteElemObjOppositeSimplexCategoryOpMkNonDegenerateOfFinite, CategoryTheory.coyoneda_preservesLimits, CategoryTheory.Sheaf.classifier_Ω₀, CategoryTheory.Comon.ComonToMonOpOp_map, SheafOfModules.pushforwardComp_inv_app_val_app, CategoryTheory.Sheaf.instHasSubobjectClassifierTypeOfEssentiallySmall, CategoryTheory.GrothendieckTopology.pseudofunctorOver_toPrelaxFunctor_toPrelaxFunctorStruct_toPrefunctor_obj_str_comp_hom_app, AlgebraicGeometry.AffineScheme.Γ_preservesLimits, CategoryTheory.Limits.reflectsLimits_leftOp, CategoryTheory.GrothendieckTopology.Point.toPresheafFiberMap_naturality_assoc, TopCat.Presheaf.germ_ext, CategoryTheory.Presheaf.isSheaf_coherent_of_projective_comp, AlgebraicGeometry.Scheme.Spec_map_presheaf_map_eqToHom, AlgebraicGeometry.Scheme.Opens.toSpecΓ_appTop_assoc, SSet.id_app, CategoryTheory.yonedaMonObj_map, CategoryTheory.Functor.IsCoverDense.sheafIso_inv_hom, CategoryTheory.Sheaf.isConstant_iff_forget, CategoryTheory.ShortComplex.homologyMap'_op, SSet.spine_arrow, CategoryTheory.instRepresentablyFlatOppositeOpOfRepresentablyCoflat, CategoryTheory.Cat.opFunctor_obj, AlgebraicGeometry.exists_eq_pow_mul_of_is_compact_of_quasi_separated_space_aux, SSet.instFiniteTensorObj, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_hom_naturality_assoc, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivCounitIso_inv_app_hom, CategoryTheory.shrinkYonedaObjObjEquiv_map_app, CategoryTheory.Subfunctor.ofSection_obj, CategoryTheory.uliftYonedaEquiv_naturality, CategoryTheory.OverPresheafAux.OverArrows.map₂_val, CategoryTheory.SimplicialObject.δ_comp_σ_self'_assoc, CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheafObjObjIso.hom_map_assoc, CategoryTheory.Functor.OneHypercoverDenseData.isEquivalence, CategoryTheory.Limits.opHomCompWhiskeringLimYonedaIsoCocones_hom_app_app_app, SSet.stdSimplex.δ_objEquiv_symm_apply, CategoryTheory.Limits.reflectsColimitsOfShape_rightOp, CategoryTheory.Presheaf.classifier_χ, CategoryTheory.Subpresheaf.isSeparated, CategoryTheory.SimplicialObject.Homotopy.h_comp_σ_succ_of_lt_assoc, CategoryTheory.Limits.reflectsLimit_leftOp, HomologicalComplex.instQuasiIsoMapOppositeSymmUnopFunctorOp, CategoryTheory.instHasImagesSheafType, CategoryTheory.Sheaf.hasFilteredColimitsOfSize, CategoryTheory.CosimplicialObject.Augmented.leftOpRightOpIso_inv_left, CategoryTheory.Limits.Fork.unop_op_ι, SSet.RelativeMorphism.ofSimplex₀_map, CategoryTheory.GrothendieckTopology.toSheafification_app, CategoryTheory.Limits.FormalCoproduct.powerBifunctor_obj, CategoryTheory.plusPlusSheaf_map_hom, AlgebraicTopology.DoldKan.map_Hσ, TopCat.Sheaf.extend_hom_app, CategoryTheory.ObjectProperty.isSeparating_unop_iff, SSet.stdSimplex.toSSetObj_app_const_zero, CategoryTheory.OverPresheafAux.YonedaCollection.yonedaEquivFst_eq, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_inv_app_shift, CategoryTheory.op_inv_braiding, CategoryTheory.Pretriangulated.TriangleOpEquivalence.counitIso_hom_app_hom₁, CategoryTheory.presheaf_mono_of_mono, CategoryTheory.Limits.opParallelPairIso_hom_app_zero, AlgebraicGeometry.IsAffineOpen.primeIdealOf_isMaximal_of_isClosed, CategoryTheory.Limits.walkingCospanOpEquiv_inverse_obj, AlgebraicGeometry.Scheme.fromSpecStalk_toSpecΓ, AlgebraicGeometry.Scheme.ker_toSpecΓ, TopCat.Presheaf.map_germ_eq_Γgerm, CategoryTheory.instPreservesFiniteLimitsFunctorOppositeSheafReflectorSheafToPresheaf, CategoryTheory.GrothendieckTopology.instFullSheafTypeUliftYoneda, AlgebraicGeometry.Scheme.IdealSheafData.ideal_bot, SSet.stdSimplex.ι₀_whiskerLeft_toSSetObjI_μ_assoc, CategoryTheory.Functor.coreprW_hom_app, CategoryTheory.Limits.coneOpEquiv_inverse_obj, AlgebraicGeometry.Spec.toLocallyRingedSpace_obj, CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence_counitIso, CategoryTheory.ShortComplex.unopFunctor_obj, PresheafOfModules.pushforward₀_obj_obj_isAddCommGroup, CategoryTheory.Nerve.instIsStrictSegalObjCatTruncatedOfNatNatNerveFunctor₂, PartialOrder.mem_nerve_degenerate_of_eq, AlgebraicTopology.DoldKan.compatibility_Γ₂N₁_Γ₂N₂_natTrans, SSet.Augmented.stdSimplex_map_right, SSet.Subcomplex.preimage_range, SSet.Subcomplex.toImage_app_coe, CategoryTheory.Limits.walkingSpanOpEquiv_inverse_obj, PresheafOfModules.toSheaf_map_sheafificationAdjunction_counit_app, CategoryTheory.Limits.cospanUnop_inv_app, AlgebraicGeometry.Scheme.ker_of_isAffine, CategoryTheory.Limits.Cowedge.condition_assoc, CategoryTheory.sheafification_map, CategoryTheory.Functor.natTransEquiv_symm_apply_app, CategoryTheory.GrothendieckTopology.PreservesSheafification.le, SSet.PtSimplex.MulStruct.δ_succ_succ_map, AlgebraicGeometry.IsAffineOpen.algebraMap_Spec_obj, AlgebraicGeometry.isBasis_basicOpen, CategoryTheory.Comon.monoidal_whiskerLeft_hom, CategoryTheory.Coyoneda.colimitCocone_ι_app, CategoryTheory.Limits.reflectsColimitsOfSize_rightOp, CategoryTheory.Pretriangulated.instIsHomologicalOppositeAddCommGrpCatObjFunctorPreadditiveYoneda, SheafOfModules.pushforwardNatTrans_id, topCatToSheafCompHausLike_map_hom_app, CategoryTheory.Presheaf.functorEnrichedHomCoyonedaObjEquiv_naturality, SSet.Subcomplex.homOfLE_comp_assoc, CategoryTheory.Limits.piConst_obj_map, CommMonCat.coyonedaForget_inv_app_app, Condensed.instPreservesLimitsOfShapeOppositeProfiniteDiscreteCarrierToTopTotallyDisconnectedSpaceOfFinite, PartialOrder.mem_nerve_nonDegenerate_iff_injective, CategoryTheory.GrothendieckTopology.plusFunctor_obj, CategoryTheory.ShortComplex.unopMap_τ₁, CategoryTheory.Equalizer.Presieve.Arrows.compatible_iff, SSet.prodStdSimplex.objEquiv_apply_snd, SSet.Subcomplex.unionProd.symmIso_hom, CategoryTheory.Limits.hasColimit_rightOp_iff_hasLimit, CategoryTheory.Pretriangulated.TriangleOpEquivalence.counitIso_inv_app_hom₃, AlgebraicGeometry.Scheme.Γevaluation_naturality_apply, CategoryTheory.Adjunction.Triple.op_adj₂, SSet.Subcomplex.instEpiToImage, CategoryTheory.NatTrans.removeUnop_app, SSet.hoFunctor.unitHomEquiv_eq, AlgebraicGeometry.locallyOfFiniteType_iff, SheafOfModules.restrictScalars_obj_val, AlgebraicGeometry.SheafedSpace.comp_c_app, CategoryTheory.ShortComplex.LeftHomologyMapData.op_φQ, CategoryTheory.NatTrans.removeOp_id, AlgebraicGeometry.Flat.flat_and_surjective_iff_faithfullyFlat_of_isAffine, PresheafOfModules.mono_iff_surjective, CategoryTheory.Square.op_X₃, HomotopicalAlgebra.instHasFactorizationOppositeTrivialCofibrationsFibrationsOfCofibrationsTrivialFibrations, CategoryTheory.GrothendieckTopology.Point.skyscraperPresheafHomEquiv_naturality_right_assoc, TopCat.Presheaf.SheafConditionEqualizerProducts.w, CategoryTheory.Limits.coend.map_comp_assoc, CategoryTheory.Localization.isoOfHom_op_inv, CategoryTheory.ShortComplex.op_pOpcycles_opcyclesOpIso_hom_assoc, CategoryTheory.sheafifyMap_id, AlgebraicTopology.DoldKan.Γ₀_obj_map, CategoryTheory.instFaithfulMonFunctorOppositeMonCatYonedaMon, CategoryTheory.Equivalence.sheafCongrPreregular_unitIso_hom_app_hom_app, CategoryTheory.cosimplicialSimplicialEquiv_functor_obj_obj, CategoryTheory.CostructuredArrow.toStructuredArrow'_obj, instPreservesFiniteCoproductsSheafTypeUliftYonedaOfPreservesFiniteProductsOppositeObjFunctorIsSheaf, PresheafOfModules.freeObj_map, SheafOfModules.instSmallElemForallObjCompModuleCatCarrierOppositeRingCatObjFunctorIsSheafPresheafOfModulesForgetEvaluationForgetLinearMapIdCarrierSections, CategoryTheory.isCoseparating_op_iff, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_presheaf_map_top_assoc, SSet.S.mk_map_eq_iff_of_mono, SSet.Subcomplex.image_le_iff, AlgebraicGeometry.liftCoborder_app, AlgebraicGeometry.LocallyRingedSpace.coe_toΓSpecSheafedSpace_hom_base_hom_apply_asIdeal, CategoryTheory.SimplicialObject.Augmented.const_map_left, CategoryTheory.simplicialCosimplicialEquiv_unitIso_hom_app, CategoryTheory.Limits.reflectsLimits_of_unop, CategoryTheory.Limits.pullbackIsoOpPushout_inv_fst_assoc, AlgebraicGeometry.Scheme.Opens.toSpecΓ_preimage_basicOpen, TopCat.Presheaf.isSheaf_iff_isSheaf_comp, AlgebraicTopology.DoldKan.N₁Γ₀_hom_app_f_f, SSet.Augmented.stdSimplex_obj_left, CategoryTheory.Functor.sheafPushforwardContinuousCompSheafToPresheafIso_inv_app_app, CategoryTheory.yonedaMon_naturality, CategoryTheory.MorphismProperty.MapFactorizationData.op_Z, SSet.hom_ext_iff, CategoryTheory.GrothendieckTopology.yonedaULiftEquiv_symm_naturality_left, CategoryTheory.Limits.colimitYonedaHomIsoLimit_π_apply, CategoryTheory.NatTrans.rightOpWhiskerRight, AlgebraicGeometry.isIntegral_appTop_of_universallyClosed, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.ofRestrict_invApp_apply, CategoryTheory.Presheaf.isLocallyInjective_whisker, AlgebraicGeometry.SheafedSpace.id_hom_c_app, CategoryTheory.cokernelUnopOp_hom, SSet.instIsStableUnderCoproductsMonomorphismsOfHasCoproductsType, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_w, AlgebraicGeometry.Scheme.zeroLocus_def, SSet.Edge.exists_of_simplex, CategoryTheory.Subfunctor.Subpresheaf.mem_ofSection_obj, SSet.stdSimplex.σ_objMk₁_of_le, CategoryTheory.WithInitial.opEquiv_inverse_obj, HomologicalComplex.instIsCorepresentableCompEvalObjOppositeFunctorTypeCoyonedaOp, AlgebraicGeometry.LocallyRingedSpace.toΓSpec_continuous, CategoryTheory.Limits.walkingCospanOpEquiv_unitIso_inv_app, AlgebraicGeometry.Spec_map_localization_isIso, CategoryTheory.unop_whiskerRight, CategoryTheory.Limits.coconeUnopOfConeEquiv_counitIso, AlgebraicGeometry.StructureSheaf.isLocalizedModule_toPushforwardStalkAlgHom_aux, CategoryTheory.Presheaf.coherentExtensiveEquivalence_functor_obj_obj, CategoryTheory.SimplicialObject.Augmented.const_obj_right, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv_assoc, CategoryTheory.Pseudofunctor.DescentData.instIsIsoαCategoryObjLocallyDiscreteOppositeCatMkOpHom, CategoryTheory.GrothendieckTopology.W_adj_unit_app, CategoryTheory.Sheaf.cartesianMonoidalCategorySnd_hom, CategoryTheory.Limits.preservesLimitsOfSize_leftOp, CategoryTheory.Limits.coneOfCoconeLeftOp_π_app, CategoryTheory.MorphismProperty.presheaf_mono_of_le, CategoryTheory.ObjectProperty.instSmallOppositeOp, CommAlgCat.lift_unop_hom, AlgebraicGeometry.ΓSpec.adjunction_counit_app, SSet.S.le_iff, CategoryTheory.Sheaf.instHasFunctorialFactorizationLocallySurjectiveLocallyInjective, AlgebraicGeometry.exists_eq_pow_mul_of_is_compact_of_quasi_separated_space_aux_aux, CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheafObj_condition_assoc, CategoryTheory.Limits.coneOfCoconeUnop_π, CategoryTheory.Equivalence.rightOp_inverse_map, AlgebraicGeometry.StructureSheaf.algebraMap_self_map, PresheafOfModules.limitCone_π_app_app, AlgebraicGeometry.tilde.isoTop_hom, AlgebraicTopology.DoldKan.inclusionOfMooreComplexMap_comp_PInfty_assoc, CategoryTheory.OverPresheafAux.unitAuxAuxAux_inv, CategoryTheory.ShortComplex.RightHomologyMapData.unop_φK, AlgebraicGeometry.Scheme.restrict_presheaf_obj, CategoryTheory.additive_yonedaObj', SSet.hasDimensionLT_iff, CategoryTheory.Pseudofunctor.toDescentDataAsCoalgebra_map_hom, CategoryTheory.Limits.isColimitOfConeOfCoconeLeftOp_desc, CategoryTheory.yonedaAddMonObjIsoOfRepresentableBy_inv_app_hom_apply, CategoryTheory.nerveMap_app_mk₁, CategoryTheory.OverPresheafAux.yonedaCollectionPresheafToA_app, CategoryTheory.Limits.SequentialProduct.functorMap_commSq_aux, CategoryTheory.Presheaf.isSheaf_comp_of_isSheaf, CategoryTheory.Functor.leftOpRightOpIso_hom_app, AugmentedSimplexCategory.equivAugmentedSimplicialObject_inverse_obj_obj, ModuleCat.Tilde.toOpen_res, CategoryTheory.OverPresheafAux.counitForward_val_snd, CategoryTheory.Sheaf.isLocallySurjective_iff_epi', CategoryTheory.ParametrizedAdjunction.homEquiv_symm_naturality_one_assoc, CategoryTheory.SimplicialObject.augmentOfIsTerminal_hom_app, SSet.stdSimplex.faceSingletonIso_zero_hom_comp_ι_eq_δ_assoc, CategoryTheory.op_leftUnitor, CategoryTheory.Idempotents.instIsIdempotentCompleteSimplicialObject, AlgebraicGeometry.germ_comp_stalkToFiberRingHom, CategoryTheory.Over.opEquivOpUnder_inverse_map, AlgebraicGeometry.Scheme.IdealSheafData.mem_supportSet_iff, CategoryTheory.Equalizer.Presieve.isSheafFor_singleton_iff, CategoryTheory.Limits.coconeUnopOfConeEquiv_functor_obj, CategoryTheory.Projective.projective_iff_preservesEpimorphisms_coyoneda_obj, SheafOfModules.instAdditiveSheafAddCommGrpCatToSheaf, CategoryTheory.Limits.SequentialProduct.cone_π_app_comp_Pi_π_neg_assoc, CategoryTheory.Presheaf.isLocallyInjective_toSheafify, AlgebraicGeometry.Scheme.Opens.toSpecΓ_appTop, commBialgCatEquivComonCommAlgCat_functor_obj_unop_X, AlgebraicGeometry.Scheme.Γ_def, CommAlgCat.mul_op_of_unop_hom, CategoryTheory.Idempotents.DoldKan.equivalence_counitIso, CategoryTheory.Pretriangulated.Opposite.contractibleTriangleIso_inv_hom₃, CategoryTheory.presheafHom_map_app, CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheafObjObjIso_inv_naturality, AlgebraicGeometry.Scheme.IdealSheafData.mem_support_iff, CondensedMod.epi_iff_surjective_on_stonean, CategoryTheory.Limits.opParallelPairIso_hom_app_one, CategoryTheory.typeEquiv_unitIso_hom_app, AlgebraicGeometry.PresheafedSpace.GlueData.ι_image_preimage_eq, AlgebraicGeometry.Spec.full, CategoryTheory.Equivalence.symmEquivFunctor_map, SSet.instFiniteObjOppositeSimplexCategoryTensorObj, CategoryTheory.Presheaf.instPreservesFiniteProductsOppositeObjFunctorIsSheafCoherentTopology, CategoryTheory.OverPresheafAux.MakesOverArrow.map₁, CategoryTheory.Functor.rightOpId_inv_app, CategoryTheory.Equivalence.sheafCongrPrecoherent_counitIso_inv_app_hom_app, CategoryTheory.Limits.isLimitConeOfCoconeUnop_lift, CategoryTheory.faithful_preadditiveCoyoneda, InfiniteGalois.mulEquivToLimit_symm_continuous, SSet.horn₃₁.desc.multicofork_π_two_assoc, IsOpenMap.pullbackIso_inv_app_app, CategoryTheory.Ind.isIndObject_inclusion_obj, CategoryTheory.Limits.isColimitOfConeUnopOfCocone_desc, CategoryTheory.Limits.yonedaCompLimIsoCocones_hom_app_app, SSet.ι₁_app_snd_apply, PresheafOfModules.instEpiModuleCatCarrierObjOppositeRingCatApp, AlgebraicGeometry.LocallyOfFiniteType.finiteType_appLE, AlgebraicTopology.DoldKan.Γ₂N₂ToKaroubiIso_hom_app, CategoryTheory.Limits.coend.condition_assoc, AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_stalkMap_toSpec_assoc, AlgebraicGeometry.Spec.map_appLE, CategoryTheory.Limits.instHasColimitDiscreteOppositeCompInverseOppositeOpFunctor, CategoryTheory.Functor.rightOp_obj, SSet.stdSimplex.objMk₁_injective, CategoryTheory.Limits.preservesLimitsOfShape_op, AlgebraicGeometry.Scheme.Hom.finitePresentation_appLE, CategoryTheory.Functor.mapTriangleOpCompTriangleOpEquivalenceFunctorApp_hom_hom₁, CategoryTheory.Limits.spanUnop_hom_app, smoothSheafCommRing.ι_evalHom, CategoryTheory.shrinkYonedaObjObjEquiv_map_app_assoc, AlgebraicGeometry.Γ_restrict_isLocalization, CategoryTheory.Subfunctor.ofSection_eq_range', CategoryTheory.simplicialCosimplicialEquiv_counitIso_hom_app_app, LightCondMod.LocallyConstant.instFullModuleCatSheafLightProfiniteCoherentTopologyConstantSheaf, AlgebraicGeometry.Scheme.Hom.appIso_inv_app_presheafMap, CategoryTheory.faithful_preadditiveYoneda, CategoryTheory.Yoneda.yoneda_faithful, AlgebraicTopology.DoldKan.HigherFacesVanish.comp_Hσ_eq_zero, CategoryTheory.Limits.ProductsFromFiniteCofiltered.finiteSubproductsCocone_π_app_eq_sum, CategoryTheory.Sheaf.cartesianMonoidalCategorySnd_val, CategoryTheory.Pretriangulated.Opposite.contractibleTriangleIso_hom_hom₃, CategoryTheory.representablyFlat_op_iff, CategoryTheory.Sheaf.χ_hom, CategoryTheory.Limits.instHasPushoutOppositeOpOfHasPullback, lightProfiniteToLightCondSetIsoTopCatToLightCondSet_inv_app_hom_app_hom_hom, AlgebraicGeometry.iSup_basicOpen_of_span_eq_top, AlgebraicGeometry.Scheme.Hom.app_eq, CategoryTheory.linearYoneda_map_app, CategoryTheory.SimplicialObject.augmentOfIsTerminal_left, AlgebraicGeometry.RingedSpace.res_zero, CategoryTheory.Sheaf.classifier_Ω, CategoryTheory.Pseudofunctor.CoGrothendieck.ι_map_base, CategoryTheory.OverPresheafAux.yonedaCollectionFunctor_obj, AlgebraicGeometry.PresheafedSpace.restrictStalkIso_hom_eq_germ, CategoryTheory.GrothendieckTopology.W_of_preservesSheafification, CategoryTheory.SimplicialObject.δ_naturality, AlgebraicGeometry.Scheme.Hom.comp_appLE, AlgebraicGeometry.Scheme.Modules.restrictFunctorComp_hom_app_app, CategoryTheory.Limits.end_.condition, SSet.horn.spineId_map_hornInclusion, CategoryTheory.GrothendieckTopology.Point.Hom.presheafFiber_comp_assoc, SSet.Truncated.HomotopyCategory.mk_surjective, TopCat.Presheaf.germ_res_assoc, AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_stalkMap_toSpec, CategoryTheory.Limits.preservesColimitsOfSize_op, CategoryTheory.GrothendieckTopology.yonedaULiftEquiv_symm_naturality_right, CategoryTheory.Sheaf.instMonoAppArrowPLocallySurjectiveLocallyInjectiveFunctorialLocallySurjectiveInjectiveFactorization, SSet.instFiniteCoprod, CommRingCat.moduleCatRestrictScalarsPseudofunctor_map, HomologicalComplex.opcyclesOpIso_hom_naturality_assoc, CompHausLike.LocallyConstant.counit_app_hom_app, CategoryTheory.Functor.sheafAdjunctionCocontinuous_counit_app_hom, AlgebraicGeometry.Spec.map_preimage_unop, CategoryTheory.Limits.opCoproductIsoProduct'_inv_comp_inj, CategoryTheory.Pseudofunctor.DescentData'.comp_pullHom'', SSet.horn₃₁.desc.multicofork_π_zero_assoc, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_hom_app, SSet.Subcomplex.image_monotone, SSet.horn_obj, CategoryTheory.sectionsFunctorNatIsoCoyoneda_inv_app_coe, CategoryTheory.NatIso.op_isoWhiskerRight, CategoryTheory.instMonoFunctorOppositeHomFullSubcategoryIsSheafOfHasWeakSheafifyOfSheaf, TopCat.Presheaf.germToPullbackStalk_stalkPullbackHom, CondensedSet.epi_iff_surjective_on_stonean, SSet.finite_of_isColimit, CategoryTheory.preadditiveYonedaObj_obj_isModule, AlgebraicGeometry.Scheme.Modules.restrictFunctorComp_inv_app_app, AlgebraicGeometry.ΓSpecIso_obj_hom, CategoryTheory.Presieve.IsSheafFor.comp_iff_of_preservesPairwisePullbacks, CategoryTheory.ComposableArrows.opEquivalence_functor_obj_map, SSet.Truncated.StrictSegal.spine_δ_arrow_lt, CategoryTheory.SimplicialObject.σ_def, CategoryTheory.Presheaf.coherentExtensiveEquivalence_inverse_obj_obj, SSet.leftUnitor_inv_app_apply, SSet.ι₀_fst_assoc, SSet.ι₁_comp, CategoryTheory.Comon.monoidal_leftUnitor_hom_hom, CategoryTheory.ShortComplex.HomologyData.unop_left, AlgebraicGeometry.Scheme.SpecΓIdentity_app, CategoryTheory.Functor.representable_preservesLimit, CategoryTheory.Presheaf.restrictedULiftYoneda_map_app, SSet.skeleton_obj_eq_top, SSet.Subcomplex.yonedaEquiv_coe, CategoryTheory.Pseudofunctor.isPrestackFor_ofArrows_iff, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.inv_naturality_assoc, CategoryTheory.hoFunctor.preservesFiniteProducts, CategoryTheory.RelCat.unopFunctor_comp_opFunctor_eq, AlgebraicGeometry.Spec.germ_stalkMapIso_hom_assoc, CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan_inv_app_app_apply_eq_id, PresheafOfModules.presheaf_map_apply_coe, AlgebraicGeometry.Scheme.IdealSheafData.le_ofIdeals_iff, CategoryTheory.CosimplicialObject.Augmented.leftOp_left_map, CategoryTheory.GrothendieckTopology.OneHypercoverFamily.IsSheafIff.fac, AlgebraicGeometry.Scheme.Hom.preimage_basicOpen_top, InfiniteGalois.isOpen_mulEquivToLimit_image_fixingSubgroup, CategoryTheory.Limits.spanOp_inv_app, CategoryTheory.Presheaf.isSeparated_iff_subsingleton, AlgebraicGeometry.FormallyUnramified.formallyUnramified_of_affine_subset, CategoryTheory.Equivalence.sheafCongrPrecoherent_counitIso_hom_app_hom_app, PresheafOfModules.instReflectsIsomorphismsSheafOfModulesSheafAddCommGrpCatToSheaf_1, CategoryTheory.GrothendieckTopology.toSheafify_comp_sheafifyCompIso_inv_assoc, CategoryTheory.Functor.ofOpSequence_obj, LightCondensed.isColimitLocallyConstantPresheafDiagram_desc_apply, AlgebraicGeometry.StructureSheaf.IsLocalization.to_basicOpen, CategoryTheory.Equivalence.rightOp_unitIso_hom_app, SSet.OneTruncation₂.ofNerve₂.natIso_hom_app_obj, CategoryTheory.Abelian.IsGrothendieckAbelian.OppositeModuleEmbedding.full_embedding, AlgebraicGeometry.Scheme.Opens.germ_stalkIso_inv_assoc, SSet.Subcomplex.homOfLE_ι_assoc, CategoryTheory.SheafOfTypes.finitary_extensive, CategoryTheory.Presieve.isSheafFor_arrows_iff, CategoryTheory.Sheaf.instIsGrothendieckAbelian, HomologicalComplex.quasiIsoAt_unopFunctor_map_iff, CategoryTheory.RanIsSheafOfIsCocontinuous.fac_assoc, AlgebraicGeometry.formallySmooth_stalkMap_iff, CategoryTheory.preservesLimits_preadditiveCoyoneda_obj, CategoryTheory.Functor.IsCoverDense.restrictHomEquivHom_naturality_left_assoc, AlgebraicTopology.DoldKan.PInfty_comp_map_mono_eq_zero, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.Hom.comm, CategoryTheory.MorphismProperty.IsInvertedBy.rightOp, AlgebraicGeometry.Scheme.Modules.restrict_map, CategoryTheory.Pseudofunctor.CoGrothendieck.map_obj_fiber, TopCat.Sheaf.objSupIsoProdEqLocus_hom_fst, CategoryTheory.Groupoid.invEquivalence_inverse_obj, CategoryTheory.linearYoneda_obj_obj_isAddCommGroup, CategoryTheory.Limits.preservesFiniteColimits_of_leftOp, AlgebraicGeometry.injective_germ_basicOpen, CategoryTheory.Limits.preservesFiniteLimits_of_rightOp, CategoryTheory.GrothendieckTopology.plusMap_plusLift, AlgebraicGeometry.instPreservesColimitsOfShapeOppositeCommRingCatSchemeDiscreteSpecOfFinite, CommAlgCat.fst_unop_hom, CategoryTheory.Sheaf.toImage_hom, CategoryTheory.isIso_iff_isIso_yoneda_map, CategoryTheory.GrothendieckTopology.Point.skyscraperSheafAdjunction_homEquiv_symm_apply, smoothSheafCommRing.evalHom_germ, CategoryTheory.essImage_yonedaGrp, CategoryTheory.ParametrizedAdjunction.homEquiv_symm_naturality_three, CategoryTheory.Sheaf.natTransΓRes_app, CategoryTheory.Limits.Fork.op_pt, SSet.instFiniteObjOppositeSimplexCategoryOfFinite, CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheafObjObjIso.hom_mapPreimage, AlgebraicGeometry.HasRingHomProperty.iff_appLE, PresheafOfModules.Derivation.congr_d, CategoryTheory.cosimplicialSimplicialEquiv_inverse_obj, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_map_assoc, CategoryTheory.cokernel.π_unop, CategoryTheory.GrothendieckTopology.yonedaEquiv_naturality', CategoryTheory.Sheaf.image_obj, CategoryTheory.equivYoneda'_hom_hom, CategoryTheory.GrothendieckTopology.uliftYonedaEquiv_uliftYoneda_map, AlgebraicGeometry.StructureSheaf.toOpen_comp_comap_assoc, CategoryTheory.MonoidalCategory.MonoidalRightAction.rightActionOfOppositeRightAction_actionHom, CategoryTheory.SimplicialObject.isoCoskOfIsCoskeletal_hom, SSet.RelativeMorphism.le_preimage, CategoryTheory.Functor.map_shift_unop, CategoryTheory.LocalizerMorphism.LeftResolution.opFunctor_obj, SSet.prod_map_fst, AlgebraicGeometry.IsAffineOpen.self_le_basicOpen_union_iff, CategoryTheory.Presieve.FamilyOfElements.Compatible.functorPushforward, CategoryTheory.Limits.hasFiniteColimits_opposite, CategoryTheory.Functor.IsCoverDense.presheafIso_hom_app, AlgebraicGeometry.exists_appTop_π_eq_of_isAffine_of_isLimit, SheafOfModules.Presentation.quasicoherentData_presentation, TopCat.Presheaf.app_surjective_of_injective_of_locally_surjective, CategoryTheory.sheafifyMap_sheafifyLift, AlgebraicGeometry.LocallyRingedSpace.GlueData.ι_isoSheafedSpace_inv, SSet.Truncated.id_app, CategoryTheory.Join.opEquiv_functor_map_op_edge, CategoryTheory.Limits.walkingParallelPairOpEquiv_functor, AlgebraicGeometry.StructureSheaf.const_mul_cancel, CategoryTheory.Limits.preservesLimitsOfSize_of_unop, CategoryTheory.yonedaMonObj_obj_coe, CategoryTheory.Presheaf.app_localPreimage, CategoryTheory.Functor.IsDenseSubsite.sheafifyHomEquivOfIsEquivalence_naturality_left, SSet.RelativeMorphism.map_eq_of_mem, CategoryTheory.Limits.coconeUnopOfConeEquiv_functor_map_hom, CategoryTheory.typeEquiv_counitIso_hom_app_hom_app, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.mk₀_f_comp_biprodAddEquiv_symm_biprodIsoProd_hom, HomologicalComplex.opcyclesOpIso_hom_toCycles_op, CategoryTheory.Pseudofunctor.DescentData.Hom.comm_assoc, SSet.whiskerRight_app_apply, topToLocale_obj, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.image_preimage_is_empty, CategoryTheory.Functor.IsDenseSubsite.mapPreimage_comp_map, CategoryTheory.Functor.instIsRepresentableCompOppositeOpObjTypeYonedaObjRightAdjointObjIsDefined, CategoryTheory.Injective.injective_iff_preservesEpimorphisms_preadditive_yoneda_obj', CategoryTheory.Presheaf.functorToRepresentables_obj, CategoryTheory.Limits.hasLimit_rightOp_iff_hasColimit, CategoryTheory.MonoidalCategory.MonoidalLeftAction.leftActionOfOppositeLeftAction_actionObj, CategoryTheory.unop_inv_leftUnitor, ContinuousMap.yonedaPresheaf_map, CategoryTheory.Functor.op_commShiftIso_hom_app_assoc, CategoryTheory.Functor.FullyFaithful.compYonedaCompWhiskeringLeftMaxRight_inv_app_app, SSet.Subcomplex.unionProd.ι₂_ι, SSet.StrictSegal.spineToSimplex_map, CategoryTheory.instLocallySmallFullSubcategoryFunctorOppositeTypeIsIndObject, SSet.rightUnitor_inv_app_apply, CategoryTheory.isIso_op, CategoryTheory.ShortComplex.RightHomologyData.op_K, CategoryTheory.yoneda_obj_obj, CategoryTheory.Join.opEquiv_functor_map_op_inclRight, PresheafOfModules.forgetToPresheafModuleCatObjMap_apply, CategoryTheory.GrothendieckTopology.Point.toPresheafFiberMap_w, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_zero_unitIso_inv_app, CategoryTheory.uliftYonedaEquiv_comp, SSet.stdSimplex.face_obj, CategoryTheory.Pretriangulated.shift_opShiftFunctorEquivalence_counitIso_inv_app_assoc, SheafOfModules.pushforwardCongr_hom_app_val_app, SSet.stdSimplex.nonDegenerateEquiv_apply_apply, SSet.Truncated.HomotopyCategory.BinaryProduct.left_unitality, CategoryTheory.Limits.coconeOpEquiv_inverse_map, CategoryTheory.GrothendieckTopology.Point.presheafFiberMapCocone_pt, CategoryTheory.sheafifyLift_id_toSheafify, SSet.modelCategoryQuillen.horn_ι_mem_J, CategoryTheory.evalEquiv_apply, CategoryTheory.hoFunctor.isIso_prodComparison_of_stdSimplex, CategoryTheory.Limits.Cofork.op_π_app_zero, AlgebraicGeometry.Scheme.Γevaluation_naturality_assoc, AlgebraicGeometry.Scheme.kerAdjunction_counit_app, CategoryTheory.Square.unop_X₃, CategoryTheory.opOpEquivalence_inverse, CategoryTheory.GrothendieckTopology.Point.presheafFiberOfIsCofilteredCocone_pt, CategoryTheory.Limits.IndObjectPresentation.toCostructuredArrow_obj_right_as, CategoryTheory.GrothendieckTopology.plusCompIso_whiskerLeft_assoc, CategoryTheory.Presheaf.freeYoneda_map, CategoryTheory.GrothendieckTopology.pointBotFunctor_map_hom, PresheafOfModules.pullback_assoc, CategoryTheory.Limits.opSpan_hom_app, CategoryTheory.MonoidalCategory.MonoidalLeftAction.leftActionOfOppositeLeftAction_actionHom_unop, CategoryTheory.Limits.π_comp_colimitLeftOpIsoUnopLimit_inv_assoc, CategoryTheory.Limits.widePushoutShapeOp_map, CategoryTheory.GrothendieckTopology.Point.instIsIsoδFunctorOppositePresheafFiber, CategoryTheory.GrothendieckTopology.instIsIsoFunctorOppositeSheafSheafComposeNatTransPlusPlusAdjunction, AlgebraicGeometry.Scheme.IdealSheafData.coe_support_ofIdealTop, AlgebraicGeometry.isIso_pushoutSection_of_isCompact_of_flat_right_of_ringHomFlat, SSet.stdSimplex.map_apply, SSet.instHasDimensionLTToSSetBotSubcomplex, CategoryTheory.Cat.opEquivalence_unitIso, CategoryTheory.GrothendieckTopology.overMapPullback_assoc_assoc, CategoryTheory.Sheaf.isSheaf_of_isLimit, AlgebraicTopology.AlternatingFaceMapComplex.ε_app_f_zero, HomologicalComplex.unop_X, CategoryTheory.Equivalence.sheafCongr.unitIso_hom_app_hom_app, AlgebraicGeometry.Scheme.component_nontrivial, CategoryTheory.whiskering_linearYoneda, TopCat.Presheaf.presheafEquivOfIso_unitIso_hom_app_app, CategoryTheory.Limits.isColimitCoconeUnopOfCone_desc, CategoryTheory.NatTrans.op_whiskerRight, LightCondensed.isoFinYonedaComponents_inv_comp, HomologicalComplex.opFunctor_map_f, AlgebraicGeometry.Scheme.zeroLocus_iInf_of_nonempty, CategoryTheory.Limits.coconeOpEquiv_functor_obj, CategoryTheory.LocalizerMorphism.LeftResolution.unop_w, CategoryTheory.ShortComplex.opEquiv_functor, CategoryTheory.Sheaf.cartesianMonoidalCategoryWhiskerLeft_hom, SSet.Truncated.spine_map_vertex, CategoryTheory.Iso.op_trans, CategoryTheory.FunctorToTypes.rightAdj_map_app, AlgebraicGeometry.Scheme.AffineZariskiSite.generate_presieveOfSections_mem_grothendieckTopology, AlgebraicGeometry.ΓSpec.toOpen_comp_locallyRingedSpaceAdjunction_homEquiv_app, CategoryTheory.GrothendieckTopology.Point.tensorHom_comp_toPresheafFiber_μ, AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap_app, AlgebraicGeometry.ΓSpec.adjunction_unit_app, CategoryTheory.ShortComplex.HomologyMapData.op_right, CategoryTheory.Functor.sheafPushforwardContinuousId_hom_app_hom_app, CategoryTheory.Functor.homObjFunctor_map_app, SSet.ι₀_snd, CategoryTheory.MonoidalCategory.DayConvolution.corepresentableBy₂'_homEquiv, AlgebraicGeometry.Scheme.basicOpen_pow, CategoryTheory.Limits.preservesLimitsOfSize_rightOp, CategoryTheory.Sheaf.instHasFiniteColimits, CategoryTheory.sheafToPresheaf_δ, CategoryTheory.Subfunctor.image_isFinite, CategoryTheory.monoidalUnopUnop_ε, AlgebraicGeometry.Scheme.Opens.toSpecΓ_top, AlgebraicGeometry.HasRingHomProperty.appLE, SSet.Truncated.Edge.CompStruct.tensor_simplex_fst, CategoryTheory.MorphismProperty.LeftFraction.unop_f, CategoryTheory.Functor.isRepresentedBy_iff, CategoryTheory.Sheaf.Hom.mono_iff_presheaf_mono, CategoryTheory.instIsSplitEpiOppositeOpOfIsSplitMono, CategoryTheory.ShortComplex.fromOpcycles_op_cyclesOpIso_inv_assoc, CategoryTheory.IsPushout.unop_iff, AlgebraicGeometry.Scheme.Hom.ker_apply, SSet.PtSimplex.MulStruct.δ_succ_castSucc_map, CategoryTheory.Injective.instEnoughProjectivesOppositeOfEnoughInjectives, CategoryTheory.Limits.limitUnopIsoUnopColimit_inv_comp_π_assoc, CategoryTheory.Subfunctor.le_sheafify, CategoryTheory.Limits.preservesFiniteLimits_leftOp, TopCat.Sheaf.existsUnique_gluing, SSet.Homotopy.h₀, AlgebraicGeometry.Scheme.IdealSheafData.ideal_comap_of_isOpenImmersion, CategoryTheory.Limits.pullbackIsoUnopPushout_inv_fst, CategoryTheory.Limits.whiskeringLimYonedaIsoCones_inv_app_app, CategoryTheory.whiskering_linearCoyoneda₂, CategoryTheory.Presheaf.freeYonedaHomEquiv_symm_comp, CategoryTheory.Hom.addEquivCongrRight_symm_apply, CategoryTheory.typeEquiv_unitIso_inv_app, CategoryTheory.Functor.IsLeftAdjoint.rightOp, CategoryTheory.NatTrans.unop_whiskerRight, classifyingSpaceUniversalCover_map, AlgebraicGeometry.AffineSpace.SpecIso_inv_appTop_coord, CategoryTheory.MorphismProperty.RightFractionRel.op, CategoryTheory.instRepresentablyCoflatOppositeOpOfRepresentablyFlat, CategoryTheory.Limits.Cone.equiv_hom_fst, CategoryTheory.IsPullback.unop_iff, AlgebraicGeometry.AffineSpace.functor_obj_map, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_inv_app'_assoc, CategoryTheory.hoFunctor.instIsLeftAdjointSSetCatHoFunctor, CategoryTheory.MorphismProperty.instHasTwoOutOfThreePropertyOppositeOp, CategoryTheory.NatTrans.op_comp_assoc, CategoryTheory.Limits.endFunctor_obj, CategoryTheory.ParametrizedAdjunction.inr_arrowHomEquiv_symm_apply_left_assoc, SSet.StrictSegal.spineInjective, CategoryTheory.Limits.FormalCoproduct.cechIsoCechNerve_inv_app, CategoryTheory.GrothendieckTopology.Cover.multicospanComp_inv_app, AlgebraicGeometry.germ_stalkClosedPointIso_hom_assoc, CategoryTheory.ShortComplex.rightHomologyMap'_op, CategoryTheory.ShortComplex.opcyclesOpIso_hom_toCycles_op_assoc, CategoryTheory.MonoidalCategory.MonoidalRightAction.rightActionOfOppositeRightAction_actionObj, CategoryTheory.GrothendieckTopology.yonedaOpCompCoyoneda_inv_app_app, Condensed.isColimitLocallyConstantPresheaf_desc_apply, SSet.degenerate_le_preimage, CategoryTheory.Functor.leftOpRightOpEquiv_functor_obj_obj, CategoryTheory.Limits.colimitCoyonedaHomIsoLimitUnop_π_apply, SSet.Truncated.HomotopyCategory.BinaryProduct.inverseCompFunctorIso_inv_app, CompHausLike.LocallyConstant.functorToPresheaves_map_app, AlgebraicGeometry.Scheme.AffineZariskiSite.restrictIsoSpec_hom_app, CategoryTheory.factorThruImage_comp_imageUnopOp_inv, SSet.Subcomplex.mem_nonDegenerate_iff, AlgebraicGeometry.PresheafedSpace.ColimitCoconeIsColimit.desc_c_naturality, AugmentedSimplexCategory.equivAugmentedSimplicialObjectFunctorCompToArrowIso_hom_app_right, CategoryTheory.isRegularEpiCategory_sheaf, CategoryTheory.SimplicialObject.instHasLimitsOfShape, CategoryTheory.FunctorToTypes.rightAdj_map_app_app, CategoryTheory.Join.opEquiv_inverse_obj_right_op, CategoryTheory.GrothendieckTopology.uliftYonedaCompSheafToPresheaf_inv_app_app, CategoryTheory.Limits.limitRightOpIsoOpColimit_hom_comp_ι, CategoryTheory.ShortComplex.opcyclesOpIso_hom_naturality, CategoryTheory.Limits.multispanIndexCoend_snd, CategoryTheory.NatIso.op_symm, CategoryTheory.Sieve.sieveOfUliftSubfunctor_apply, CategoryTheory.yonedaEquiv_yoneda_map, CategoryTheory.Limits.opProdIsoCoprod_inv_inl, CategoryTheory.OverPresheafAux.unitForward_naturality₂, SSet.RelativeMorphism.botEquiv_apply, CategoryTheory.Limits.isLimitOfCoconeLeftOpOfCone_lift, AlgebraicGeometry.Scheme.evaluation_eq_zero_iff_notMem_basicOpen, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctor_obj_pt, CommMonCat.coyonedaType_obj_obj_coe, CategoryTheory.GrothendieckTopology.plusFunctorWhiskerLeftIso_hom_app, HomotopicalAlgebra.fibrations_eq_unop, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.s₀_comp_δ₁_assoc, AlgebraicGeometry.Scheme.stalkMap_germ_assoc, CategoryTheory.Pseudofunctor.CoGrothendieck.Hom.congr, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_inv_eq_germ_apply, TopCat.Presheaf.SheafConditionEqualizerProducts.piInters.hom_ext_iff, CategoryTheory.Pretriangulated.Opposite.UnopUnopCommShift.iso_hom_app_assoc, CategoryTheory.Limits.preservesFiniteColimits_leftOp, SSet.mem_skeleton_obj_iff_of_nonDegenerate, CategoryTheory.MorphismProperty.IsStableUnderBaseChange.op, CategoryTheory.Limits.BinaryFan.unop_mk, CategoryTheory.Subobject.functor_obj, CategoryTheory.CategoryOfElements.costructuredArrowYonedaEquivalence_counitIso, CategoryTheory.LocalizerMorphism.instIsLocalizedEquivalenceOppositeOpOp, AlgebraicGeometry.SheafedSpace.ofRestrict_hom_c_app, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.inv_invApp, CategoryTheory.GrothendieckTopology.Point.Hom.sheafFiber_comp, CategoryTheory.shrinkYonedaMon_obj, AlgebraicTopology.inclusionOfMooreComplex_app, PresheafOfModules.sheafification_map, CategoryTheory.Limits.opCompYonedaSectionsEquiv_apply_app, CategoryTheory.MorphismProperty.instHasOfPostcompPropertyOppositeOpOfHasOfPrecompProperty, TopCat.Presheaf.whiskerIsoMapGenerateCocone_inv_hom, CategoryTheory.Pretriangulated.preadditiveYoneda_homologySequenceδ_apply, SSet.StrictSegal.spineToSimplex_interval, CategoryTheory.Idempotents.isIdempotentComplete_iff_opposite, AlgebraicTopology.DoldKan.Γ₂_obj_X_map, CategoryTheory.toPresheafToSheafCompComposeAndSheafify_app, SimplicialObject.Splitting.toKaroubiNondegComplexIsoN₁_inv_f_f, CategoryTheory.IsPushout.op_iff, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.inv_naturality_assoc, CategoryTheory.forgetEnrichmentOppositeEquivalence_unitIso, AlgebraicGeometry.Scheme.Hom.appIso_inv_app, InfiniteGalois.proj_adjoin_singleton_val, TopCat.Presheaf.stalk_mono_of_mono, SSet.Subcomplex.image_comp, CategoryTheory.Subfunctor.Subpresheaf.IsGeneratedBy.mem, SSet.Truncated.Edge.map_associator_hom, CategoryTheory.CategoryOfElements.costructuredArrowYonedaEquivalence_inverse, CategoryTheory.SimplicialObject.augment_right, CategoryTheory.Limits.isIndObject_iff, CategoryTheory.Limits.hasColimitsOfShape_op_of_hasLimitsOfShape, CategoryTheory.Functor.IsRepresentable.iff_exists_isRepresentedBy, CategoryTheory.MorphismProperty.LeftFraction.op_f, TopCat.Presheaf.map_germ_eq_Γgerm_assoc, AlgebraicGeometry.Scheme.Opens.germ_stalkIso_hom, CategoryTheory.Limits.pushoutIsoOpPullback_inv_fst, HomologicalComplex.instQuasiIsoAtOppositeMapSymmOpFunctorOp, TopologicalSpace.Opens.op_map_id_obj, SSet.stdSimplex.yonedaEquiv_symm_app_objEquiv_symm, CategoryTheory.SimplicialObject.augment_hom_zero, CategoryTheory.isSeparator_iff_faithful_coyoneda_obj, PresheafOfModules.toFreeYonedaCoproduct_fromFreeYonedaCoproduct_assoc, SSet.degenerate_zero, CategoryTheory.SimplicialObject.δ_comp_σ_of_gt'_assoc, CategoryTheory.Limits.coconeUnopOfConeEquiv_inverse_map, CategoryTheory.ShortComplex.RightHomologyData.unop_f', TopCat.Presheaf.app_surjective_of_stalkFunctor_map_bijective, CategoryTheory.Limits.BinaryFan.op_mk, CategoryTheory.Sieve.uliftNatTransOfLe_comm, SSet.horn₂₀.sq, AlgebraicGeometry.RingedSpace.isUnit_of_isUnit_germ, CategoryTheory.Functor.op_iff, AlgebraicGeometry.Scheme.zeroLocus_radical, CategoryTheory.uliftYoneda_obj_map, CategoryTheory.Functor.LeibnizAdjunction.adj_counit_app_right, TopCat.Sheaf.interUnionPullbackConeLift_right, AlgebraicGeometry.Scheme.Modules.restrictFunctorCongr_inv_app_app, LightProfinite.Extend.functor_map, CategoryTheory.regularTopology.mapToEqualizer_eq_comp, CategoryTheory.Groupoid.invEquivalence_functor_map, CategoryTheory.Presheaf.final_toCostructuredArrow_comp_pre, CategoryTheory.GrothendieckTopology.uliftYonedaCompSheafToPresheaf_hom_app_app, CompHausLike.LocallyConstant.functor_map_hom, CategoryTheory.instPreservesFiniteLimitsFunctorOppositeSheafLeftAdjointSheafToPresheaf, SheafOfModules.pullback_comp_id, CategoryTheory.yoneda_preservesLimitsOfShape, AlgebraicGeometry.Scheme.Hom.map_appLE_assoc, SSet.horn₃₂.exists_desc, CategoryTheory.ComposableArrows.opEquivalence_functor_map_app, CategoryTheory.ShortComplex.SnakeInput.op_L₀, CategoryTheory.GrothendieckTopology.map_yonedaEquiv', CategoryTheory.GrothendieckTopology.W_inverseImage_whiskeringLeft, AlgebraicGeometry.Scheme.AffineZariskiSite.presieveOfSections_surjective, CategoryTheory.sheafBotEquivalence_unitIso, CategoryTheory.MonoidalCategory.MonoidalRightAction.rightActionOfOppositeRightAction_actionUnitIso, SSet.Truncated.Edge.src_eq, CategoryTheory.Functor.initial_op_of_final, CategoryTheory.Presieve.preservesTerminal_of_isSheaf_for_empty, CategoryTheory.Sieve.natTransOfLe_comm, AugmentedSimplexCategory.equivAugmentedSimplicialObject_counitIso_inv_app_left_app, AlgebraicGeometry.Scheme.Modules.pseudofunctor_mapComp_hom_τl, CategoryTheory.kernelUnopOp_hom, SSet.stdSimplex.mem_face_iff, CategoryTheory.SimplicialObject.Augmented.rightOp_right_map, AlgebraicGeometry.StructureSheaf.const_one, SSet.hoFunctor.preservesTerminal, CategoryTheory.TwoSquare.guitartExact_op_iff, SSet.Truncated.rightExtensionInclusion_right_as, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_hom_app, AlgebraicGeometry.Scheme.Opens.ι_appIso, CategoryTheory.NatTrans.op_whiskerLeft_assoc, CategoryTheory.Equivalence.unop_counitIso, PresheafOfModules.surjective_of_epi, CategoryTheory.eHomFunctor_map_app, CategoryTheory.Idempotents.DoldKan.equivalence_inverse, CategoryTheory.Pretriangulated.Opposite.contractibleTriangleIso_inv_hom₁, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_toSpecΓ, AlgebraicGeometry.StructureSheaf.globalSectionsIso_hom, CategoryTheory.SimplicialThickening.functor_map, SheafOfModules.instPreservesFiniteLimitsFunctorOppositeAddCommGrpCatCompSheafToSheafSheafToPresheaf, CategoryTheory.uliftYoneda_obj_obj, CategoryTheory.ShortComplex.homologyOpIso_hom_naturality_assoc, CategoryTheory.OverPresheafAux.counitAuxAux_inv, CategoryTheory.MorphismProperty.IsMultiplicative.op, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_δ_assoc, AlgebraicGeometry.Scheme.Hom.map_appLE'_assoc, SSet.stdSimplex.coe_edge_down_toOrderHom, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.section_app_comp_hom_app, CategoryTheory.preservesFiniteColimits_preadditiveYonedaObj_of_injective, AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap_app', AlgebraicGeometry.IsAffineOpen.toSpecΓ_isoSpec_inv, CategoryTheory.Functor.IsDenseSubsite.sheafifyHomEquivOfIsEquivalence_naturality_right, CategoryTheory.ComposableArrows.opEquivalence_inverse_obj, AlgebraicGeometry.Scheme.Hom.appIso_inv_naturality, CategoryTheory.Limits.reflectsColimitsOfSize_of_leftOp, CategoryTheory.SimplicialObject.δ_comp_δ'_assoc, AlgebraicGeometry.Scheme.IdealSheafData.map_ideal, AlgebraicTopology.DoldKan.N₂Γ₂_inv_app_f_f, SSet.Subcomplex.unionProd.ι₁_ι, CategoryTheory.Limits.opProductIsoCoproduct'_comp_self, SimplexCategory.toTopHomeo_symm_naturality_apply, SSet.stdSimplex.faceSingletonIso_one_hom_comp_ι_eq_δ, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_presheaf_map, CategoryTheory.instIsEquivalenceOppositeOpOp, CategoryTheory.SimplicialObject.δ_comp_σ_self', CategoryTheory.Limits.π_comp_colimitOpIsoOpLimit_inv_assoc, CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinset_obj_map, SSet.instIsStableUnderCobaseChangeMonomorphisms, AlgebraicGeometry.Scheme.basicOpen_add_le, AlgebraicGeometry.Scheme.Hom.comp_app_assoc, AlgebraicGeometry.Scheme.comp_app, CategoryTheory.yonedaMon_map_app, CategoryTheory.SimplicialObject.Homotopy.precomp_h, CategoryTheory.shrinkYonedaEquiv_naturality, CategoryTheory.Injective.projective_iff_injective_op, AlgebraicGeometry.exists_of_res_zero_of_qcqs_of_top, AlgebraicGeometry.Scheme.zeroLocus_map, AlgebraicGeometry.Scheme.Hom.finite_app, TopCat.Presheaf.stalk_hom_ext_iff, LightCondensed.isoLocallyConstantOfIsColimit_inv, CategoryTheory.GrothendieckTopology.plusFunctorWhiskerRightIso_inv_app, AlgebraicGeometry.Scheme.AffineZariskiSite.coequifibered_iff_forall_isLocalizationAway, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_zeroLocus, CategoryTheory.Functor.opUnopEquiv_functor, AlgebraicTopology.DoldKan.inclusionOfMooreComplexMap_comp_PInfty, CompHausLike.LocallyConstantModule.functor_obj_obj_obj_isAddCommGroup_zero_apply, AlgebraicGeometry.PresheafedSpace.restrictStalkIso_inv_eq_germ_apply, AlgebraicGeometry.Scheme.evaluation_naturality, CategoryTheory.Limits.IsIndObject.finallySmall, CategoryTheory.GrothendieckTopology.toPlus_comp_plusCompIso_inv, CategoryTheory.Presheaf.imageSieve_apply, CategoryTheory.ShortComplex.LeftHomologyData.op_Q, AugmentedSimplexCategory.equivAugmentedSimplicialObjectFunctorCompDropIso_hom_app_app, Alexandrov.principals_obj, CategoryTheory.yonedaEquiv_naturality', Condensed.comp_val, CategoryTheory.Functor.instIsRepresentableObjOppositeTypeYoneda, AlgebraicGeometry.ΓSpec.left_triangle, AlgebraicGeometry.Scheme.Modules.toOpen_fromTildeΓ_app_assoc, CompHausLike.LocallyConstantModule.functor_obj_obj_obj_isAddCommGroup_add_apply, AlgebraicGeometry.Scheme.Hom.ideal_ker_le, CategoryTheory.OverPresheafAux.restrictedYonedaObj_map, AlgebraicGeometry.AffineSpace.comp_homOfVector, CondensedMod.hom_naturality_apply, AlgebraicGeometry.smooth_iff, CategoryTheory.isDetector_iff_reflectsIsomorphisms_coyoneda_obj, CategoryTheory.Pseudofunctor.DescentData'.comp_hom_assoc, CategoryTheory.monoidalUnopUnop_δ, CategoryTheory.CategoryOfElements.costructuredArrowULiftYonedaEquivalence_functor_obj, SSet.ι₁_snd_assoc, SimplicialObject.Splitting.ofIso_isColimit', PresheafOfModules.toSheaf_map_sheafificationHomEquiv_symm, CategoryTheory.Limits.walkingSpanOpEquiv_counitIso_hom_app, CategoryTheory.SimplicialObject.Truncated.sk.full, SSet.horn.edge_coe, HomologicalComplex.isSupportedOutside_op_iff, HomologicalComplex.unopEquivalence_unitIso, CategoryTheory.GrothendieckTopology.plusMap_comp_assoc, AlgebraicGeometry.Scheme.Hom.appIso_hom', TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv_inverse, AlgebraicTopology.DoldKan.HigherFacesVanish.comp_δ_eq_zero, AlgebraicGeometry.Scheme.zeroLocus_inf, CategoryTheory.Limits.preservesColimitsOfSize_of_unop, CategoryTheory.Presieve.preservesProduct_of_isSheafFor, CategoryTheory.WithTerminal.opEquiv_functor_obj, CategoryTheory.Limits.compCoyonedaSectionsEquiv_symm_apply_coe, CategoryTheory.Enriched.Functor.functorHom_whiskerLeft_natTransEquiv_symm_app, CategoryTheory.cokernel.π_op, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_homEquiv_apply', CategoryTheory.Pretriangulated.Opposite.OpOpCommShift.iso_hom_app_assoc, CategoryTheory.Limits.coneOfSectionCompYoneda_pt, CategoryTheory.PreOneHypercover.IsStronglySheafFor.map_amalgamate, CategoryTheory.Adjunction.corepresentableBy_homEquiv, AlgebraicGeometry.Scheme.IdealSheafData.ofIdeals_mono, AlgebraicGeometry.Scheme.instIsOpenImmersionToSpecΓOfIsQuasiAffine, CategoryTheory.ObjectProperty.instSmallOppositeOp_1, AlgebraicGeometry.Scheme.homOfLE_appTop, CategoryTheory.SimplicialObject.Homotopy.singularChainComplexFunctor_map_homology_eq_of_simplicialHomotopy, ContinuousMap.yonedaPresheaf_obj, CategoryTheory.Equalizer.Presieve.sheaf_condition, CategoryTheory.Limits.hasLimit_leftOp_iff_hasColimit, TopCat.toSheafCompHausLike_obj_obj, CategoryTheory.cosimplicialSimplicialEquiv_functor_map_app, CategoryTheory.Limits.isLimitConeUnopOfCocone_lift, SSet.horn₃₁.ι₂_desc, CategoryTheory.NatIso.unop_symm, CategoryTheory.Presheaf.isLocallyInjective_whisker_iff, CategoryTheory.Cat.opFunctor_map, CategoryTheory.Limits.isIndObject_colimit, AlgebraicGeometry.germ_injective_of_isIntegral, CategoryTheory.Functor.representable_preservesLimits, AlgebraicGeometry.Scheme.isoSpec_inv_naturality_assoc, CategoryTheory.Equivalence.inverseFunctorObjIso_inv, AlgebraicGeometry.StructureSheaf.smul_const, TopCat.Presheaf.pushforward_obj_obj, CategoryTheory.RanIsSheafOfIsCocontinuous.fac', AlgebraicTopology.DoldKan.N₂Γ₂ToKaroubiIso_hom_app, AlgebraicGeometry.IsAffineOpen.fromSpec_app_self, AlgebraicGeometry.Spec.toPresheafedSpace_map_op, SSet.modelCategoryQuillen.cofibration_of_mono, CategoryTheory.Pseudofunctor.toDescentDataAsCoalgebra_obj_hom, CategoryTheory.Limits.IndObjectPresentation.yoneda_isColimit_desc, PresheafOfModules.injective_of_mono, CategoryTheory.Limits.instHasLimitOppositeDiscreteOpFunctor, CategoryTheory.GrothendieckTopology.Point.skyscraperSheafFunctor_map_hom, SSet.OneTruncation₂.nerveEquiv_symm_apply_map, CategoryTheory.Presheaf.tautologicalCocone'_pt, AlgebraicGeometry.isClosedImmersion_iff_isAffineHom, AlgebraicGeometry.instPreservesColimitsOfShapeOppositeCommRingCatSchemeDiscreteWalkingPairSpec, CategoryTheory.whiskering_preadditiveYoneda, CategoryTheory.Pretriangulated.commShiftIso_unopUnop_inv_app, CategoryTheory.Limits.opCospan_hom_app, CategoryTheory.Presheaf.FamilyOfElementsOnObjects.IsCompatible.familyOfElements_apply, CategoryTheory.Limits.preservesFiniteColimits_op, SSet.Truncated.StrictSegal.spine_δ_arrow_gt, HomologicalComplex.exactAt_op_iff, PresheafOfModules.isoMk_inv_app, CategoryTheory.isSheaf_pointwiseColimit, IsOpenMap.pullbackObjIso_inv_app, CategoryTheory.Limits.coyonedaOpColimitIsoLimitCoyoneda'_hom_comp_π, CategoryTheory.lan_preservesFiniteLimits_of_preservesFiniteLimits, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.app_inv_app'_assoc, CategoryTheory.GrothendieckTopology.Point.toPresheafFiberOfIsCofiltered_naturality_assoc, CategoryTheory.Functor.unopOpIso_hom_app, CategoryTheory.uliftYonedaEquiv_uliftYoneda_map, CategoryTheory.Comon.MonOpOpToComonObj_comon_comul, HomologicalComplex.opEquivalence_counitIso, AlgebraicGeometry.AffineSpace.hom_ext_iff, CategoryTheory.Limits.preservesFiniteProducts_leftOp, CategoryTheory.Limits.coneLeftOpOfCoconeEquiv_inverse_obj, CategoryTheory.PresheafOfGroups.OneCocycle.ev_symm, CategoryTheory.instFaithfulFunctorOppositeTypeShrinkYoneda, CategoryTheory.yoneda_obj_map, CategoryTheory.Limits.PreservesFiniteLimitsOfIsFilteredCostructuredArrowYonedaAux.functorToInterchangeIso_hom_app_app, SSet.stdSimplex.isoNerve_hom_app_apply, CategoryTheory.ShortComplex.instHasHomologyOppositeOp, CategoryTheory.Cat.opFunctorInvolutive_hom_app_toFunctor_map, AlgebraicGeometry.ΓSpec.unop_locallyRingedSpaceAdjunction_counit_app', SSet.Subcomplex.mem_degenerate_iff, AlgebraicGeometry.LocallyRingedSpace.stalkMap_germ_assoc, CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheafMap_presheafObjObjIso_hom, TopCat.Presheaf.app_bijective_of_stalkFunctor_map_bijective, CategoryTheory.full_linearCoyoneda, CategoryTheory.Pseudofunctor.DescentData.ofObj_obj, AlgebraicGeometry.Scheme.Hom.preimage_basicOpen, CategoryTheory.Limits.FormalCoproduct.cechFunctor_obj, CategoryTheory.RelCat.opEquivalence_functor, AlgebraicTopology.DoldKan.compatibility_N₂_N₁_karoubi, CategoryTheory.IsGrothendieckAbelian.preservesColimit_coyoneda_obj_of_mono, topCatToSheafCompHausLike_obj, Condensed.epi_iff_locallySurjective_on_compHaus, instIsMonHomOppositeCommAlgCatOpOfHomToAlgHomBialgHom, CategoryTheory.Pretriangulated.Opposite.contractible_distinguished, AlgebraicTopology.DoldKan.QInfty_f_naturality_assoc, CategoryTheory.Functor.op_comp_isSheaf_of_types, CategoryTheory.Cat.opFunctorInvolutive_hom_app_toFunctor_obj, AlgebraicTopology.DoldKan.MorphComponents.postComp_a, AlgebraicGeometry.Scheme.IdealSheafData.subschemeι_app, AlgebraicGeometry.Scheme.Hom.appLE_appIso_inv, HomologicalComplex.fromOpcycles_op_cyclesOpIso_inv_assoc, CategoryTheory.Limits.isIndObject_limit_of_discrete_of_hasLimitsOfShape, CategoryTheory.GrothendieckTopology.Point.W_isInvertedBy_presheafFiber, SSet.stdSimplex.faceSingletonComplIso_hom_ι, CategoryTheory.isDetector_unop_iff, CategoryTheory.Limits.coneOfCoconeLeftOp_pt, CategoryTheory.Sheaf.adjunction_counit_app_val, CategoryTheory.GrothendieckTopology.pseudofunctorOver_mapId_hom_toNatTrans_app_hom_app, CategoryTheory.ObjectProperty.instIsClosedUnderLimitsOfShapeOppositeOpOfIsClosedUnderColimitsOfShape_1, AlgebraicGeometry.LocallyRingedSpace.comp_ring_hom_ext, TopCat.Presheaf.pullbackPushforwardAdjunction_unit_pullback_map_germToPullbackStalk, CategoryTheory.cokernelOpUnop_inv, CategoryTheory.SheafOfTypes.adhesive, CategoryTheory.Presheaf.isLeftKanExtension_along_uliftYoneda_iff, CategoryTheory.coyonedaEquiv_naturality, SheafOfModules.id_val, CategoryTheory.SimplicialObject.δ_comp_δ', AlgebraicGeometry.Scheme.isoSpec_Spec_hom, SSet.instHasDimensionLETensorObjHAddNat, CategoryTheory.CategoryOfElements.fromCostructuredArrow_obj_snd, Opens.mayerVietorisSquare_X₂, CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheaf_map, SSet.ι₁_app_fst, LightCondensed.isLocallySurjective_iff_locallySurjective_on_lightProfinite, SSet.skeleton_succ, CategoryTheory.Pseudofunctor.CoGrothendieck.ext_iff, CategoryTheory.GrothendieckTopology.yonedaEquiv_comp, AlgebraicGeometry.Scheme.basicOpen_res_eq, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.instEpiSheafAddCommGrpCatGShortComplex, CategoryTheory.GrothendieckTopology.Point.skyscraperPresheafHomEquiv_naturality_left_symm_assoc, CategoryTheory.coyonedaEquiv_apply, SSet.Subcomplex.iSup_ofSimplex_nonDegenerate_eq_top, CategoryTheory.Sieve.EffectiveEpimorphic.iff_forall_isSheafFor_yoneda, CategoryTheory.Limits.reflectsFiniteLimits_leftOp, PresheafOfModules.pushforward_obj_map_apply, CategoryTheory.ObjectProperty.isDetecting_unop_iff, AlgebraicGeometry.StructureSheaf.isLocalizedModule_toPushforwardStalkAlgHom, PresheafOfModules.instIsIsoFunctorSheafOfModulesCounitSheafificationAdjunction, CategoryTheory.MonoidalCategory.MonoidalRightAction.rightActionOfOppositeRightAction_actionAssocIso_unop, CategoryTheory.Equivalence.sheafCongr.inverse_obj_obj_obj, CategoryTheory.Equivalence.sheafCongrPreregular_inverse_obj_obj_obj, CategoryTheory.Limits.compCoyonedaSectionsEquiv_apply_app, CategoryTheory.Limits.kernelOrderHom_coe, SSet.ι₁_snd, AlgebraicGeometry.Scheme.Hom.app_invApp', CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_zero_unitIso_hom_app, CategoryTheory.Equivalence.rightOp_inverse_obj, CategoryTheory.Pretriangulated.op_distinguished, CategoryTheory.Functor.rightOpLeftOpIso_inv_app, AlgebraicGeometry.isNoetherian_iff_of_finite_iSup_eq_top, CategoryTheory.shrinkYonedaMon_map_app_shrinkYonedaObjObjEquiv_symm, SSet.Truncated.StrictSegal.spineInjective, AlgebraicGeometry.Scheme.image_basicOpen, CategoryTheory.Enriched.FunctorCategory.enrichedComp_π_assoc, CategoryTheory.DinatTrans.compNatTrans_app, HomologicalComplex.homologyOp_hom_naturality_assoc, AlgebraicGeometry.exists_appTop_map_eq_zero_of_isAffine_of_isLimit, AlgebraicGeometry.tilde.isIso_toOpen_top, CategoryTheory.SimplicialObject.Augmented.hom_ext_iff, CategoryTheory.Presheaf.isLocallyInjective_forget_iff, CategoryTheory.regularTopology.isLocallySurjective_iff, PartialOrder.mem_range_nerve_σ_iff, CategoryTheory.StructuredArrow.toCostructuredArrow_map, LightCondensed.forget_map_hom_app, CategoryTheory.Pseudofunctor.DescentData.hom_comp_assoc, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.invApp_app, TopCat.Presheaf.SubmonoidPresheaf.map, CategoryTheory.Limits.FormalCoproduct.cechIsoCechNerveApp_inv_π_assoc, CategoryTheory.Abelian.extFunctor_obj, CategoryTheory.ShortComplex.cyclesOpIso_inv_naturality, CategoryTheory.Limits.isIndObject_pi, CategoryTheory.Limits.coneRightOpOfCocone_π, CategoryTheory.Equalizer.Presieve.Arrows.FirstObj.ext_iff, CategoryTheory.Sheaf.isSeparating, CategoryTheory.Sheaf.isGrothendieckAbelian_of_essentiallySmall, SimplicialObject.Splitting.πSummand_comp_cofan_inj_id_comp_PInfty_eq_PInfty, CategoryTheory.kernel.ι_unop, LightCondensed.instPreservesLimitsOfShapeOppositeLightProfiniteDiscreteObjFinite, CategoryTheory.Limits.FormalCoproduct.cechIsoCechNerveApp_hom_π_assoc, CategoryTheory.ShortComplex.Splitting.unop_r, CategoryTheory.CostructuredArrow.overEquivPresheafCostructuredArrow_functor_map_toOverCompYoneda, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.app_invApp_assoc, CategoryTheory.NatTrans.equifibered_op_iff, SSet.stdSimplex.objMk₁_of_castSucc_lt, SSet.hasDimensionLT_of_isColimit, CategoryTheory.Comon.monoidal_whiskerRight_hom, CategoryTheory.Functor.IsCoverDense.Types.appHom_restrict, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.app_invApp, CategoryTheory.OverPresheafAux.counitBackward_counitForward, AlgebraicTopology.DoldKan.map_PInfty_f, AlgebraicGeometry.PresheafedSpace.restrictStalkIso_hom_eq_germ_apply, CategoryTheory.Comma.unopFunctor_obj, AlgebraicGeometry.HasRingHomProperty.appTop, CategoryTheory.Sheaf.cond, CategoryTheory.Limits.colimitHomIsoLimitYoneda_hom_comp_π_assoc, CategoryTheory.faithful_linearCoyoneda, HomologicalComplex.unopSymm_X, CategoryTheory.SimplicialObject.Augmented.rightOpLeftOpIso_inv_left_app, AlgebraicGeometry.LocallyRingedSpace.basicOpen_zero, CategoryTheory.Functor.functorHom_ext_iff, CategoryTheory.op_braiding, CategoryTheory.Presheaf.w, CategoryTheory.ObjectProperty.IsConservativeFamilyOfPoints.jointlyReflectIsomorphisms_type, SimplicialObject.Split.toKaroubiNondegComplexFunctorIsoN₁_hom_app_f_f, CategoryTheory.GrothendieckTopology.yonedaEquiv_symm_map, CategoryTheory.CategoryOfElements.costructuredArrowULiftYonedaEquivalence_functor_map, CategoryTheory.unop_neg, CategoryTheory.Functor.IsDenseSubsite.sheafEquiv_inverse, TopCat.Sheaf.existsUnique_gluing', SSet.Path.map_arrow, CategoryTheory.Pretriangulated.Opposite.commShift_natTrans_op_int, CategoryTheory.Functor.rightOp_faithful, AlgebraicGeometry.RingedSpace.isUnit_res_of_isUnit_germ, CategoryTheory.Limits.Wedge.condition, CategoryTheory.Sheaf.ΓObjEquivSections_naturality_symm, CategoryTheory.Pseudofunctor.CoGrothendieck.ι_obj_fiber, CategoryTheory.Abelian.extFunctor_map_app, CategoryTheory.Functor.rightOp_full, CategoryTheory.Limits.hasColimit_leftOp_iff_hasLimit, CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheafObjObjIso.hom_mapPreimage_assoc, SSet.Truncated.HomotopyCategory.BinaryProduct.iso_hom_toFunctor, CategoryTheory.Iso.unop_hom_inv_id_app, CategoryTheory.Coyoneda.naturality, CategoryTheory.GrothendieckTopology.sheafification_obj, Condensed.isoLocallyConstantOfIsColimit_inv, CategoryTheory.Presheaf.isLocallySurjective_iff_range_sheafify_eq_top, CategoryTheory.MonoidalCategory.MonoidalLeftAction.oppositeLeftAction_actionHomLeft_op, CategoryTheory.Sheaf.mono_of_isLocallyInjective, CategoryTheory.regularTopology.parallelPair_pullback_initial, CategoryTheory.ShortComplex.instHasLeftHomologyOppositeOpOfHasRightHomology, AlgebraicGeometry.Scheme.isNilpotent_iff_basicOpen_eq_bot, CategoryTheory.monoidalOpOp_μ, CategoryTheory.Sheaf.isLocallyInjective_iff_injective, CategoryTheory.Limits.hasEqualizers_opposite, CategoryTheory.cosimplicialSimplicialEquiv_counitIso_inv_app_app, CategoryTheory.Limits.IndObjectPresentation.toCostructuredArrow_obj_left, AlgebraicGeometry.IsIntegral.component_integral, CategoryTheory.Pseudofunctor.isStackFor_iff, PresheafOfModules.forgetToPresheafModuleCatObjObj_coe, CategoryTheory.Limits.pushoutIsoUnopPullback_inl_hom, CategoryTheory.ComposableArrows.opEquivalence_unitIso_hom_app, CategoryTheory.Equivalence.congrLeftFunctor_obj, CategoryTheory.Presheaf.isSeparator, CategoryTheory.SimplicialObject.augment_hom_app, CategoryTheory.Functor.W_map_of_adjunction_of_isContinuous, CategoryTheory.Sheaf.classifier_χ₀, AlgebraicGeometry.StructureSheaf.const_zero, CategoryTheory.ShortComplex.unop_X₁, CategoryTheory.Pretriangulated.mem_distTriang_op_iff, AlgebraicGeometry.Scheme.Hom.comp_appLE_assoc, CategoryTheory.TwoSquare.natTrans_op, TopCat.presheafToTypes_map, CategoryTheory.SimplicialObject.isCoskeletal_iff, CategoryTheory.SimplicialObject.Augmented.rightOpLeftOpIso_hom_left_app, CategoryTheory.Functor.sheafPushforwardContinuous_obj_obj_obj, AlgebraicGeometry.Scheme.isPullback_toSpecΓ_toSpecΓ, CategoryTheory.ShortComplex.unopMap_τ₂, CategoryTheory.Functor.opUnopEquiv_inverse, HomologicalComplex.cyclesOpIso_inv_naturality, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_map_bijective, CategoryTheory.yonedaGrp_naturality_assoc, AddCommMonCat.coyoneda_map_app, AlgebraicGeometry.Scheme.instSubsingletonCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensBot, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_hom_eq_germ_apply, CategoryTheory.Sheaf.ΓRes_map_assoc, AlgebraicTopology.DoldKan.N₂Γ₂_compatible_with_N₁Γ₀, AlgebraicGeometry.Scheme.ΓSpecIso_naturality, CategoryTheory.Under.mapFunctor_map, SSet.whiskerLeft_app_apply, CategoryTheory.Adjunction.unop_counit, CategoryTheory.Limits.coconeOfConeRightOp_ι, CategoryTheory.Limits.Fork.op_π, CategoryTheory.PresheafOfGroups.OneCochain.mul_ev, CategoryTheory.Limits.preservesLimitsOfSize_of_rightOp, AlgebraicGeometry.Scheme.restrictFunctorΓ_hom_app, CategoryTheory.Pseudofunctor.bijective_toDescentData_map_iff, AlgebraicGeometry.Scheme.Modules.restrictFunctorCongr_hom_app_app, AlgebraicGeometry.Scheme.Modules.pseudofunctor_mapComp_hom_τr, AlgebraicGeometry.Scheme.Γ_map_op, CategoryTheory.Iso.unop_symm, HomologicalComplex.opcyclesOpIso_hom_naturality, CategoryTheory.Functor.instIsRepresentableCompOppositeUliftFunctor, CategoryTheory.Presieve.FamilyOfElements.isAmalgamation_iff_ofArrows, AlgebraicGeometry.instIsIsoSchemeAppUnitOppositeCommRingCatAdjunctionOfIsAffine, CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily_compatible, CategoryTheory.Limits.isFiltered_costructuredArrow_yoneda_of_preservesFiniteLimits, TopCat.Sheaf.comp_app, AlgebraicGeometry.Scheme.Modules.restrictFunctorId_inv_app_app, smoothSheafCommRing.ι_forgetStalk_inv_apply, CategoryTheory.PreOneHypercover.multicospanIndex_left, CategoryTheory.GrothendieckTopology.W.monoidal, PresheafOfModules.unitHomEquiv_apply_coe, CategoryTheory.Equivalence.symmEquiv_functor, CategoryTheory.Limits.coconeRightOpOfConeEquiv_inverse_obj, Condensed.lanPresheafNatIso_hom_app, CategoryTheory.Limits.FormalCoproduct.cochainComplexFunctor_obj_d, CategoryTheory.Equivalence.symmEquivInverse_obj_functor, CategoryTheory.Pseudofunctor.CoGrothendieck.ι_map_fiber, CategoryTheory.isEventuallyConstant_of_isArtinianObject, CategoryTheory.Limits.widePushoutShapeOp_obj, AlgebraicGeometry.SheafedSpace.Γ_map, AlgebraicTopology.DoldKan.Γ₀.Obj.mapMono_on_summand_id, CategoryTheory.Sheaf.ab5ofSize, CategoryTheory.Sieve.functorInclusion_app, CategoryTheory.ShortComplex.opEquiv_counitIso, CategoryTheory.Pretriangulated.shiftFunctorZero_op_hom_app, Opens.coe_mayerVietorisSquare_X₁, CategoryTheory.GrothendieckTopology.uliftYonedaEquiv_symm_app_apply, SSet.comp_app, AlgebraicGeometry.Scheme.Modules.pushforwardComp_hom_app_app, CategoryTheory.OverPresheafAux.yonedaCollectionPresheaf_obj, TopCat.Presheaf.IsSheaf.isSheafPreservesLimitPairwiseIntersections, CategoryTheory.MonoidalCategory.MonoidalRightAction.rightActionOfOppositeRightAction_actionObj_unop, CategoryTheory.SimplicialObject.δ_comp_σ_succ', CategoryTheory.Pseudofunctor.presheafHom_map, AlgebraicTopology.AlternatingFaceMapComplex.obj_d_eq, CategoryTheory.prodOpEquiv_unitIso_inv_app, PresheafOfModules.freeObj_obj, CategoryTheory.nerveFunctor_map, CategoryTheory.Limits.IndObjectPresentation.toCostructuredArrow_map_left, CategoryTheory.Sheaf.ΓObjEquivHom_naturality_symm, SSet.stdSimplex.σ_objEquiv_symm_apply, AlgebraicGeometry.germ_stalkClosedPointIso_hom, CategoryTheory.Limits.coneOpEquiv_functor_obj, CategoryTheory.Join.opEquiv_inverse_map_inclLeft_op, TopCat.Presheaf.isLocallySurjective_iff, PresheafOfModules.inverseImage_W_toPresheaf_eq_inverseImage_isomorphisms, CategoryTheory.Pretriangulated.opShiftFunctorEquivalenceSymmHomEquiv_apply, TopCat.Sheaf.IsFlasque.epi_of_shortExact, SSet.Augmented.stdSimplex_obj_right, fintypeToFinBoolAlgOp_map, CategoryTheory.Limits.coconeRightOpOfConeEquiv_unitIso, LightCondensed.forget_obj_obj_map, CategoryTheory.Limits.multispanIndexCoend_left, AlgebraicGeometry.Scheme.toSpecΓ_isoSpec_inv, CategoryTheory.MorphismProperty.LeftFraction.op_s, AlgebraicGeometry.IsFinite.instHasAffinePropertyAndIsAffineFiniteCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTopHomAppTop, AlgebraicTopology.normalizedMooreComplex_objD, CategoryTheory.Limits.coneOpEquiv_counitIso, CategoryTheory.Functor.final_op_of_initial, SheafOfModules.toSheaf_obj_obj, CategoryTheory.Limits.coneUnopOfCoconeEquiv_functor_map_hom, CategoryTheory.MonoidalCategory.DayConvolutionUnit.rightUnitorCorepresentingIso_hom_app_app, CategoryTheory.Pseudofunctor.DescentData'.pullHom'₁₃_eq_pullHom_of_chosenPullback₃_assoc, SheafOfModules.pushforwardCongr_inv_app_val_app, CategoryTheory.Functor.PullbackObjObj.ofHasPullback_π, CategoryTheory.MonoidalCategory.MonoidalLeftAction.leftActionOfOppositeLeftAction_actionUnitIso, HomologicalComplex.instHasHomologyOppositeObjSymmOpFunctorOp, CategoryTheory.SimplicialObject.Homotopy.refl_h, AlgebraicGeometry.Scheme.Modules.mapPresheaf_app, AugmentedSimplexCategory.equivAugmentedSimplicialObject_functor_map_right, CategoryTheory.preservesLimits_preadditiveYoneda_obj, CategoryTheory.Adjunction.rightOp_counit, CategoryTheory.instPreservesFiniteLimitsFunctorOppositeSheafPresheafToSheaf, SSet.StrictSegal.spineToSimplex_spine, CategoryTheory.MorphismProperty.IsInvertedBy.leftOp, SSet.Subcomplex.prod_obj, SimplicialObject.Split.cofan_inj_naturality_symm_assoc, CategoryTheory.Functor.IsCoverDense.Types.appIso_inv, CategoryTheory.Limits.reflectsLimitsOfSize_rightOp, CommAlgCat.snd_unop_hom, SimplicialObject.Splitting.ofIso_ι, PresheafOfModules.Sheafify.smul_add, CategoryTheory.isCodetector_iff_reflectsIsomorphisms_yoneda_obj, CategoryTheory.Limits.pullbackIsoOpPushout_inv_fst, AlgebraicGeometry.PresheafedSpace.GlueData.f_invApp_f_app_assoc, CategoryTheory.MonoidalCategory.DayConvolution.corepresentableBy_homEquiv_symm_apply, CategoryTheory.monoidalOpOp_η, CategoryTheory.Comon.monoidal_associator_hom_hom, CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheafMap_presheafObjObjIso_hom_assoc, CategoryTheory.Limits.PreservesFiniteLimitsOfIsFilteredCostructuredArrowYonedaAux.isoAux_hom_app, typeToBoolAlgOp_map, AlgebraicGeometry.Scheme.Hom.preservesLocalization_normalizationDiagramMap, CategoryTheory.op_inv_leftUnitor, CategoryTheory.Hom.mulEquivCongrRight_symm_apply, SSet.Truncated.Edge.map_whiskerLeft, CategoryTheory.Functor.op_commShiftIso_inv_app_assoc, CategoryTheory.CostructuredArrow.toStructuredArrow'_map, CategoryTheory.Pretriangulated.TriangleOpEquivalence.counitIso_inv_app_hom₂, CategoryTheory.RetractArrow.op_i_right, SSet.horn₃₂.ι₃_desc, AlgebraicGeometry.ΓSpec.isIso_locallyRingedSpaceAdjunction_counit, SSet.comp_const, SSet.Truncated.HomotopyCategory.BinaryProduct.inverse_map_mkHom_homMk_id, CategoryTheory.Limits.reflectsLimitsOfSize_op, CategoryTheory.Sheaf.hasSeparator, CategoryTheory.additive_coyonedaObj', CategoryTheory.Subfunctor.range_eq_ofSection', SimplicialObject.opFunctor_obj_δ, CategoryTheory.unop_add, CategoryTheory.instFaithfulSheafSheafCompose, AlgebraicGeometry.Proj.awayToSection_comp_appLE, CategoryTheory.Limits.reflectsFiniteLimits_of_leftOp, CategoryTheory.Functor.FullyFaithful.homNatIso'_hom_app_down, CategoryTheory.SimplicialObject.Augmented.const_map_right, CategoryTheory.GrothendieckTopology.Point.instIsMonoidalFunctorOppositeHomPresheafToSheafCompSheafFiberIso, CategoryTheory.Functor.leftOpId_inv_app, CategoryTheory.Functor.PreOneHypercoverDenseData.multicospanMap_app, CategoryTheory.SimplicialObject.σ_comp_σ, CategoryTheory.Limits.pullbackIsoOpPushout_inv_snd, SheafOfModules.conjugateEquiv_pullbackComp_inv, AlgebraicGeometry.Spec.sheafedSpaceMap_hom_c_app, AlgebraicGeometry.IsAffineOpen.basicOpen_basicOpen_is_basicOpen, AlgebraicTopology.DoldKan.whiskerLeft_toKaroubi_N₂Γ₂_hom, SSet.const_app, AlgebraicGeometry.PresheafedSpace.GlueData.ι_isOpenEmbedding, TopCat.Presheaf.Pushforward.comp_inv_app, RingCat.moduleCatRestrictScalarsPseudofunctor_mapId, AlgebraicGeometry.Scheme.instFullOppositeIdealSheafDataOverSubschemeFunctor, TopCat.toSSetObj₀Equiv_apply, CategoryTheory.regularTopology.isSheaf_yoneda_obj, CategoryTheory.MonoidalCategory.MonoidalRightAction.oppositeRightAction_actionAssocIso_op, CategoryTheory.Equivalence.sheafCongrPreregular_inverse_map_hom_app, CategoryTheory.Sheaf.Hom.mono_of_presheaf_mono, CategoryTheory.Sheaf.instHasExactColimitsOfShapeOfHasFiniteLimitsOfPreservesColimitsOfShapeFunctorOppositeSheafToPresheaf, SSet.Truncated.HomotopyCategory.BinaryProduct.mapHomotopyCategory_prod_id_comp_inverse, AlgebraicGeometry.Scheme.evaluation_naturality_apply, AlgebraicGeometry.Spec.toSheafedSpace_obj, CategoryTheory.Functor.RepresentableBy.equivUliftYonedaIso_symm_apply_homEquiv, CategoryTheory.MorphismProperty.MapFactorizationData.op_i, AlgebraicGeometry.Scheme.toSpecΓ_appTop, CategoryTheory.Limits.coendFunctor_obj, SSet.Truncated.Edge.map_snd, CategoryTheory.ShortComplex.unop_X₃, CategoryTheory.SimplicialObject.instIsIsoAppUnitTruncatedCoskAdj, CategoryTheory.sheafToPresheafCompCoyonedaCompWhiskeringLeftSheafToPresheaf_app_app, CategoryTheory.Limits.end_.map_id, AlgebraicGeometry.Scheme.Hom.appIso_inv_naturality_assoc, CategoryTheory.leftDualFunctor_map, SheafOfModules.inl_freeSumIso_hom_assoc, AlgebraicTopology.DoldKan.MorphComponents.id_φ, SSet.stdSimplex.instFiniteObjOppositeSimplexCategory, AlgebraicGeometry.Scheme.Hom.appLE_map_assoc, SSet.StrictSegal.isPointwiseRightKanExtensionAt.fac_aux₂, CategoryTheory.GrothendieckTopology.sheafToPresheaf_map_sheafComposeNatTrans_eq_sheafifyCompIso_inv, CategoryTheory.Limits.walkingParallelPairOpEquiv_unitIso_inv_app_zero, CategoryTheory.ParametrizedAdjunction.homEquiv_symm_naturality_two, CategoryTheory.SimplicialObject.δ_comp_δ'', CategoryTheory.plusPlusSheaf_obj_obj, CategoryTheory.instSmallCarrierObjOppositeMonCatMonFunctorYonedaMon, PresheafOfModules.freeAdjunctionUnit_app, CategoryTheory.Presheaf.isLocallySurjective_iff_range_sheafify_eq_top', CategoryTheory.GrothendieckTopology.toPlusNatTrans_app, CategoryTheory.op_rightUnitor, smoothSheaf.obj_eq, CategoryTheory.Presieve.FamilyOfElements.singletonEquiv_symm_apply, AlgebraicTopology.normalizedMooreComplex_map, CategoryTheory.Functor.IsLocalization.op_iff, CategoryTheory.MorphismProperty.LeftFraction.unop_s, CategoryTheory.shrinkYonedaObjObjEquiv_obj_map_assoc, CategoryTheory.Functor.unop_obj, CategoryTheory.Pretriangulated.triangleOpEquivalence_counitIso, AlgebraicGeometry.PresheafedSpace.comp_c_app_assoc, AugmentedSimplexCategory.equivAugmentedSimplicialObject_functor_map_left_app, CategoryTheory.Presheaf.isLocallySurjective_iff_whisker_forget, SSet.degenerate_app_apply, CategoryTheory.SimplicialObject.Homotopy.h_comp_σ_succ_of_lt, CategoryTheory.Presheaf.isSheaf_iff_isLimit, CategoryTheory.sheafComposeNatTrans_app_uniq, CategoryTheory.Functor.ofOpSequence_map_homOfLE_succ, CategoryTheory.Sieve.shrinkFunctorUliftFunctorIso_inv_ι, CategoryTheory.Under.opEquivOpOver_functor_obj, CategoryTheory.MonoidalCategory.MonoidalLeftAction.leftActionOfOppositeLeftAction_actionHomLeft_unop, CategoryTheory.Presheaf.coherentExtensiveEquivalence_functor_map_hom, SSet.Subcomplex.preimage_obj, AlgebraicGeometry.instIsRightAdjointCommRingCatOppositeSchemeΓ, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.isoRestrict_hom_hom_c_app, CategoryTheory.OverPresheafAux.OverArrows.yonedaArrow_val, AlgebraicGeometry.IsAffineOpen.iSup_basicOpen_eq_self_iff, SSet.Subcomplex.image_top, AlgebraicGeometry.Scheme.Hom.comp_appIso, CategoryTheory.Limits.hasPushouts_opposite, CategoryTheory.Pretriangulated.triangleOpEquivalence_functor, SSet.Truncated.HomotopyCategory.BinaryProduct.functor_comp_inverse, CompHausLike.LocallyConstantModule.functorToPresheaves_obj_map, CategoryTheory.Enriched.Functor.natTransEquiv_symm_app_app_apply, AlgebraicGeometry.Scheme.basicOpen_one, CategoryTheory.Limits.isIndObject_limit_comp_yoneda, CategoryTheory.Sheaf.instHasFiniteLimits, CategoryTheory.Limits.preservesColimits_leftOp, CategoryTheory.Square.op_X₁, CategoryTheory.MonoidalCategory.MonoidalLeftAction.oppositeLeftAction_actionObj, SheafOfModules.pushforwardPushforwardAdj_unit_app_val_app, PresheafOfModules.evaluation_preservesLimit, HomologicalComplex.extend_op_d, SSet.Truncated.HomotopyCategory.descOfTruncation_map_homMk, PresheafOfModules.presheaf_obj_coe, CategoryTheory.Pseudofunctor.DescentData.pullFunctorObj_obj, CategoryTheory.NatTrans.op_app, AlgebraicTopology.DoldKan.Γ₀_map_app, CategoryTheory.Limits.isFiltered_costructuredArrow_yoneda_iff_nonempty_preservesFiniteLimits, CategoryTheory.uliftYoneda_obj_map_down, CategoryTheory.SimplicialObject.comp_left_app, AugmentedSimplexCategory.equivAugmentedSimplicialObject_counitIso_hom_app_right, CategoryTheory.Limits.PushoutCocone.op_π_app, AlgebraicGeometry.Scheme.Hom.appIso_inv_appLE_assoc, CategoryTheory.NatTrans.op_comp, SSet.Truncated.Edge.exists_of_simplex, SSet.Subcomplex.prod_le_unionProd, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_invApp, SSet.Truncated.HomotopyCategory.BinaryProduct.inverse_map_mkHom_homMk_homMk, SSet.Truncated.Edge.map_id, CategoryTheory.Functor.Elements.instHasInitialObjOppositeTypeFlipShrinkYonedaOp, CategoryTheory.Subfunctor.IsGeneratedBy.iSup_eq, AlgebraicGeometry.Scheme.Modules.Hom.id_app, SSet.RelativeMorphism.comm_assoc, CategoryTheory.Adjunction.rightOp_eq, CategoryTheory.Limits.Cone.equiv_inv_π, CategoryTheory.rightDualFunctor_map, SSet.orderEmbeddingN_apply, CategoryTheory.Idempotents.DoldKan.N_obj, CategoryTheory.isCofilteredOrEmpty_op_of_isFilteredOrEmpty, CategoryTheory.ShortComplex.hasRightHomology_iff_unop, CategoryTheory.Equivalence.sheafCongrPreregular_inverse_obj_obj_map, CategoryTheory.instFinitaryExtensiveSheafOfHasPullbacksOfHasSheafify, AlgebraicGeometry.targetAffineLocally_affineAnd_iff, CompHausLike.LocallyConstant.adjunction_left_triangle, AlgebraicGeometry.StructureSheaf.algebraMap_obj_top_bijective, AlgebraicTopology.DoldKan.toKaroubiCompN₂IsoN₁_hom_app, CategoryTheory.extensiveTopology.isLocallySurjective_iff, SSet.Subcomplex.instMonoToRange, AlgebraicGeometry.Scheme.Hom.appIso_inv_app_assoc, CategoryTheory.NatIso.unop_whiskerRight, CategoryTheory.Limits.π_comp_colimitOpIsoOpLimit_inv, CategoryTheory.Functor.leftOpRightOpEquiv_inverse_map, TopCat.Sheaf.pushforward_obj_val, HomotopicalAlgebra.weakEquivalences_unop_iff, CategoryTheory.SimplicialObject.Truncated.trunc_obj_map, CategoryTheory.Join.InclLeftCompRightOpOpEquivFunctor_hom_app, CategoryTheory.Limits.hasProductsOfShape_opposite, HomologicalComplex.opInverse_obj, CategoryTheory.Cat.opFunctorInvolutive_inv_app_toFunctor_map, CategoryTheory.Functor.leftOpRightOpEquiv_unitIso_hom_app, CategoryTheory.Limits.reflectsLimitsOfShape_of_unop, CategoryTheory.Limits.hasFiniteCoproducts_opposite, CategoryTheory.Presheaf.isSheaf_iff_multiequalizer, SheafOfModules.fullyFaithfulForget_preimage_val, HomologicalComplex.unopSymm_d, AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalHom, SSet.Subcomplex.unionProd.ι₁_ι_assoc, CategoryTheory.Presheaf.uliftYonedaAdjunction_homEquiv_app, CategoryTheory.Sheaf.isLocallyInjective_forget, PresheafOfModules.instReflectsIsomorphismsSheafOfModulesSheafAddCommGrpCatToSheaf, SheafOfModules.pullback_id_comp, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_self, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_appLE_assoc, TopCat.Sheaf.sections_exact_of_left_exact, CategoryTheory.Limits.ProductsFromFiniteCofiltered.finiteSubproductsCone_pt, CategoryTheory.Sieve.uliftNatTransOfLe_app_down_coe, CategoryTheory.Sheaf.truth_hom, PresheafOfModules.homMk_app, SimplicialObject.Split.Hom.comm, AlgebraicGeometry.Scheme.Γ_map, SSet.modelCategoryQuillen.cofibration_iff, CategoryTheory.full_preadditiveCoyoneda, CategoryTheory.Functor.IsCoverDense.restrictHomEquivHom_naturality_right_assoc, AlgebraicGeometry.AffineScheme.forgetToScheme_map, CategoryTheory.SimplicialObject.whiskering_map_app_app, CategoryTheory.ParametrizedAdjunction.arrowHomEquiv_apply_right_fst_assoc, SheafOfModules.restrictScalars_map_val, CategoryTheory.SimplicialObject.Augmented.rightOp_right_obj, SSet.Truncated.hoFunctor₂_naturality, CochainComplex.homotopyOp_hom_eq, CategoryTheory.cones_obj_map_app, CategoryTheory.Limits.colimitHomIsoLimitYoneda'_inv_comp_π_assoc, AlgebraicGeometry.Scheme.zeroLocus_eq_univ_iff_subset_nilradical_of_isCompact, CategoryTheory.OverPresheafAux.MakesOverArrow.map₂, SSet.instHasDimensionLTTensorObjHAddNat, AlgebraicTopology.DoldKan.QInfty_f_naturality, CategoryTheory.Limits.walkingSpanOpEquiv_unitIso_inv_app, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.instIsIsoCommRingCatInvApp, CategoryTheory.NatTrans.removeRightOp_app, SSet.horn₃₂.ι₀_desc, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_naturality_apply, CategoryTheory.instReflectsIsomorphismsSheafSheafCompose, CategoryTheory.SimplicialObject.Augmented.rightOpLeftOpIso_hom_right, TopCat.Presheaf.stalkFunctor_map_germ_apply, HomologicalComplex.isStrictlySupported_op_iff, CategoryTheory.sheafificationNatIso_inv_app_hom, TopCat.Presheaf.germ_res_apply', CategoryTheory.toSheafify_plusPlusIsoSheafify_hom, CategoryTheory.isRegularEpi_unop_iff_isRegularMono, AlgebraicGeometry.Scheme.LocalRepresentability.yoneda_toGlued_yonedaGluedToSheaf_assoc, CategoryTheory.Square.opFunctor_map_τ₁, CategoryTheory.Functor.instIsEquivalenceOppositeOp, CategoryTheory.Limits.opCospan_inv_app, AugmentedSimplexCategory.equivAugmentedSimplicialObjectFunctorCompToArrowIso_hom_app_left, CategoryTheory.Limits.limitRightOpIsoOpColimit_hom_comp_ι_assoc, CategoryTheory.ObjectProperty.isClosedUnderColimitsOfShape_op_iff_unop, CategoryTheory.Functor.WellOrderInductionData.Extension.map_succ, SSet.Subcomplex.homOfLE_ι, CategoryTheory.LocalizerMorphism.RightResolution.unopFunctor_obj, CategoryTheory.Limits.FormalCoproduct.cechIsoCechNerveApp_hom_π, AlgebraicGeometry.Scheme.Modules.Hom.isIso_iff_isIso_app, CategoryTheory.SimplicialObject.Truncated.rightExtensionInclusion_right_as, Condensed.isoFinYonedaComponents_inv_comp, CategoryTheory.Functor.sheafPushforwardCocontinuousCompSheafToPresheafIso_inv, CategoryTheory.Sheaf.coneΓ_pt, AlgebraicGeometry.Spec_zeroLocus_eq_zeroLocus, SSet.face_le_horn, SSet.Subcomplex.image_preimage_le, CategoryTheory.Functor.PullbackObjObj.mapArrowRight_comp, AlgebraicGeometry.Scheme.Opens.ι_appTop, CategoryTheory.Limits.reflectsColimit_op, AlgebraicGeometry.Scheme.Modules.Hom.comp_app, CategoryTheory.unopHom_apply, CategoryTheory.Limits.hasLimit_of_hasColimit_leftOp, HomologicalComplex.cyclesOpIso_hom_naturality_assoc, CategoryTheory.Limits.coconeRightOpOfConeEquiv_inverse_map, Condensed.instFinalOppositeDiscreteQuotientCarrierToTopTotallyDisconnectedSpaceCostructuredArrowFintypeCatProfiniteOpToProfiniteOpPtAsLimitConeFunctorOp, CategoryTheory.Adjunction.compPreadditiveYonedaIso_hom_app_app_apply, CategoryTheory.Limits.IndObjectPresentation.yoneda_ι_app, CompHausLike.LocallyConstantModule.functor_obj_obj_obj_isAddCommGroup_nsmul_apply, CategoryTheory.SimplicialObject.δ_comp_δ_self, CategoryTheory.ShortComplex.RightHomologyData.op_H, SSet.Truncated.liftOfStrictSegal.hσ'₀, AddCommGrpCat.coyonedaType_obj_map, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.coalgebraEquivalence_functor_obj_a, CategoryTheory.Square.opFunctor_obj, CategoryTheory.hoFunctor.preservesBinaryProducts, CategoryTheory.Limits.preservesFiniteProducts_op, CategoryTheory.instPreservesColimitsOfShapeSheafExtensiveTopologyFunctorOppositeSheafToPresheafOfPreservesFiniteProductsColim, CategoryTheory.Functor.PullbackObjObj.π_iso_of_iso_right_inv, CategoryTheory.Limits.Cone.equiv_hom_snd, CategoryTheory.preservesLimits_preadditiveYonedaObj, CategoryTheory.Limits.end_.hom_ext_iff, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_snd, CategoryTheory.Limits.isLimitOfCoconeRightOpOfCone_lift, PresheafOfModules.toPresheaf_map_app_apply, CategoryTheory.Functor.RepresentableBy.homEquiv_comp, CategoryTheory.Equivalence.symmEquivInverse_obj_counitIso_inv, CategoryTheory.HasDetector.hasCodetector_op, CategoryTheory.yonedaCommGrpGrp_obj, AlgebraicGeometry.Scheme.map_basicOpen, CategoryTheory.Limits.FormalCoproduct.cechFunctor_map_app, CategoryTheory.isFiltered_op_of_isCofiltered, AddCommGrpCat.coyonedaType_obj_obj_coe, CondensedSet.continuous_coinducingCoprod, CategoryTheory.Limits.IsIndObject.instIsClosedUnderIsomorphismsFunctorOppositeType, CategoryTheory.Sieve.sieveOfSubfunctor_apply, CategoryTheory.Limits.PullbackCone.unop_ι_app, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIso_inv_app_hom, CategoryTheory.Limits.reflectsFiniteLimits_op, CategoryTheory.Limits.instHasPullbackOppositeOpOfHasPushout, CategoryTheory.Square.unop_f₁₃, AlgebraicGeometry.Proj.zero_apply, SSet.Truncated.StrictSegal.spineToSimplex_vertex, CategoryTheory.GrothendieckTopology.Point.skyscraperPresheafHomEquiv_naturality_right, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_inv_naturality_assoc, PresheafOfModules.Derivation'.d_app, CategoryTheory.instFaithfulMonFunctorOppositeMonCatyonedaAddMon, PresheafOfModules.neg_app, CategoryTheory.CosimplicialObject.Augmented.leftOpRightOpIso_hom_right_app, CategoryTheory.Limits.FormalCoproduct.evalOp_map_app, CategoryTheory.Adjunction.op_unit, AlgebraicGeometry.PresheafedSpace.GlueData.componentwise_diagram_π_isIso, AlgebraicGeometry.isAffine_affineScheme, CategoryTheory.Functor.IsCoverDense.Types.presheafIso_inv_app, CategoryTheory.GrothendieckTopology.overMapPullbackComp_inv_app_hom_app, CategoryTheory.ShortComplex.opMap_τ₂, CategoryTheory.Pretriangulated.mem_distTriang_op_iff', CategoryTheory.cosimplicialSimplicialEquiv_unitIso_hom_app, CategoryTheory.Join.inclRightCompOpEquivInverse_hom_app_op, AlgebraicGeometry.exists_affineOpens_le_appLE_of_appLE, StalkSkyscraperPresheafAdjunctionAuxs.germ_fromStalk_assoc, CategoryTheory.Functor.RepresentableBy.isRepresentedBy, SSet.Truncated.HomotopyCategory.BinaryProduct.inverse_comp_functor, CategoryTheory.Functor.IsLeftAdjoint.leftOp, CategoryTheory.Functor.representable_preservesLimitsOfShape, AlgebraicGeometry.mono_pushoutSection_of_isCompact_of_flat_left_of_ringHomFlat, CategoryTheory.GrothendieckTopology.uliftYoneda_obj_obj_map_down, CategoryTheory.Limits.cospanOp_hom_app, CategoryTheory.Sheaf.isConstant_iff_isIso_counit_app', CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_w_apply, AlgebraicGeometry.Scheme.isoSpec_inv_naturality, CategoryTheory.Limits.hasColimit_of_hasLimit_rightOp, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.coalgebraEquivalence_inverse_map_hom, CategoryTheory.NatIso.op_associator, CategoryTheory.SheafOfTypes.balanced, AlgebraicGeometry.IsAffineOpen.toSpecΓ_fromSpec_assoc, SSet.stdSimplex.objMk₁_of_le_castSucc, CategoryTheory.coherentTopology.isSheaf_yoneda_obj, TopCat.Sheaf.objSupIsoProdEqLocus_inv_snd, AlgebraicTopology.DoldKan.P_f_naturality_assoc, AlgebraicGeometry.StructureSheaf.const_algebraMap, CategoryTheory.ObjectProperty.limitsOfShape_eq_unop_colimitsOfShape, SimplexCategory.toTopHomeo_naturality, CategoryTheory.preservesFiniteLimits_presheafToSheaf, CategoryTheory.Limits.coend.ι_desc, AlgebraicGeometry.Scheme.Hom.id_appIso, CategoryTheory.op_hom_braiding, CategoryTheory.MorphismProperty.rightFractionRel_op_iff, CategoryTheory.GrothendieckTopology.diagram_map, CategoryTheory.Sieve.uliftFunctorInclusion_app, CategoryTheory.CategoryOfElements.costructuredArrowULiftYonedaEquivalence_counitIso, smoothSheafCommRing.ι_forgetStalk_inv_assoc, skyscraperPresheafCocone_ι_app, AlgebraicTopology.DoldKan.map_P, CategoryTheory.Equivalence.inverseFunctorObj'_hom_app, CategoryTheory.Limits.IndObjectPresentation.toCostructuredArrow_obj_hom, CategoryTheory.Presheaf.hasSeparator, AlgebraicGeometry.Scheme.Modules.pseudofunctor_mapId_hom_τr, CategoryTheory.GrothendieckTopology.pointBotPresheafFiberIso_inv, CategoryTheory.Functor.op_commShiftIso_inv_app, CategoryTheory.Arrow.augmentedCechNerve_left, AlgebraicGeometry.instQuasiSeparatedToSpecΓOfQuasiSeparatedSpaceCarrierCarrierCommRingCat, CategoryTheory.Comma.unopFunctorCompSnd_inv_app, PresheafOfModules.map_comp, AlgebraicGeometry.IsAffineOpen.basicOpenSectionsToAffine_isIso, CategoryTheory.cocones_obj_map_app, SSet.Subcomplex.image_eq_range, PresheafOfModules.instPreservesLimitsOfShapeFunctorOppositeAbToPresheaf, CategoryTheory.coyonedaPreservesLimitsOfShape, SSet.RelativeMorphism.comm, CategoryTheory.typeEquiv_counitIso_inv_app_hom_app, SSet.Truncated.StrictSegal.spine_δ_vertex_ge, AlgebraicGeometry.Scheme.IdealSheafData.subschemeFunctor_obj, CategoryTheory.ShiftedHom.opEquiv'_symm_apply, CompHausLike.LocallyConstantModule.functor_obj_obj_obj_isAddCommGroup_neg_apply, SheafOfModules.inr_freeSumIso_hom_assoc, AlgebraicGeometry.PresheafedSpace.map_id_c_app, SSet.StrictSegal.spineToSimplex_edge, TopCat.Presheaf.germ_exist, CategoryTheory.yonedaFunctor_preservesLimits, CategoryTheory.Idempotents.DoldKan.η_inv_app_f, AlgebraicGeometry.exists_smooth_of_formallySmooth_stalk, AlgebraicGeometry.Scheme.comp_appTop_assoc, CategoryTheory.GrothendieckTopology.W_whiskerLeft_iff, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.coalgebraEquivalence_counitIso_hom_app_f, RingCat.moduleCatRestrictScalarsPseudofunctor_map, CategoryTheory.typeEquiv_inverse_map, CategoryTheory.GrothendieckTopology.diagramPullback_app, AlgebraicGeometry.SheafedSpace.GlueData.ι_isoPresheafedSpace_inv, CategoryTheory.Over.opEquivOpUnder_counitIso, CategoryTheory.Limits.coconeOfConeRightOp_pt, CategoryTheory.Limits.FormalCoproduct.cechIsoAugmentedCechNerve_hom_right, CategoryTheory.Limits.widePullbackShapeOp_obj, TopCat.Presheaf.SheafConditionEqualizerProducts.res_π, CategoryTheory.SimplicialObject.Augmented.whiskering_map_app_left, SSet.PtSimplex.MulStruct.δ_map_of_lt, AlgebraicGeometry.Scheme.local_affine, TopCat.Presheaf.germ_res, CategoryTheory.Limits.coneRightOpOfCoconeEquiv_inverse_map, CategoryTheory.Limits.instPreservesColimitsOfShapeOppositeDiscrete, HomotopicalAlgebra.trivialCofibrations_op, CategoryTheory.instIsRegularMonoOppositeOpOfIsRegularEpi, CategoryTheory.yonedaAddMonObj_map, AlgebraicGeometry.PresheafedSpace.GlueData.f_invApp_f_app, AlgebraicGeometry.isIso_pushoutSection_of_iSup_eq, CategoryTheory.OverPresheafAux.counitAux_hom, SheafOfModules.pushforward_map_val, CategoryTheory.instPreservesFilteredColimitsOfSizeObjOppositeFunctorTypeCoyonedaOpOfIsFinitelyPresentable, CategoryTheory.Subfunctor.isSheaf_iff, AlgebraicGeometry.Scheme.IdealSheafData.ideal_top, CategoryTheory.Functor.leftOp_full, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.s_comp_σ, SSet.Truncated.StrictSegal.spineToSimplex_edge, CategoryTheory.MorphismProperty.op_inverseImage, CategoryTheory.CommSq.LiftStruct.opEquiv_symm_apply, CategoryTheory.sheafToPresheafCompYonedaCompWhiskeringLeftSheafToPresheaf_hom_app_app_hom, CategoryTheory.Pretriangulated.instCommShiftOppositeOpOpEquivalenceInt, TopCat.Presheaf.instIsLocalizationCarrierObjOppositeOpensCarrierCommRingCatObjLocalizationPresheaf, CategoryTheory.Functor.leftOp_additive, CategoryTheory.GrothendieckTopology.Point.instIsLeftAdjointSheafSheafFiber, CategoryTheory.GrothendieckTopology.overMapPullbackCongr_eq_eqToIso, SSet.Truncated.spine_arrow, AugmentedSimplexCategory.equivAugmentedSimplicialObject_counitIso_hom_app_left_app, CategoryTheory.CategoryOfElements.fromCostructuredArrow_map_coe, SSet.stdSimplex.instHasDimensionLEObjSimplexCategoryMk, AlgebraicGeometry.instIsAffineObjOppositeCommRingCatSchemeSpec, AlgebraicGeometry.Scheme.Hom.etale_appLE, CategoryTheory.Equivalence.leftOp_inverse_map, PresheafOfModules.evaluation_preservesColimitsOfSize, CategoryTheory.Functor.sheafPushforwardCocontinuousCompSheafToPresheafIso_hom, CompHausOpToFrame.faithful, CategoryTheory.Equivalence.inverseFunctorMapIso_symm_eq_isoInverseOfIsoFunctor, SSet.stdSimplex.monotone_apply, CategoryTheory.isoSheafify_hom, CategoryTheory.Sheaf.cartesianMonoidalCategoryFst_hom, CategoryTheory.Pretriangulated.Opposite.commShift_adjunction_op_int, SSet.horn.faceι_ι, CategoryTheory.Square.opFunctor_map_τ₂, SSet.StrictSegal.isPointwiseRightKanExtensionAt.fac_aux₃, CategoryTheory.MonoidalCategory.MonoidalRightAction.oppositeRightAction_actionRight_op, AlgebraicTopology.DoldKan.degeneracy_comp_PInfty_assoc, CategoryTheory.extensiveTopology.presheafIsLocallySurjective_iff, AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalHom, SheafOfModules.pushforwardComp_hom_app_val_app, CategoryTheory.Limits.reflectsColimit_leftOp, CategoryTheory.RetractArrow.unop_r_right, SSet.Truncated.liftOfStrictSegal.hδ'₁, AlgebraicGeometry.Scheme.isoSpec_Spec, CategoryTheory.Pretriangulated.TriangleOpEquivalence.unitIso_inv_app, AlgebraicTopology.DoldKan.toKaroubiCompN₂IsoN₁_inv_app, CategoryTheory.unop_leftUnitor, CategoryTheory.GrothendieckTopology.Point.instIsIsoMapFunctorOppositePresheafFiberToSheafify, AlgebraicGeometry.Scheme.Hom.appLE_congr, CategoryTheory.Functor.PullbackObjObj.mapArrowLeft_left, CategoryTheory.yonedaGrpObj_map, CategoryTheory.Functor.unop_map, CategoryTheory.eHomFunctor_obj_map, TopCat.Presheaf.Pushforward.id_hom_app, CategoryTheory.unop_braiding, SSet.S.equivElements_apply_fst, CategoryTheory.Limits.isLimitConeOfCoconeRightOp_lift, CategoryTheory.Presheaf.isLocallySurjective_comp, TopCat.Presheaf.pullback_obj_obj_ext_iff, SSet.Subcomplex.ofSimplex_le_iff, CategoryTheory.forgetEnrichmentOppositeEquivalence_counitIso, CategoryTheory.MorphismProperty.RightFraction.op_f, HomotopicalAlgebra.instIsStableUnderRetractsOppositeWeakEquivalences, CategoryTheory.cosimplicialToSimplicialAugmented_obj, CategoryTheory.GrothendieckTopology.Point.Hom.presheafFiber_id, CategoryTheory.Functor.RepresentableBy.homEquiv_unop_comp, CondensedMod.LocallyConstant.instFaithfulSheafCompHausCoherentTopologyTypeConstantSheaf, AlgebraicGeometry.Scheme.IdealSheafData.isLocalization_away, CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence.functor_map_left_right, SSet.N.le_iff_exists_mono, AlgebraicGeometry.Scheme.ofRestrict_toLRSHom_c_app, AlgebraicGeometry.Scheme.LocalRepresentability.glueData_V, CategoryTheory.yonedaGrpObjIsoOfRepresentableBy_hom, CategoryTheory.Pseudofunctor.DescentData'.isoMk_hom_hom, CategoryTheory.Limits.hasLimit_op_of_hasColimit, SheafOfModules.unit_val, CategoryTheory.OverPresheafAux.unitBackward_unitForward, CategoryTheory.MonoidalCategory.MonoidalRightAction.rightActionOfOppositeRightAction_actionHom_unop, AugmentedSimplexCategory.equivAugmentedSimplicialObject_functor_obj_hom_app, CategoryTheory.regularTopology.equalizerCondition_iff_isIso_lift, CategoryTheory.Adjunction.compYonedaIso_inv_app_app, PartialOrder.mem_nerve_nonDegenerate_iff_strictMono, CategoryTheory.GrothendieckTopology.preservesSheafification_iff_of_adjunctions_of_hasSheafCompose, CategoryTheory.yonedaGrpObjIsoOfRepresentableBy_inv, CategoryTheory.Comon.Comon_EquivMon_OpOp_unitIso, PresheafOfModules.evaluation_preservesFiniteLimits, LightCondSet.topCatAdjunctionUnit_hom_app_apply, AlgebraicGeometry.IsClosedImmersion.hasAffineProperty, SheafOfModules.Presentation.map_relations_I, SSet.N.mk'_surjective, CategoryTheory.SimplicialObject.Augmented.drop_obj, CategoryTheory.Join.opEquiv_functor_obj_op_left, SSet.Subcomplex.image_ofSimplex, CategoryTheory.Functor.IsRightAdjoint.op, PresheafOfModules.instPreservesLimitsOfSizeFunctorOppositeAbToPresheaf, CategoryTheory.Equivalence.sheafCongrPreregular_functor_map_hom_app, CategoryTheory.Equalizer.Presieve.Arrows.sheaf_condition, CategoryTheory.Functor.hom_obj, CategoryTheory.Sheaf.adjunction_unit_app_val, AlgebraicGeometry.Scheme.preimage_opensRange_toSpecΓ, SSet.Truncated.HomotopyCategory.homToNerveMk_app_one, AlgebraicGeometry.Proj.fromOfGlobalSections_toSpecZero, CategoryTheory.Coyoneda.colimitCoconeIsColimit_desc, SSet.stdSimplex.δ_one_toSSetObjI, TopCat.Presheaf.app_injective_of_stalkFunctor_map_injective, CategoryTheory.Join.opEquiv_inverse_obj_left_op, CategoryTheory.SimplicialObject.equivalenceRightToLeft_right, CategoryTheory.Limits.SequentialProduct.cone_π_app_comp_Pi_π_pos, CategoryTheory.Functor.PullbackObjObj.mapArrowLeft_comp_assoc, CategoryTheory.Functor.opInv_map, CategoryTheory.Limits.isIndObject_yoneda, AlgebraicGeometry.Scheme.Spec.algebraMap_residueFieldIso_inv, SSet.prodStdSimplex.objEquiv_map_apply, SheafOfModules.instIsLeftAdjointOverOverRingCatPushforwardIdSheafOver, SSet.Subcomplex.le_iff_of_hasDimensionLT, CategoryTheory.ShortComplex.SnakeInput.op_L₃, topCatOpToFrm_obj_coe, CategoryTheory.Limits.hasColimit_op_iff_hasLimit, AlgebraicGeometry.IsAffineOpen.exists_basicOpen_le, CategoryTheory.Limits.PreservesFiniteLimitsOfIsFilteredCostructuredArrowYonedaAux.flipFunctorToInterchange_inv_app_app, SSet.stdSimplex.objEquiv_symm_mem_nonDegenerate_iff_mono, SSet.Truncated.Edge.mk'_edge, SimplicialObject.Splitting.cofan_inj_comp_app, CategoryTheory.Limits.coneUnopOfCoconeEquiv_functor_obj, CategoryTheory.ParametrizedAdjunction.homEquiv_naturality_three, CategoryTheory.Presheaf.instIsLeftKanExtensionOppositeObjFunctorTypeYonedaYonedaMap, AlgebraicGeometry.Scheme.eq_zeroLocus_of_isClosed_of_isAffine, CategoryTheory.Limits.limitOpIsoOpColimit_inv_comp_π_assoc, LightProfinite.Extend.functorOp_map, CategoryTheory.Presieve.FamilyOfElements.map_apply, PresheafOfModules.sub_app, TopCat.Sheaf.interUnionPullbackConeLift_left, AlgebraicTopology.NormalizedMooreComplex.d_squared, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_hom_naturality_assoc, CategoryTheory.NatIso.unop_leftUnitor, CategoryTheory.Pretriangulated.Opposite.functor_isTriangulated_op, CategoryTheory.Functor.const.opObjOp_hom_app, HomotopicalAlgebra.cofibrations_eq_unop, AlgebraicGeometry.IsAffineOpen.comap_primeIdealOf_appLE, CategoryTheory.Sieve.yonedaFamily_fromCocone_compatible, CategoryTheory.NatIso.unop_associator, CategoryTheory.Limits.coneOfCoconeRightOp_pt, instIsRightAdjointSSetTopCatToSSet, TopCat.subpresheafToTypes_map_coe, CategoryTheory.MorphismProperty.RightFraction.op_map, CategoryTheory.GrothendieckTopology.W.whiskerLeft, PresheafOfModules.colimitPresheafOfModules_map, CategoryTheory.ShortComplex.opcyclesOpIso_inv_naturality, AlgebraicGeometry.Scheme.Hom.appLE_appIso_inv_assoc, AlgebraicGeometry.Scheme.toSpecΓ_image_zeroLocus, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.s₀_comp_δ₁, CategoryTheory.GrothendieckTopology.uliftYoneda_obj_obj_obj, CategoryTheory.toSheafification_app, SSet.nonDegenerate_iff_of_isIso, CategoryTheory.Pretriangulated.Opposite.OpOpCommShift.iso_inv_app_assoc, AlgebraicGeometry.LocallyRingedSpace.evaluation_naturality, CategoryTheory.SimplicialObject.cechNerve_obj, CategoryTheory.MorphismProperty.IsInvertedBy.op, Alexandrov.lowerCone_π_app, CategoryTheory.SimplicialObject.δ_comp_σ_of_le, CategoryTheory.OverPresheafAux.unitAuxAux_inv_app_fst, AlgebraicTopology.DoldKan.Γ₀.map_app, CategoryTheory.Limits.IndObjectPresentation.ofCocone_I, AlgebraicGeometry.PresheafedSpace.stalkMap_germ_apply, CategoryTheory.GrothendieckTopology.Plus.exists_of_sep, instPreservesColimitSheafTypeYonedaOfPreservesLimitOppositeOpObjFunctorIsSheaf, CategoryTheory.Limits.ι_comp_colimitRightOpIsoUnopLimit_hom, CategoryTheory.Comma.opFunctorCompSnd_hom_app, CategoryTheory.Limits.reflectsColimit_of_leftOp, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_map_d, CategoryTheory.op_hom_associator, CategoryTheory.Presheaf.isSheaf_iff_preservesFiniteProducts_of_projective, topToLocale_map, CategoryTheory.opOpEquivalence_unitIso, instPreservesColimitsOfShapeSheafTypeYonedaOfPreservesLimitsOfShapeOppositeObjFunctorIsSheaf, CategoryTheory.Limits.hasProducts_opposite, CategoryTheory.WithInitial.opEquiv_counitIso_inv_app, CategoryTheory.GrothendieckTopology.OneHypercoverFamily.IsSheafIff.fac', CategoryTheory.Pretriangulated.TriangleOpEquivalence.functor_obj, CategoryTheory.Equivalence.sheafCongrPreregular_counitIso_inv_app_hom_app, CategoryTheory.OverPresheafAux.unitAuxAux_inv_app_snd_coe, CategoryTheory.Limits.widePushoutShapeOpEquiv_counitIso, CategoryTheory.hoFunctor.isIso_prodComparison_stdSimplex, CategoryTheory.Limits.instHasLimitDiscreteOppositeCompInverseOppositeOpFunctor, CategoryTheory.NatTrans.unop_id, AlgebraicGeometry.toSpecΓ_SpecMap_ΓSpecIso_inv_assoc, CategoryTheory.ParametrizedAdjunction.homEquiv_symm_naturality_two_assoc, CategoryTheory.Equalizer.Presieve.w, CategoryTheory.Limits.coneLeftOpOfCocone_π_app, AlgebraicTopology.DoldKan.natTransP_app, AlgebraicGeometry.Scheme.Modules.pseudofunctor_obj_obj, AlgebraicGeometry.Scheme.homOfLE_app, AlgebraicGeometry.SheafedSpace.Γ_obj_op, CategoryTheory.HasSeparator.hasCoseparator_op, PresheafOfModules.Elements.fromFreeYoneda_app_apply, CategoryTheory.Equivalence.congrLeftFunctor_map, CategoryTheory.Equalizer.Presieve.Arrows.compatible_iff_of_small, SSet.Truncated.comp_app_assoc, CategoryTheory.Coyoneda.colimitCocone_pt, SSet.Subcomplex.homOfLE_comp, CategoryTheory.unop_epi_iff, CategoryTheory.Subfunctor.Subpresheaf.IsGeneratedBy.image, SSet.Truncated.Edge.map_tensorHom, CategoryTheory.Subfunctor.sheafify_isSheaf, PresheafOfModules.pushforward_obj_obj, AlgebraicGeometry.Scheme.kerFunctor_map, CategoryTheory.Limits.FormalCoproduct.cech_map, CategoryTheory.Limits.walkingParallelPairOpEquiv_unitIso_inv_app_one, CategoryTheory.Limits.coneUnopOfCoconeEquiv_inverse_obj, CategoryTheory.Join.inclLeftCompOpEquivInverse_inv_app_op, CategoryTheory.Limits.reflectsColimit_rightOp, SSet.Truncated.StrictSegal.spineToSimplex_arrow, AlgebraicGeometry.Scheme.Modules.instIsIsoAbApp, AlgebraicGeometry.Scheme.Hom.naturality_assoc, CategoryTheory.Limits.hasLimit_rightOp_of_hasColimit, SSet.horn.edge₃_coe_down, SheafOfModules.Presentation.mapRelations_mapGenerators, AlgebraicGeometry.IsAffineOpen.fromSpec_app_self_apply, CategoryTheory.op_mono_iff, HomologicalComplex.cyclesOpIso_hom_naturality, AlgebraicGeometry.PresheafedSpace.colimit_presheaf, TopCat.Presheaf.pushforward_map_app', AlgebraicTopology.DoldKan.N₁_obj_X, CategoryTheory.shrinkYonedaGrp_obj_map_shrinkYonedaGrpObjObjEquiv_symm, AlgebraicGeometry.Scheme.Hom.iInf_ker_openCover_map_comp_apply, SSet.Truncated.rightExtensionInclusion_left, CategoryTheory.op_hom_rightUnitor, CategoryTheory.Pseudofunctor.IsStack.essSurj_of_sieve, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.sheafCondition_iff_bijective_toPullbackObj, SimplicialObject.Splitting.σ_comp_πSummand_id_eq_zero_assoc, CategoryTheory.Functor.RepresentableBy.uniqueUpToIso_hom, CategoryTheory.GrothendieckTopology.OneHypercoverFamily.IsSheafIff.fac'_assoc, AlgebraicGeometry.eq_zero_of_basicOpen_eq_bot, CategoryTheory.Functor.PreOneHypercoverDenseData.multicospanMapIso_inv, AlgebraicGeometry.Scheme.homOfLE_appLE, AlgebraicTopology.map_alternatingFaceMapComplex, CategoryTheory.SimplicialObject.δ_comp_σ_self, preservesFilteredColimits_coyoneda, CategoryTheory.Adjunction.Triple.leftToRight_op, LightCondensed.hom_ext_iff, CategoryTheory.uliftYoneda_map_app, CategoryTheory.GrothendieckTopology.Point.toPresheafFiberMap_presheafFiberMapObjIso_hom, CategoryTheory.Yoneda.yoneda_full, CategoryTheory.sheafificationIso_inv_hom, CompHausLike.LocallyConstantModule.functorToPresheaves_obj_obj_carrier, CategoryTheory.Presieve.isSheafFor_over_map_op_comp_ofArrows_iff, CategoryTheory.Functor.map_opShiftFunctorEquivalence_counitIso_hom_app_unop, CategoryTheory.Iso.unop_inv, CategoryTheory.SimplicialThickening.SimplicialCategory.assoc, SSet.stdSimplex.const_down_toOrderHom, CategoryTheory.Functor.uliftYonedaReprXIso_hom_app, TopCat.Presheaf.pushforward_obj_map, CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence_unitIso, CategoryTheory.Limits.opProdIsoCoprod_hom_fst, SSet.Subcomplex.topIso_inv_ι_assoc, CategoryTheory.NatIso.op_leftUnitor, AlgebraicGeometry.Scheme.inv_appTop, CategoryTheory.ComposableArrows.opEquivalence_functor_obj_obj, CategoryTheory.Limits.FormalCoproduct.cochainComplexFunctor_obj_X, CategoryTheory.Limits.reflectsFiniteCoproducts_op, AlgebraicGeometry.isLocalization_away_of_isAffine, CategoryTheory.Presieve.isSheafFor_over_map_op_comp_iff, CategoryTheory.Adjunction.leftOp_counit, CategoryTheory.Hom.addEquivCongrRight_apply, AlgebraicGeometry.Scheme.Hom.isIso_app, CategoryTheory.equivYoneda_hom_app, AlgebraicGeometry.Scheme.GlueData.sheafValGluedMk_val, CategoryTheory.Equivalence.sheafCongr_counitIso, CategoryTheory.Presheaf.instIsCardinalPresentableFunctorOppositeTypeObjUliftYonedaOfHasColimitsOfSize, CategoryTheory.Functor.mapTriangleOpCompTriangleOpEquivalenceFunctorApp_hom_hom₃, AlgebraicGeometry.Scheme.basicOpen_appLE, TopCat.Presheaf.presheafEquivOfIso_unitIso_inv_app_app, CategoryTheory.Limits.coend.map_comp, PresheafOfModules.instFaithfulSheafOfModulesCompObjFunctorOppositeRingCatIsSheafForgetRestrictScalars, SSet.horn₂₁.isPushout, CategoryTheory.Limits.isLimitOfCoconeOfConeUnop_lift, CategoryTheory.Presieve.FamilyOfElements.IsAmalgamation.compPresheafMap, CategoryTheory.GrothendieckTopology.uliftYonedaIsoYoneda_hom_app_hom_app, SheafOfModules.relationsOfIsCokernelFree_I, CategoryTheory.Limits.walkingCospanOpEquiv_functor_map, CategoryTheory.Injective.instOppositeOpOfProjective, CategoryTheory.SimplicialObject.δ_comp_σ_of_gt_assoc, CategoryTheory.NatTrans.coequifibered_op_iff, AlgebraicGeometry.instIsRightAdjointCommRingCatOppositeLocallyRingedSpaceΓ, CategoryTheory.Equivalence.symmEquiv_counitIso, CategoryTheory.Functor.map_opShiftFunctorEquivalence_counitIso_inv_app_unop_assoc, CategoryTheory.Presieve.FamilyOfElements.compPresheafMap_comp, AlgebraicGeometry.Scheme.LocalRepresentability.instIsIsoSheafZariskiTopologyTypeYonedaGluedToSheaf, AlgebraicGeometry.Scheme.LocalRepresentability.glueData_f, CategoryTheory.Limits.pushoutIsoOpPullback_inr_hom, CategoryTheory.Coyoneda.isIso, CategoryTheory.Functor.const.opObjOp_inv_app, AlgebraicGeometry.structureSheafInType.add_apply, CategoryTheory.Presheaf.isLocallyInjective_comp_iff, CategoryTheory.Functor.CorepresentableBy.equivUliftCoyonedaIso_symm_apply_homEquiv, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.s_comp_δ_assoc, CategoryTheory.Projective.projective_iff_preservesEpimorphisms_preadditiveCoyoneda_obj, AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex_comp_inclusionOfMooreComplexMap, CategoryTheory.Limits.has_cofiltered_limits_op_of_has_filtered_colimits, CategoryTheory.ObjectProperty.IsDetecting.isIso_iff_of_mono, SSet.Subcomplex.fromPreimage_app_coe, CategoryTheory.FunctorToTypes.functorHomEquiv_apply_app, AlgebraicTopology.DoldKan.MorphComponents.preComp_φ, SSet.horn_eq_iSup, AlgebraicGeometry.morphismRestrict_appTop, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_δ, CategoryTheory.Functor.sheafPushforwardContinuous_obj_obj_map, AlgebraicGeometry.Scheme.IdealSheafData.ofIdealTop_ideal, AlgebraicTopology.DoldKan.HigherFacesVanish.comp_P_eq_self_assoc, SSet.Subcomplex.homOfLE_app_val, CategoryTheory.MonoidalCategory.MonoidalLeftAction.leftActionOfOppositeLeftAction_actionAssocIso_unop, CategoryTheory.ShortComplex.LeftHomologyData.op_p, CategoryTheory.Limits.coyonedaOpColimitIsoLimitCoyoneda_inv_comp_π, SSet.Subcomplex.unionProd.ι₂_ι_assoc, CategoryTheory.Limits.piConst_obj_obj, AlgebraicTopology.DoldKan.Γ₀_obj_obj, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_hom_naturality, instIsCommMonObjOppositeCommAlgCatXUnopMonObjCommBialgCatFunctorCommBialgCatEquivComonCommAlgCatOfIsCocommCarrier, LightCondensed.underlying_obj, CategoryTheory.Presieve.IsSheafFor.functorInclusion_comp_extend, CategoryTheory.Square.op_X₂, CategoryTheory.Equivalence.op_functor, CategoryTheory.cosimplicialSimplicialEquiv_unitIso_inv_app, AlgebraicTopology.DoldKan.Q_f_naturality_assoc, AlgebraicGeometry.Scheme.LocalRepresentability.glueData_t', CategoryTheory.Limits.preservesColimitsOfSize_leftOp, TopCat.Presheaf.stalkPushforward_germ, SSet.Truncated.spine_vertex, CategoryTheory.instFinallySmallOppositeOfInitiallySmall, CategoryTheory.yonedaMap_app_apply, CategoryTheory.Functor.toSheafify_pullbackSheafificationCompatibility, CategoryTheory.Limits.BinaryCofan.op_mk, CategoryTheory.Sheaf.Ω_obj, CategoryTheory.Sheaf.ΓHomEquiv_naturality_right_symm, AlgebraicGeometry.SheafedSpace.comp_hom_c_app', SSet.yonedaEquiv_const, CategoryTheory.Pseudofunctor.toDescentDataAsCoalgebraCompCoalgebraEquivalenceFunctorIso_hom_app_f, CategoryTheory.Functor.IsCoverDense.Types.naturality_apply, SSet.stdSimplex.obj₀Equiv_symm_mem_face_iff, SimplicialObject.opFunctorCompOpFunctorIso_hom_app_app, SSet.prodStdSimplex.nonDegenerateEquiv₁_fst, CategoryTheory.GrothendieckTopology.preservesLimitsOfSize_yoneda, SSet.range_eq_iSup_sigma_ι, SSet.Subcomplex.BicartSq.isPushout, CategoryTheory.forgetEnrichmentOppositeEquivalence_inverse, CategoryTheory.Arrow.augmentedCechNerve_right, AlgebraicGeometry.ΓSpecIso_hom_stalkClosedPointIso_inv, germ_skyscraperPresheafStalkOfSpecializes_hom_assoc, CategoryTheory.NatTrans.rightOpWhiskerRight_assoc, SSet.Subcomplex.range_comp, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_apply, CategoryTheory.Limits.reflectsColimitsOfShape_op, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctor_obj_π_app, SSet.Subcomplex.PairingCore.nonDegenerate₂, CategoryTheory.Sheaf.coneΓ_π_app, CategoryTheory.SimplicialObject.Truncated.trunc_map_app, SSet.prodStdSimplex.instFiniteTensorObjObjSimplexCategoryStdSimplexMk, SSet.stdSimplex.objEquiv_symm_comp, CategoryTheory.SimplicialObject.Augmented.w₀_assoc, CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheafObj_condition, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_basicOpen, CategoryTheory.ShortComplex.HomologyData.op_left, TopCat.Presheaf.germ_stalkPullbackHom_assoc, TopCat.subsheafToTypes_obj, CategoryTheory.PresheafOfGroups.OneCochain.ev_precomp, CategoryTheory.Limits.FormalCoproduct.instPreservesLimitOppositeDiscreteFunctorCompOpObjFunctorEvalOp, TopCat.Presheaf.whiskerIsoMapGenerateCocone_hom_hom, CategoryTheory.SimplicialObject.cechNerve_map, AlgebraicGeometry.LocallyRingedSpace.Γ_map, CategoryTheory.GrothendieckTopology.Point.skyscraperPresheafHomEquiv_symm_apply, CategoryTheory.shrinkYoneda_obj_map_shrinkYonedaObjObjEquiv_symm, CategoryTheory.GrothendieckTopology.Cover.multicospanComp_hom_app, AlgebraicGeometry.Flat.flat_of_affine_subset, AlgebraicGeometry.isCompactOpen_iff_eq_basicOpen_union, CategoryTheory.instHasSubobjectClassifierFunctorOppositeTypeOfEssentiallySmall, TopCat.Presheaf.presheafEquivOfIso_functor_obj_obj, AlgebraicGeometry.PresheafedSpace.componentwiseDiagram_obj, AlgebraicGeometry.AffineSpace.functor_obj_obj, SSet.Subcomplex.eqToIso_inv, CategoryTheory.Sheaf.ΓObjEquivHom_naturality, CategoryTheory.SimplicialObject.whiskering_obj_obj_obj, AlgebraicGeometry.Proj.germ_map_sectionInBasicOpen, CategoryTheory.Functor.IsRepresentedBy.uliftYonedaIso_hom, CategoryTheory.Limits.end_.map_comp, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.s_comp_σ_assoc, CategoryTheory.GrothendieckTopology.isoToPlus_inv, CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinsetLimitCone_cone_pt, CategoryTheory.Sheaf.ΓRes_map, AlgebraicTopology.DoldKan.HigherFacesVanish.comp_P_eq_self, AlgebraicGeometry.Spec.algebraMap_stalkIso_inv, SimplicialObject.opEquivalence_unitIso, CategoryTheory.Functor.PreOneHypercoverDenseData.multicospanIndex_right, CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinset_map_app, SheafOfModules.pushforwardCongr_symm, CategoryTheory.Idempotents.DoldKan.Γ_obj_map, CategoryTheory.Functor.const.unop_functor_op_obj_map, CategoryTheory.kernelUnopUnop_hom, CategoryTheory.Functor.leftOpComp_inv_app, CategoryTheory.Functor.instIsCorepresentableCompObjOppositeTypeCoyonedaOpObjLeftAdjointObjIsDefined, CategoryTheory.MonObj.ofRepresentableBy_mul, CategoryTheory.Comma.opFunctorCompFst_hom_app, CategoryTheory.Sieve.shrinkFunctorUliftFunctorIso_inv_ι_assoc, CategoryTheory.GrothendieckTopology.Point.sheafFiberComapIso_hom_app, LightCondSet.hom_naturality_apply, CategoryTheory.GrothendieckTopology.Point.toPresheafFiberMap_naturality_apply, PresheafOfModules.instAdditiveModuleCatCarrierObjOppositeRingCatEvaluation, SSet.Truncated.mapHomotopyCategory_obj, CategoryTheory.ShortComplex.homologyOpIso_inv_naturality_assoc, CategoryTheory.monoidalOpOp_ε, AlgebraicGeometry.LocallyRingedSpace.stalkMap_germ_apply, CategoryTheory.regularTopology.equalizerCondition_w', CommAlgCat.toUnit_unop_hom, SSet.Subcomplex.prod_monotone, SSet.ι₀_comp_assoc, CommGrpCat.coyonedaForget_hom_app_app_hom, CategoryTheory.GrothendieckTopology.map_uliftYonedaEquiv', CategoryTheory.Limits.preservesFiniteProducts_rightOp, AlgebraicGeometry.HasAffineProperty.affineAnd_iff, CategoryTheory.Limits.limitCompYonedaIsoCocone_inv, PresheafOfModules.Hom.naturality_assoc, SSet.Subcomplex.instMonoι, CategoryTheory.GrothendieckTopology.Point.instIsSiftedOppositeElementsFiber, CategoryTheory.nerve_map, AlgebraicGeometry.IsOpenImmersion.app_eq_appIso_inv_app_of_comp_eq, CategoryTheory.Limits.Fork.op_ι_app, AlgebraicGeometry.AffineSpace.functor_map_app, CategoryTheory.hasCoseparator_op_iff, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.c_iso, AlgebraicGeometry.Scheme.IdealSheafData.ker_subschemeι_app, SheafOfModules.add_val, AlgebraicGeometry.IsOpenImmersion.ΓIso_inv, CategoryTheory.SimplicialObject.Augmented.const_obj_left, AlgebraicGeometry.Scheme.Modules.Hom.sub_app, PresheafOfModules.ofPresheaf_obj_carrier, CategoryTheory.RelCat.opEquivalence_counitIso, CategoryTheory.ShortComplex.rightHomologyMap_op, AlgebraicGeometry.Scheme.LocalRepresentability.yoneda_toGlued_yonedaGluedToSheaf, SSet.Truncated.Edge.map_edge, CategoryTheory.instFullGrpFunctorOppositeGrpCatYonedaGrp, AlgebraicGeometry.LocallyRingedSpace.restrict_presheaf_map, AlgebraicGeometry.AffineScheme.Spec_essSurj, CategoryTheory.coyonedaEquiv_coyoneda_map, CategoryTheory.Limits.reflectsColimits_op, CategoryTheory.CategoryOfElements.costructuredArrow_yoneda_equivalence_naturality, CategoryTheory.ShortComplex.Homotopy.op_h₃, CategoryTheory.Presheaf.imageSieve_eq_sieveOfSection, CategoryTheory.Functor.mapTriangleOpCompTriangleOpEquivalenceFunctorApp_inv_hom₂, AlgebraicGeometry.ΓSpec.adjunction_homEquiv_apply, CategoryTheory.Equivalence.rightOp_functor_obj, SSet.horn₃₂.ι₁_desc_assoc, CategoryTheory.Sheaf.cartesianMonoidalCategoryWhiskerRight_val, AlgebraicGeometry.Proj.add_apply, CategoryTheory.sheafifyMap_comp, AlgebraicGeometry.PresheafedSpace.Γ_obj, CategoryTheory.GrothendieckTopology.plusFunctor_map, CategoryTheory.Presieve.EffectiveEpimorphic.iff_forall_isSheafFor_yoneda, CategoryTheory.Limits.coneOfSectionCompYoneda_π, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctor_map_hom, CategoryTheory.Functor.instFaithfulOppositeTypeRestrictedULiftYonedaOfIsDense, CategoryTheory.shrinkYonedaObjObjEquiv_symm_comp, CategoryTheory.op_add, CategoryTheory.Comma.opFunctorCompFst_inv_app, CategoryTheory.Limits.isIndObject_iff_preservesFiniteLimits, CategoryTheory.Functor.IsRightAdjoint.leftOp, CategoryTheory.Limits.coend.ι_map, PresheafOfModules.sheafifyMap_val, CategoryTheory.isCodetector_op_iff, CategoryTheory.SimplicialObject.augmentOfIsTerminal_right, CategoryTheory.CostructuredArrow.toStructuredArrow_obj, CategoryTheory.GrothendieckTopology.sheafifyMap_comp, SSet.skeletonOfMono_zero, SheafOfModules.instFaithfulSheafAddCommGrpCatToSheaf, AlgebraicGeometry.Scheme.Hom.ι_toNormalization_assoc, CategoryTheory.ObjectProperty.IsConservativeFamilyOfPoints.jointlyFaithful, CategoryTheory.Functor.OneHypercoverDenseData.isSheaf_iff.fac_assoc, CategoryTheory.Pseudofunctor.toDescentData_obj, CategoryTheory.Presheaf.classifier_truth, CategoryTheory.Presheaf.IsSheaf.existsUnique_amalgamation_ofArrows, CommRingCat.equalizer_ι_isLocalHom', HomotopicalAlgebra.trivialFibrations_eq_unop, CategoryTheory.Presheaf.isStrongGenerator, CategoryTheory.Abelian.instAdditiveOppositeFunctorAddCommGrpCatExtFunctor, SSet.Truncated.Edge.id_edge, CategoryTheory.Limits.coconeUnopOfCone_pt, CategoryTheory.ParametrizedAdjunction.unit_whiskerRight_map_assoc, CategoryTheory.Presheaf.isPullback_χ_truth, CategoryTheory.SimplicialObject.δ_naturality_assoc, CategoryTheory.simplicialCosimplicialEquiv_inverse_obj, CategoryTheory.Limits.pullbackIsoOpPushout_hom_inl, CategoryTheory.Limits.Concrete.surjective_π_app_zero_of_surjective_map, SSet.ι₀_comp, CategoryTheory.ShortComplex.Homotopy.unop_h₁, CategoryTheory.Comma.unopFunctor_map, CategoryTheory.Limits.walkingCospanOpEquiv_functor_obj, CategoryTheory.Limits.coconeLeftOpOfConeEquiv_unitIso, TopCat.Presheaf.germ_stalkPullbackInv, CategoryTheory.Limits.IndObjectPresentation.ofCocone_isColimit, CategoryTheory.preservesHomology_preadditiveYonedaObj_of_injective, AlgebraicTopology.DoldKan.natTransPInfty_f_app, instPreservesFiniteProductsOppositeYonedaPresheafOfPreservesFiniteCoproductsTopCat, CategoryTheory.ShortComplex.quasiIso_opMap, CategoryTheory.Limits.hasLimits_op_of_hasColimits, CondensedSet.isDiscrete_tfae, CategoryTheory.Idempotents.DoldKan.Γ_obj_obj, CategoryTheory.Equivalence.precoherent_isSheaf_iff, SSet.horn₃₂.desc.multicofork_π_zero_assoc, CategoryTheory.linearCoyoneda_obj_map, CategoryTheory.Equivalence.symmEquiv_unitIso, CategoryTheory.ShortComplex.SnakeInput.op_v₂₃, AlgebraicGeometry.Scheme.Hom.fromNormalization_app_assoc, AlgebraicGeometry.Scheme.isoSpec_inv_toSpecΓ_assoc, CategoryTheory.simplicialCosimplicialEquiv_functor_map_app, AlgebraicGeometry.Proj.fromOfGlobalSections_morphismRestrict, CategoryTheory.Functor.leftOpRightOpEquiv_counitIso_hom_app_app, SSet.Homotopy.h₀_assoc, CompHausLike.LocallyConstantModule.functor_obj_obj_obj_isModule_smul_apply, CategoryTheory.OverPresheafAux.restrictedYonedaObj_obj, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverse_obj_pt, CategoryTheory.OverPresheafAux.MakesOverArrow.app, ContinuousMap.piComparison_fac, CategoryTheory.Limits.isColimitOfConeLeftOpOfCocone_desc, CategoryTheory.Limits.walkingSpanOpEquiv_functor_map, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.invApp_app_apply, AlgebraicGeometry.Scheme.IdealSheafData.ideal_iInf, AlgebraicGeometry.stalkMap_toStalk_apply, AlgebraicGeometry.IsAffineOpen.primeIdealOf_eq_map_closedPoint, CategoryTheory.Limits.unop_zero, CategoryTheory.Limits.hasCoproductsOfShape_opposite, CategoryTheory.Square.unop_f₂₄, CategoryTheory.ShortComplex.SnakeInput.op_v₀₁, CategoryTheory.Adjunction.leftOp_eq, CategoryTheory.Presheaf.IsSheaf.amalgamate_map_assoc, CategoryTheory.CategoryOfElements.costructuredArrowYonedaEquivalence_unitIso, CategoryTheory.SimplicialObject.δ_comp_σ_succ, PresheafOfModules.comp_toPresheaf_map_sheafifyHomEquiv'_symm_hom, SSet.IsStrictSegal.segal, CategoryTheory.Limits.coneRightOpOfCocone_pt, CategoryTheory.ShortComplex.rightHomologyFunctorOpNatIso_hom_app, CategoryTheory.Equivalence.sheafCongr.counitIso_inv_app_hom_app, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIso_hom_app_hom, CategoryTheory.OverPresheafAux.OverArrows.app_val, CategoryTheory.SimplicialObject.Homotopy.ToChainHomotopy.hom_eq_zero, AlgebraicGeometry.Scheme.stalkMap_germ_apply, AlgebraicGeometry.Scheme.IdealSheafData.range_glueDataObjι, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.id_hom, CategoryTheory.Presieve.FamilyOfElements.isAmalgamation_singleton_iff, CategoryTheory.IsDetecting.isIso_iff_of_mono, CategoryTheory.Functor.opHom_obj, AlgebraicTopology.NormalizedMooreComplex.map_f, AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap_app_assoc, AlgebraicGeometry.Scheme.Hom.instIsIsoCommRingCatAppObjOpensOpensFunctor, CategoryTheory.yonedaCommGrpGrpObj_obj_coe, CategoryTheory.Limits.reflectsLimitsOfShape_of_leftOp, CategoryTheory.ShortComplex.opEquiv_inverse, CategoryTheory.Subfunctor.range_eq_ofSection, CategoryTheory.orderDualEquivalence_counitIso, CategoryTheory.linearYoneda_obj_additive, CategoryTheory.sheafBotEquivalence_inverse_obj_obj, CategoryTheory.Iso.unop_hom_inv_id_app_assoc, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_hom_app_assoc, CategoryTheory.CommSq.op, AlgebraicGeometry.Scheme.Modules.pushforwardId_hom_app_app, CategoryTheory.Limits.hasLimit_op_iff_hasColimit, CommAlgCat.one_op_of_unop_hom, SSet.opEquivalence_unitIso, AlgebraicGeometry.Scheme.Hom.germ_stalkMap, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.fromBiprod_biprodIsoProd_inv_apply, TopCat.Sheaf.IsFlasque.structured_arrows_elements_sheaf_chains_bounded, PresheafOfModules.restriction_app, CategoryTheory.ShiftedHom.opEquiv_symm_apply_comp, CategoryTheory.yonedaPairing_map, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_fst, CategoryTheory.Limits.preservesLimits_op, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIsoApp_hom_hom, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_inv_app, AlgebraicTopology.DoldKan.Γ₂N₂ToKaroubiIso_inv_app, AlgebraicGeometry.Scheme.toOpen_eq, CategoryTheory.NatTrans.Coequifibered.op, AlgebraicTopology.DoldKan.instMonoChainComplexNatInclusionOfMooreComplexMap, CategoryTheory.Limits.has_filtered_colimits_op_of_has_cofiltered_limits, PresheafOfModules.ι_fromFreeYonedaCoproduct_apply, CategoryTheory.GrothendieckTopology.toSheafify_comp_sheafifyCompIso_inv, CategoryTheory.Limits.FormalCoproduct.cechIsoCechNerveApp_inv_π, CategoryTheory.Limits.hasColimit_of_hasLimit_unop, CategoryTheory.Sheaf.instHasLimitsOfShape, SSet.Augmented.stdSimplex_obj_hom_app, AlgebraicGeometry.Spec.toSheafedSpace_map, CategoryTheory.ShortComplex.RightHomologyData.unop_K, CategoryTheory.Functor.IsCoverDense.iso_of_restrict_iso, CategoryTheory.GrothendieckTopology.overMapPullback_comp_id_assoc, CategoryTheory.Functor.rightOpComp_inv_app, CategoryTheory.Pseudofunctor.presheafHomObjHomEquiv_apply, TopCat.Presheaf.map_restrict, CategoryTheory.Adjunction.rightOp_unit, AlgebraicGeometry.Scheme.Modules.restrict_obj, CategoryTheory.SimplicialObject.whiskering_obj_obj_map, CategoryTheory.ObjectProperty.colimitsClosure_eq_unop_limitsClosure, CategoryTheory.Abelian.DoldKan.equivalence_inverse, CategoryTheory.Limits.coconeOpEquiv_inverse_obj, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.shortComplex_g, SSet.horn.faceι_ι_assoc, CategoryTheory.uliftYoneda_map_app_down, CategoryTheory.orderDualEquivalence_functor_obj, CategoryTheory.Comma.opEquiv_functor, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctorObj_pt, LightProfinite.instPreservesLimitsOfShapeOppositeNatForgetContinuousMapCarrierToTopAndTotallyDisconnectedSpaceSecondCountableTopology, SSet.Homotopy.h₁, CategoryTheory.Functor.opId_inv_app, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom_assoc, CategoryTheory.plusPlusSheaf_preservesZeroMorphisms, SSet.stdSimplex.δ_objMk₁_of_le, CategoryTheory.isoOpEquiv_symm_apply, SimplicialObject.Splitting.decomposition_id, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.isPushoutAddCommGrpFreeSheaf, SSet.Truncated.Path.arrow_src, CategoryTheory.Pseudofunctor.DescentData'.Hom.comm, AlgebraicGeometry.PresheafedSpace.GlueData.snd_invApp_t_app, CategoryTheory.coherentTopology.epi_π_app_zero_of_epi, SSet.spine_map_vertex, CategoryTheory.Limits.coneLeftOpOfCocone_pt, CategoryTheory.OverPresheafAux.restrictedYonedaObjMap₁_app, CategoryTheory.Sheaf.instIsLocallyInjectiveAppArrowPLocallySurjectiveLocallyInjectiveFunctorialLocallySurjectiveInjectiveFactorization, CategoryTheory.Limits.FormalCoproduct.evalOpCompInlIsoId_hom_app_app, SSet.prodStdSimplex.le_orderHomOfSimplex, PresheafOfModules.free_obj, CategoryTheory.Functor.IsCoverDense.presheafIso_inv, CategoryTheory.instPreservesLimitsOfShapeMonFunctorOppositeMonCatShrinkYonedaMon, CategoryTheory.Limits.widePullbackShapeUnop_obj, CategoryTheory.Functor.RepresentableBy.ext_iff, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.inv_naturality, CategoryTheory.Functor.const.opObjUnop_inv_app, CategoryTheory.Limits.walkingParallelPairOp_zero, CategoryTheory.flat_iff_lan_flat, SSet.RelativeMorphism.Homotopy.h₁_assoc, CategoryTheory.IndParallelPairPresentation.hg, CategoryTheory.ShortComplex.LeftHomologyData.unop_H, PresheafOfModules.sectionsMk_coe, CategoryTheory.Equivalence.rightOp_counitIso_hom_app, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_obj, CategoryTheory.Sheaf.isConstant_iff_mem_essImage, AlgebraicGeometry.Scheme.Modules.restrictAdjunction_unit_app_app, CategoryTheory.Sheaf.instPreservesFiniteLimitsFunctorOppositeSheafToPresheafOfHasFiniteLimits, CategoryTheory.yonedaPairingExt_iff, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_homEquiv_apply, CategoryTheory.sheafCompose_map_hom, CategoryTheory.op_sum, CategoryTheory.OverPresheafAux.app_unitForward, SSet.Subcomplex.prodIso_inv, AlgebraicGeometry.StructureSheaf.comapₗ_const, SSet.Truncated.liftOfStrictSegal_app_0, CategoryTheory.isCoseparator_iff_faithful_yoneda_obj, CategoryTheory.nerveFunctor.full, AlgebraicTopology.DoldKan.MorphComponents.postComp_b, CategoryTheory.instPresheafIsFiniteObjFunctorOppositeTypeYoneda, CategoryTheory.Limits.BinaryCofan.unop_mk, TopCat.Presheaf.restrictOpenCommRingCat_apply, CategoryTheory.GrothendieckTopology.plusCompIso_whiskerRight_assoc, CategoryTheory.unop_rightUnitor, CategoryTheory.Comma.opEquiv_unitIso, CategoryTheory.Pretriangulated.unop_distinguished, SSet.hasDimensionLT_iSup_iff, AlgebraicGeometry.Scheme.Hom.isIntegral_app, CategoryTheory.ObjectProperty.instIsClosedUnderLimitsOfShapeOppositeOpOfIsClosedUnderColimitsOfShape, CategoryTheory.GrothendieckTopology.yonedaEquiv_naturality, CategoryTheory.Presheaf.isLeftKanExtension_of_preservesColimits, CategoryTheory.ShortComplex.Splitting.op_s, SSet.Subcomplex.unionProd.preimage_β_inv, CategoryTheory.sheafificationAdjunction_unit_app, CategoryTheory.Limits.Fork.op_ι_app_one, CategoryTheory.Limits.reflectsColimitsOfShape_of_unop, CategoryTheory.GrothendieckTopology.Point.skyscraperPresheafFunctor_obj_map, CategoryTheory.CategoryOfElements.toCostructuredArrow_obj, CategoryTheory.OverPresheafAux.yonedaCollectionFunctor_map, CategoryTheory.preadditiveYonedaObj_obj_isAddCommGroup, CategoryTheory.Functor.IsDenseSubsite.mapPreimage_comp_map_assoc, CategoryTheory.Presheaf.uliftYonedaAdjunction_unit_app_app, CategoryTheory.ShiftedHom.opEquiv'_symm_add, AlgebraicGeometry.ΓSpecIso_inv_ΓSpec_adjunction_homEquiv, CategoryTheory.sheafificationIso_hom_hom, CategoryTheory.Functor.opUnopEquiv_unitIso, SSet.Truncated.HomotopyCategory.BinaryProduct.associativity'Iso_hom_app, SSet.Subcomplex.le_iff_contains_nonDegenerate, CategoryTheory.LocalizerMorphism.op_functor, CategoryTheory.Presheaf.imageSieve_cofanIsColimitDesc_shrinkYoneda_map, CategoryTheory.isCodetecting_op_iff, AlgebraicGeometry.RingedSpace.zeroLocus_empty_eq_univ, CategoryTheory.Functor.FullyFaithful.compYonedaCompWhiskeringLeftMaxRight_hom_app_app_down, CategoryTheory.unop_inv, AlgebraicGeometry.StructureSheaf.algebraMap_germ_assoc, TopCat.Presheaf.SubmonoidPresheaf.toLocalizationPresheaf_app, AlgebraicGeometry.StructureSheaf.res_apply, CategoryTheory.OverPresheafAux.YonedaCollection.mk_snd, CategoryTheory.Limits.hasColimit_leftOp_of_hasLimit, LightCondensed.equalizerCondition, AlgebraicGeometry.Scheme.LocalRepresentability.instIsLocallyInjectiveHomYonedaGluedToSheaf, CategoryTheory.Functor.mapPresheaf_map_c, CategoryTheory.lan_flat_of_flat, AlgebraicGeometry.Scheme.Modules.Hom.add_app, CategoryTheory.Presieve.Arrows.toCompatible_coe, SSet.stdSimplex.yonedaEquiv_map, CategoryTheory.LocalizerMorphism.nonempty_leftResolution_iff_op, CategoryTheory.Join.inclLeftCompOpEquivInverse_hom_app_op, CategoryTheory.Presheaf.coconeOfRepresentable_naturality, PresheafOfModules.Derivation.d_app, CategoryTheory.GrothendieckTopology.Point.presheafFiberMapCocone_ι_app, CategoryTheory.SimplicialObject.Homotopy.h_castSucc_comp_δ_succ_of_lt_assoc, CategoryTheory.instFaithfulSheafFunctorOppositeCompSheafComposeSheafToPresheaf, CompHausLike.LocallyConstantModule.functor_obj_obj_obj_isAddCommGroup_zsmul_apply, CategoryTheory.NatIso.op_refl, CategoryTheory.Comon.monoidal_tensorUnit_comon_comul, CategoryTheory.Limits.PullbackCone.op_inl, CategoryTheory.NatTrans.rightOp_app, SSet.RelativeMorphism.Homotopy.ofEq_h, CategoryTheory.Enriched.Functor.natTransEquiv_symm_whiskerRight_functorHom_app, CategoryTheory.Limits.instPreservesLimitsOfShapeOppositeOfIsGroupoid, CategoryTheory.Adjunction.compCoyonedaIso_hom_app_app, CategoryTheory.GrothendieckTopology.Point.Hom.sheafFiber_id, SSet.horn_obj_zero, AlgebraicTopology.DoldKan.instReflectsIsomorphismsSimplicialObjectKaroubiChainComplexNatN₁, CategoryTheory.Iso.op_refl, InfiniteGalois.toAlgEquivAux_eq_liftNormal, CategoryTheory.Functor.closedIhom_map_app, AlgebraicTopology.DoldKan.hσ'_eq, CategoryTheory.instIsIsoFunctorOppositeHomFullSubcategoryIsSheafAppSheafCounitSheafificationAdjunction, CategoryTheory.Sheaf.instHasExactLimitsOfShapeOfHasFiniteColimitsOfPreservesFiniteColimitsFunctorOppositeSheafToPresheaf, CategoryTheory.NatTrans.equifibered_unop_iff, CategoryTheory.MonoidalCategory.DayConvolutionUnit.leftUnitorCorepresentingIso_inv_app_app, CategoryTheory.Cat.opFunctorInvolutive_inv_app_toFunctor_obj, LightCondMod.hom_naturality_apply, CategoryTheory.SimplicialObject.eqToIso_refl, AlgebraicGeometry.toSpecΓ_SpecMap_ΓSpecIso_inv, PresheafOfModules.forgetToPresheafModuleCat_obj, CategoryTheory.instSmallCarrierObjOppositeGrpCatGrpFunctorYonedaGrp, SSet.mem_skeleton, CategoryTheory.GrothendieckTopology.map_yonedaULiftEquiv', AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv, SheafOfModules.unitToPushforwardObjUnit_val_app_apply, CategoryTheory.NatTrans.leftOpWhiskerRight, CategoryTheory.Limits.limitCompCoyonedaIsoCone_inv, CategoryTheory.Limits.Cofork.unop_π_app_zero, AlgebraicGeometry.instIsIsoModulesSpecOfCarrierFromTildeΓUnitOpensCarrierCarrierCommRingCatRingCatSheaf, AlgebraicTopology.DoldKan.MorphComponents.id_a, PresheafOfModules.forgetToPresheafModuleCatObj_obj, CategoryTheory.ShiftedHom.opEquiv'_zero_add_symm, CategoryTheory.Limits.coneOpEquiv_functor_map_hom, SheafOfModules.pullback_assoc, AlgebraicGeometry.Scheme.Hom.map_appLE', AlgebraicGeometry.LocallyRingedSpace.GlueData.ι_isoSheafedSpace_inv_assoc, CategoryTheory.Limits.pullbackIsoUnopPushout_hom_inl_assoc, CategoryTheory.OverPresheafAux.MakesOverArrow.of_yoneda_arrow, AlgebraicGeometry.RingedSpace.basicOpen_mul, CategoryTheory.GrothendieckTopology.Point.instHasExactColimitsOfShapeOppositeElementsFiberOfLocallySmallOfAB5OfSizeOfHasFiniteLimits, PresheafOfModules.Finite.toPresheaf_preservesFiniteColimits, CategoryTheory.Pseudofunctor.isPrestackFor_iff, CategoryTheory.Limits.preservesFiniteLimits_of_isFiltered_costructuredArrow_yoneda, SSet.horn.spineId_vertex_coe, AlgebraicGeometry.Scheme.Hom.inv_appTop, CategoryTheory.MonoidalCategory.MonoidalLeftAction.oppositeLeftAction_actionRight_op, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.coalgebraEquivalence_unitIso_hom_app_hom, CategoryTheory.ShiftedHom.opEquiv'_symm_op_opShiftFunctorEquivalence_counitIso_inv_app_op_shift, CategoryTheory.yonedaAddMon_naturality_assoc, CategoryTheory.Presieve.FamilyOfElements.Compatible.functorPullback, SSet.rightUnitor_hom_app_apply, SSet.iSup_skeletonOfMono, CategoryTheory.Presheaf.imageSieve_whisker_forget, CategoryTheory.ShortComplex.opFunctor_obj, CategoryTheory.full_preadditiveYoneda, CategoryTheory.Pretriangulated.Opposite.distinguished_cocone_triangle, AddCommMonCat.coyonedaForget_inv_app_app, CategoryTheory.SimplicialObject.instHasColimitsOfShape, AugmentedSimplexCategory.equivAugmentedSimplicialObjectFunctorCompToArrowIso_inv_app_right, CategoryTheory.Functor.map_opShiftFunctorEquivalence_unitIso_hom_app_unop_assoc, AlgebraicGeometry.Scheme.AffineZariskiSite.opensRange_relativeGluingData_map, CategoryTheory.GrothendieckTopology.uliftYonedaEquiv_apply, CategoryTheory.CompatiblePreserving.apply_map, CategoryTheory.instIsMonoidalOppositeOpOpEquivalence, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.app_invApp_assoc, instPreservesColimitSheafTypeYonedaOfPreservesLimitOppositeOpObjFunctorIsSheaf_1, SheafOfModules.forget_map, CategoryTheory.Square.IsPullback.op, localCohomology.ideal_powers_initial, CategoryTheory.Functor.IsRepresentedBy.map_bijective, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.c_iso', lightProfiniteToLightCondSetIsoTopCatToLightCondSet_hom_app_hom_app_apply, AlgebraicGeometry.Flat.flat_appLE, AlgebraicGeometry.isIntegralHom_iff, AlgebraicGeometry.SheafedSpace.GlueData.ιIsOpenImmersion, CategoryTheory.sectionsFunctorNatIsoCoyoneda_hom_app_app, CategoryTheory.GrothendieckTopology.uliftYonedaOpCompCoyoneda_hom_app_app_down, CategoryTheory.Limits.preservesFiniteColimits_of_unop, CategoryTheory.Limits.reflectsFiniteProducts_rightOp, CategoryTheory.Limits.reflectsLimitsOfSize_leftOp, CategoryTheory.Functor.PullbackObjObj.ofHasPullback_snd, CategoryTheory.Presheaf.isSheaf_functorEnrichedHom, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.s'_comp_ε_assoc, CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan.uliftYonedaEquiv_presheafHom_uliftYoneda_obj, CommGrpCat.coyoneda_obj_obj_coe, CategoryTheory.presheafToSheaf_additive, CategoryTheory.Functor.representableByUliftFunctorEquiv_apply_homEquiv, CategoryTheory.Functor.shift_map_op, CategoryTheory.Functor.Elements.shrinkYoneda_map_app_coconeπOpCompShrinkYonedaObj_ι_app, AlgebraicGeometry.RingedSpace.basicOpen_res_eq, LightCondensed.ihomPoints_symm_apply, AlgebraicGeometry.Spec_zeroLocus, AlgebraicGeometry.LocallyOfFinitePresentation.finitePresentation_appLE, CategoryTheory.Pseudofunctor.IsStackFor.isEquivalence, CategoryTheory.Functor.CorepresentableBy.equivUliftCoyonedaIso_apply, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.coalgebraEquivalence_inverse_obj_obj, CategoryTheory.nerve_obj, AlgebraicGeometry.Scheme.basicOpen_mul, LightCondensed.discrete_obj, CategoryTheory.Limits.ι_comp_colimitOpIsoOpLimit_hom, CategoryTheory.MonoidalCategory.MonoidalLeftAction.oppositeLeftAction_actionHom_op, CategoryTheory.Functor.Elements.shrinkYonedaCompWhiskeringLeftObjπCompColimIso_inv_app_apply, CategoryTheory.Presheaf.FamilyOfElementsOnObjects.IsCompatible.section_apply, SSet.Truncated.tensor_map_apply_fst, AlgebraicGeometry.Scheme.ΓSpecIso_inv_naturality, AlgebraicGeometry.Scheme.IdealSheafData.coe_support_eq_eq_iInter_zeroLocus, CategoryTheory.MonoidalCategory.DayConvolution.corepresentableBy_homEquiv_apply_app, CategoryTheory.Limits.reflectsLimitsOfShape_rightOp, PresheafOfModules.Monoidal.tensorHom_app, SSet.Truncated.HomotopyCategory.BinaryProduct.functor_obj, CategoryTheory.Pretriangulated.shiftFunctor_op_map, CategoryTheory.Limits.limitLeftOpIsoUnopColimit_inv_comp_π, CategoryTheory.SimplicialObject.Homotopy.congr_homologyMap_singularChainComplexFunctor, AlgebraicTopology.DoldKan.PInfty_f_naturality, AlgebraicGeometry.exists_of_res_eq_of_qcqs_of_top, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_map, SSet.horn.multicoequalizerDiagram, CategoryTheory.ShortComplex.RightHomologyData.op_f', CategoryTheory.RelCat.opEquivalence_inverse, CategoryTheory.Limits.hasTerminal_op_of_hasInitial, CategoryTheory.Limits.preservesLimit_of_leftOp, CategoryTheory.uliftYonedaEquiv_symm_comp_assoc, SheafOfModules.forgetPreservesLimitsOfSize, AlgebraicTopology.DoldKan.Γ₀.Obj.map_on_summand, CategoryTheory.Sieve.uliftFunctorInclusion_is_mono, SSet.horn.spineId_arrow_coe, AlgebraicGeometry.Scheme.AffineZariskiSite.PreservesLocalization.isLocallyDirected, CategoryTheory.SimplicialObject.equivalenceRightToLeft_left, AlgebraicGeometry.Scheme.Hom.finiteType_appLE, CategoryTheory.rightDualFunctor_obj, CategoryTheory.Functor.isRepresentable_comp_uliftFunctor_iff, SSet.skeletonOfMono_obj_eq_top, CategoryTheory.CategoryOfElements.costructuredArrowYonedaEquivalenceFunctorProj_hom_app, AlgebraicGeometry.exists_app_map_eq_map_of_isLimit, CategoryTheory.uliftCoyonedaEquiv_naturality, CategoryTheory.Presheaf.isSheaf_iff_isSheaf_comp, AlgebraicTopology.DoldKan.HigherFacesVanish.of_comp, CategoryTheory.Pseudofunctor.CoGrothendieck.instEssSurjαCategoryObjLocallyDiscreteOppositeCatMkOpFiberForgetInducedFunctor, CategoryTheory.Sheaf.Hom.epi_of_presheaf_epi, SimplicialObject.opFunctorCompOpFunctorIso_inv_app_app, CategoryTheory.instInitiallySmallOppositeOfFinallySmall, CategoryTheory.Limits.opProdIsoCoprod_hom_fst_assoc, CategoryTheory.Limits.coyonedaOpColimitIsoLimitCoyoneda'_inv_comp_π_assoc, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.invApp_app, CategoryTheory.op_inv_rightUnitor, CategoryTheory.Limits.end_.lift_π, CategoryTheory.OverPresheafAux.YonedaCollection.map₁_comp, CategoryTheory.Limits.hasPullback_unop_iff_hasPushout, CategoryTheory.Limits.isLimitConeLeftOpOfCocone_lift, CategoryTheory.Sheaf.isLocallyInjective_sheafToPresheaf_map_iff, PresheafOfModules.pullbackObjIsDefined_free_yoneda, RingCat.moduleCatRestrictScalarsPseudofunctor_obj, CategoryTheory.HasCoseparator.hasSeparator_op, CategoryTheory.Limits.reflectsLimitsOfShape_op, SheafOfModules.instFaithfulPresheafOfModulesObjFunctorOppositeRingCatIsSheafForget, CategoryTheory.cokernelUnopUnop_hom, CategoryTheory.sheafComposeIso_inv_fac_assoc, CondensedMod.LocallyConstant.instIsIsoCondensedSetMapForgetAppCondensedModuleCatCounitDiscreteUnderlyingAdjObjFunctor, AlgebraicGeometry.Scheme.zeroLocus_biInf_of_nonempty, SSet.iSup_range_eq_top_of_isColimit, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_add_unitIso_inv_app_eq, CategoryTheory.SimplicialObject.IsCoskeletal.isRightKanExtension, CategoryTheory.Functor.IsCoverDense.sheafCoyonedaHom_app, CondensedSet.LocallyConstant.instFaithfulCondensedTypeDiscrete, SSet.stdSimplex.objEquiv_toOrderHom_apply, CategoryTheory.Functor.sheafPullbackConstruction.preservesFiniteLimits, PresheafOfModules.toPresheaf_preservesColimitsOfShape, CategoryTheory.GrothendieckTopology.Point.instHasColimitsOfShapeOppositeElementsFiber, CategoryTheory.GrothendieckTopology.uliftYonedaEquiv_symm_naturality_left, AlgebraicGeometry.Scheme.Opens.germ_stalkIso_hom_assoc, CategoryTheory.DinatTrans.precompNatTrans_app, TopCat.Presheaf.isSheaf_on_punit_iff_isTerminal, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.s_comp_δ₀, AlgebraicGeometry.Scheme.SpecΓIdentity_inv_app, SSet.ι₁_fst, SSet.const_comp, instPreservesColimitSheafTypeUliftYonedaOfPreservesLimitOppositeOpObjFunctorIsSheaf, PresheafOfModules.instMonoModuleCatCarrierObjOppositeRingCatApp, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.const_s, AlgebraicGeometry.LocallyRingedSpace.toΓSpec_base, CategoryTheory.simplicialCosimplicialEquiv_unitIso_inv_app, CategoryTheory.opOpEquivalence_counitIso, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverse_map_hom, CategoryTheory.isoSheafify_inv, HomotopicalAlgebra.instCofibrationOppositeOpOfFibration, CategoryTheory.Limits.limit.homIso_hom, Profinite.Extend.functorOp_final, TopCat.Presheaf.isSheaf_iff_isSheaf_comp', AugmentedSimplexCategory.equivAugmentedSimplicialObject_unitIso_hom_app_app, AlgebraicGeometry.StructureSheaf.algebraMap_germ, CategoryTheory.Presieve.shrinkFunctor_ι_comp_eq_iff_isAmalgamation, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom, AlgebraicGeometry.preservesLimit_rightOp_Γ, SSet.horn₃₁.desc.multicofork_π_three_assoc, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_unit, PresheafOfModules.Derivation'.app_apply, CategoryTheory.uliftYonedaIsoYoneda_hom_app_app, CategoryTheory.IsCodetecting.isIso_iff_of_epi, CategoryTheory.RetractArrow.op_r_left, CategoryTheory.hasDetector_op_iff, AlgebraicGeometry.mono_pushoutSection_of_isCompact_of_flat_right, CategoryTheory.Functor.FullyFaithful.compUliftYonedaCompWhiskeringLeft_hom_app_app_down, SSet.Truncated.Edge.tensor_edge, AlgebraicGeometry.isLocallyNoetherian_iff_of_iSup_eq_top, CategoryTheory.Limits.isLimitConeOfCoconeLeftOp_lift, CategoryTheory.Pseudofunctor.DescentData'.descentDataEquivalence_unitIso, CategoryTheory.GrothendieckTopology.diagramFunctor_map, CategoryTheory.isSeparator_op_iff, SSet.Edge.map_edge, CategoryTheory.Sheaf.isConstant_iff_of_equivalence, CategoryTheory.Limits.coconeOpEquiv_unitIso, CategoryTheory.equivYoneda_inv_app, CategoryTheory.Sheaf.hasExactColimitsOfShape, instPreservesLimitsOfShapeSheafTypeUliftYoneda, CategoryTheory.uliftCoyonedaEquiv_symm_map_assoc, AlgebraicGeometry.StructureSheaf.instIsLocalizedModuleCarrierStalkAbPresheafOpensCarrierTopModuleStructurePresheafPrimeComplAsIdealToStalkₗ, CategoryTheory.Pretriangulated.TriangleOpEquivalence.inverse_obj, CategoryTheory.ObjectProperty.IsConservativeFamilyOfPoints.W_iff, SSet.Edge.CompStruct.map_simplex, CategoryTheory.Pretriangulated.commShiftIso_opOp_hom_app, CategoryTheory.presheafToSheafCompComposeAndSheafifyIso_inv_app, CategoryTheory.Sheaf.instHasFiniteCoproducts, CommMonCat.coyoneda_map_app, AugmentedSimplexCategory.equivAugmentedSimplicialObject_counitIso_inv_app_right, CategoryTheory.MonoidalCategory.MonoidalRightAction.rightActionOfOppositeRightAction_actionAssocIso, AlgebraicGeometry.basicOpen_eq_of_affine', AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex_homotopyInvHomId, CategoryTheory.Limits.preservesFiniteLimits_rightOp, CategoryTheory.unop_inv_braiding, CategoryTheory.ObjectProperty.isCoseparating_unop_iff, CategoryTheory.op_inv, SSet.Truncated.Path.map_vertex, AlgebraicGeometry.AffineSpace.isoOfIsAffine_hom_appTop, CategoryTheory.Pretriangulated.TriangleOpEquivalence.counitIso_hom_app_hom₂, CategoryTheory.Comma.unopFunctorCompFst_inv_app, CategoryTheory.Limits.pullbackIsoUnopPushout_inv_fst_assoc, CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheafMap_restriction, CategoryTheory.HasLiftingProperty.iff_op, CategoryTheory.Presieve.FamilyOfElements.comp_of_compatible, AlgebraicGeometry.Scheme.Opens.fromSpecStalkOfMem_toSpecΓ, SSet.PtSimplex.MulStruct.δ_castSucc_castSucc_map_assoc, CategoryTheory.GrothendieckTopology.Point.instPreservesColimitsOfShapeOppositeElementsFiberObjFunctorFlipCurriedTensor, SSet.ι₀_app_snd_apply, SSet.Truncated.cosk.faithful, CategoryTheory.Limits.IndObjectPresentation.extend_isColimit_desc_app, CategoryTheory.Limits.IndObjectPresentation.ofCocone_ι, CategoryTheory.Limits.opProdIsoCoprod_inv_inr, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.section_app_comp_hom_app_assoc, AlgebraicGeometry.Spec.coe_toTop_map_hom_apply_asIdeal, AlgebraicGeometry.Scheme.Opens.toScheme_presheaf_map, CategoryTheory.ShortComplex.leftHomologyFunctorOpNatIso_hom_app, CategoryTheory.GrothendieckTopology.Point.toPresheafFiberMap_w_assoc, AddCommMonCat.coyonedaType_obj_obj_coe, CategoryTheory.Subfunctor.Subpresheaf.image_isFinite, SSet.horn₃₁.desc.multicofork_π_zero, CategoryTheory.hasCardinalLT_arrow_op_iff, CategoryTheory.orderDualEquivalence_functor_map, AugmentedSimplexCategory.equivAugmentedSimplicialObject_unitIso_inv_app_app, CategoryTheory.Pseudofunctor.CoGrothendieck.ι_obj_base, AlgebraicGeometry.instIsDominantToSpecΓOfCompactSpaceCarrierCarrierCommRingCat, CategoryTheory.ObjectProperty.isCodetecting_unop_iff, CategoryTheory.ShortComplex.LeftHomologyMapData.unop_φH, CategoryTheory.GrothendieckTopology.Point.skyscraperPresheafAdjunction_homEquiv_apply, CategoryTheory.Functor.FullyFaithful.compUliftCoyonedaCompWhiskeringLeft_inv_app_app_down, LightCondMod.LocallyConstant.instFaithfulModuleCatSheafLightProfiniteCoherentTopologyConstantSheaf, LightCondensed.lanPresheafNatIso_hom_app, CategoryTheory.Preadditive.DoldKan.equivalence_inverse, CategoryTheory.Limits.widePullbackShapeOpEquiv_functor, SSet.mem_nonDegenerate_iff_notMem_degenerate, CategoryTheory.Limits.pushoutIsoOpPullback_inl_hom_assoc, CategoryTheory.Subfunctor.mem_ofSection_obj, CategoryTheory.GrothendieckTopology.instFaithfulSheafTypeYoneda, smoothSheafCommRing.ι_forgetStalk_hom_apply, CategoryTheory.Functor.IsDenseSubsite.mapPreimage_comp, TopCat.Sheaf.objSupIsoProdEqLocus_hom_snd, CategoryTheory.Sheaf.instIsLocallySurjectiveHomMapTypeSheafComposeForget, SSet.finite_subcomplex_top_iff, AlgebraicGeometry.stalkToFiberRingHom_germ, SSet.Truncated.StrictSegal.spine_δ_arrow_eq, CategoryTheory.Functor.IsDenseSubsite.sheafifyHomEquivOfIsEquivalence_naturality_right_assoc, AlgebraicGeometry.StructureSheaf.const_mul_cancel', AlgebraicGeometry.Scheme.isoSpec_Spec_inv, CategoryTheory.Functor.IsCoverDense.homOver_app, SSet.Truncated.spine_surjective, CategoryTheory.Groupoid.invEquivalence_functor_obj, AlgebraicGeometry.Scheme.IdealSheafData.supportSet_subset_zeroLocus, SSet.Subcomplex.preimage_max, CategoryTheory.MonoidalCategory.DayConvolution.associatorCorepresentingIso_inv_app_app, AlgebraicGeometry.instFaithfulOppositeCommRingCatLocallyRingedSpaceToLocallyRingedSpace, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_w_assoc, CategoryTheory.MonoidalCategory.MonoidalLeftAction.leftActionOfOppositeLeftAction_actionRight_unop, CategoryTheory.Functor.CorepresentableBy.uniqueUpToIso_hom, SSet.Subcomplex.mem_ofSimplex_obj, CategoryTheory.Limits.pushout.op_codiagonal, CategoryTheory.Limits.hasLimit_of_hasColimit_unop, CategoryTheory.OverPresheafAux.unitForward_unitBackward, CategoryTheory.ParametrizedAdjunction.inl_arrowHomEquiv_symm_apply_left_assoc, CategoryTheory.MonoidalCategory.MonoidalRightAction.rightActionOfOppositeRightAction_actionHomRight, CategoryTheory.FunctorToTypes.rightAdj_obj_map_app, CategoryTheory.imageUnopOp_hom_comp_image_ι, CategoryTheory.SimplicialObject.δ_comp_σ_of_gt', TopCat.Sheaf.objSupIsoProdEqLocus_inv_eq_iff, AlgebraicGeometry.IsFinite.finite_app, CategoryTheory.Limits.isColimitCoconeRightOpOfCone_desc, HomologicalComplex.instQuasiIsoOppositeMapSymmOpFunctorOp, CategoryTheory.Limits.limitUnopIsoUnopColimit_hom_comp_ι_assoc, SSet.Subcomplex.unionProd.preimage_β_hom, CategoryTheory.Limits.isColimitCoconeOfConeRightOp_desc, CategoryTheory.Limits.desc_op_comp_opCoproductIsoProduct_hom, TopCat.Sheaf.IsFlasque.of_shortExact_of_isFlasque₁₂, AlgebraicGeometry.instIsScalarTowerObjOppositeOpensCarrierTopObjFunctorTypeIsSheafGrothendieckTopologyStructureSheafInType, SSet.Subcomplex.N.mk_surjective, CategoryTheory.Sheaf.cohomologyFunctor_obj, CategoryTheory.Adjunction.Quadruple.op_adj₁, lightCondSetToTopCat_obj_carrier, SSet.Subcomplex.image_le_range, AlgebraicTopology.DoldKan.map_Q, CategoryTheory.ShortComplex.Exact.op, AlgebraicGeometry.instSubsingletonCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensOfIsEmpty, AlgebraicGeometry.exists_eq_pow_mul_of_isAffineOpen, CategoryTheory.Sheaf.mono_of_injective, CategoryTheory.GrothendieckTopology.instIsLocalizationFunctorOppositeSheafPresheafToSheafW, CategoryTheory.SimplicialObject.Truncated.whiskering_obj_obj_map, TopCat.Presheaf.IsFlasque.epi, SheafOfModules.Finite.forgetPreservesFiniteLimits, CategoryTheory.Pseudofunctor.toDescentData_map_hom, CategoryTheory.SimplicialObject.Truncated.cosk_reflective, CategoryTheory.NatIso.removeOp_inv, smoothSheafCommRing.ι_forgetStalk_hom_assoc, CategoryTheory.CommSq.LiftStruct.unop_l, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo_assoc, AlgebraicGeometry.Proj.sheafedSpaceMap_hom_c_app_hom_apply_coe, CategoryTheory.MorphismProperty.ContainsIdentities.op, AlgebraicGeometry.Proj.awayMap_awayToSection, CategoryTheory.Groupoid.invEquivalence_counitIso, SheafOfModules.Presentation.mapRelations_mapGenerators_assoc, CategoryTheory.HasLiftingProperty.iff_unop, AlgebraicGeometry.instPreservesColimitsOfShapeOppositeCommRingCatSchemeDiscretePEmptySpec, CategoryTheory.Sheaf.instIsClosedUnderLimitsOfShapeFunctorOppositeIsSheaf, CategoryTheory.GrothendieckTopology.uliftYonedaOpCompCoyoneda_app_app, AlgebraicGeometry.finite_appTop_of_universallyClosed, CategoryTheory.Square.opFunctor_map_τ₃, CategoryTheory.Presheaf.map_comp_uliftYonedaEquiv_down_assoc, SSet.Subcomplex.liftPath_arrow_coe, CategoryTheory.CategoryOfElements.costructuredArrowULiftYonedaEquivalence_inverse_obj, CategoryTheory.isCoseparating_unop_iff, AlgebraicGeometry.IsLocallyNoetherian.component_noetherian, CategoryTheory.GrothendieckTopology.Point.instPreservesFiniteLimitsFunctorOppositePresheafFiberOfLocallySmallOfHasFiniteLimitsOfAB5OfSize, SheafOfModules.forget_obj, CategoryTheory.Join.opEquiv_inverse_map_edge_op, CategoryTheory.Coyoneda.instHasColimitObjOppositeFunctorTypeCoyoneda, CategoryTheory.Presheaf.instIsCardinalLocallyPresentableFunctorOppositeOfHasPullbacks, CategoryTheory.Limits.Types.surjective_π_app_zero_of_surjective_map, CategoryTheory.Functor.IsCoverDense.Types.presheafIso_hom_app, CategoryTheory.Limits.coconeLeftOpOfCone_ι_app, CategoryTheory.MorphismProperty.RightFraction.unop_f, CategoryTheory.Limits.π_comp_colimitRightOpIsoUnopLimit_inv, CategoryTheory.Limits.ι_comp_colimitUnopIsoOpLimit_hom_assoc, CategoryTheory.GrothendieckTopology.preservesFiniteLimits_sheafification, CategoryTheory.isCoseparator_iff_faithful_preadditiveYonedaObj, CategoryTheory.yonedaGrpObj_obj_coe, CategoryTheory.LocalizerMorphism.RightResolution.unop_w, CategoryTheory.Equivalence.sheafCongrPrecoherent_functor_obj_obj_map, CategoryTheory.GrothendieckTopology.diagramFunctor_obj, AlgebraicTopology.DoldKan.N₂_obj_p_f, CategoryTheory.NatTrans.leftOpWhiskerRight_assoc, CategoryTheory.Equivalence.sheafCongrPrecoherent_inverse_obj_obj_obj, CategoryTheory.Functor.leftOpComp_hom_app, SSet.Truncated.HomotopyCategory.BinaryProduct.square, AlgebraicGeometry.IsIntegralHom.integral_app, CategoryTheory.Limits.preservesLimit_op, CondensedSet.LocallyConstant.instFullCondensedTypeDiscrete, CategoryTheory.Subfunctor.ofSection_image, CategoryTheory.Limits.preservesColimit_of_leftOp, CategoryTheory.Functor.RepresentableBy.uniqueUpToIso_inv, RingCat.moduleCatRestrictScalarsPseudofunctor_mapComp, AlgebraicGeometry.isIso_SpecMap_stakMap_localization, CategoryTheory.Limits.coend.hom_ext_iff, CategoryTheory.Functor.sheafPushforwardContinuousCompSheafToPresheafIso_hom_app_app, CategoryTheory.Functor.sieves_obj, CategoryTheory.MonoidalCategory.MonoidalRightAction.rightActionOfOppositeRightAction_actionHomLeft, CommMonCat.coyoneda_obj_obj_coe, CategoryTheory.Functor.instIsCorepresentableObjOppositeTypeCoyoneda, CommRingCat.coyonedaUnique_inv_app_hom_apply, AlgebraicGeometry.mono_pushoutSection_of_isCompact_of_flat_right_of_ringHomFlat, LightProfinite.Extend.cocone_pt, LightCondMod.epi_iff_locallySurjective_on_lightProfinite, CategoryTheory.Limits.coneLeftOpOfCoconeEquiv_functor_map_hom, CategoryTheory.GrothendieckTopology.HasSheafCompose.isSheaf, TopCat.Presheaf.SheafConditionEqualizerProducts.fork_pt, AlgebraicGeometry.PresheafedSpace.pushforwardDiagramToColimit_map, PresheafOfModules.instHasLimitModuleCatCarrierObjOppositeRingCatCompEvaluationRestrictScalarsHomMap, SSet.horn₂₀.isPushout, SSet.stdSimplex.ofSimplex_yonedaEquiv_δ, SheafOfModules.comp_val_assoc, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.invApp_app_assoc, CategoryTheory.NatTrans.Coequifibered.rightOp, CategoryTheory.Sieve.forallYonedaIsSheaf_iff_colimit, CategoryTheory.Equivalence.sheafCongrPrecoherent_inverse_obj_obj_map, CategoryTheory.Pseudofunctor.DescentData'.pullHom'_hom_self, CategoryTheory.Functor.PullbackObjObj.mapArrowRight_id, AlgebraicTopology.DoldKan.σ_comp_PInfty, CategoryTheory.Limits.preservesFiniteCoproducts_op, CategoryTheory.Sheaf.epi_of_isLocallySurjective, CategoryTheory.Limits.reflectsFiniteColimits_rightOp, TopCat.Presheaf.SheafConditionEqualizerProducts.w_apply, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_η, CategoryTheory.Pseudofunctor.DescentData.pullFunctorObjHom_eq_assoc, AlgebraicGeometry.specTargetImageFactorization_app_injective, LightCondensed.ihom_map_val_app, AlgebraicGeometry.Scheme.IsGermInjectiveAt.cond, CategoryTheory.sheafToPresheaf_η, TopCat.Presheaf.presheafEquivOfIso_counitIso_inv_app_app, AlgebraicGeometry.Scheme.IdealSheafData.zeroLocus_inter_subset_supportSet, CategoryTheory.Presheaf.isLocallySurjective_toPlus, CategoryTheory.Limits.coneLeftOpOfCoconeEquiv_functor_obj, CategoryTheory.Functor.PreOneHypercoverDenseData.multicospanIndex_fst, CategoryTheory.Limits.walkingParallelPairOpEquiv_unitIso_one, AlgebraicGeometry.PresheafedSpace.Γ_map, CategoryTheory.Limits.reflectsLimitsOfShape_of_rightOp, CategoryTheory.Limits.coconeRightOpOfCone_ι, AlgebraicTopology.NormalizedMooreComplex.obj_X, SSet.mem_skeletonOfMono_obj_iff_of_nonDegenerate, AlgebraicGeometry.ΓSpec.right_triangle, CategoryTheory.ObjectProperty.instEssentiallySmallOppositeOp, CategoryTheory.Subfunctor.range_isFinite, TopologicalSpace.Opens.op_map_comp_obj, AlgebraicGeometry.SheafedSpace.id_c_app, TopCat.Presheaf.app_isIso_of_stalkFunctor_map_iso, SSet.Truncated.Path₁.arrow_tgt, CategoryTheory.Limits.reflectsColimit_of_unop, CategoryTheory.Square.op_f₁₃, CategoryTheory.GrothendieckTopology.overMapPullbackCongr_inv_app_hom_app, CategoryTheory.Iso.unop_hom, AlgebraicGeometry.Scheme.IdealSheafData.map_ideal', CompHausLike.LocallyConstant.instIsIsoFunctorTypeUnitSheafCoherentTopologyAdjunction, CategoryTheory.Limits.coneLeftOpOfCoconeEquiv_unitIso, CategoryTheory.NatTrans.coequifibered_unop_iff, CategoryTheory.Functor.IsDenseSubsite.mapPreimage_of_eq, AlgebraicGeometry.StructureSheaf.toOpenₗ_top_bijective, CategoryTheory.Pseudofunctor.DescentData'.Hom.comm_assoc, CategoryTheory.MonoidalCategory.MonoidalLeftAction.leftActionOfOppositeLeftAction_actionObj_unop, AlgebraicGeometry.StructureSheaf.comap_id_eq_map, CategoryTheory.MonoidalClosed.internalHom_map, AlgebraicGeometry.Scheme.Hom.appIso_hom, HomotopicalAlgebra.fibrations_op, CategoryTheory.Limits.coneLeftOpOfCoconeEquiv_inverse_map, CategoryTheory.Limits.opProdIsoCoprod_hom_snd, CategoryTheory.Presieve.isSheafFor_arrows_iff_pullbacks, CategoryTheory.toSheafify_sheafifyLift, Condensed.isColimitLocallyConstantPresheafDiagram_desc_apply, SSet.S.equivElements_apply_snd, CategoryTheory.ShortComplex.shortExact_iff_unop, CategoryTheory.SimplicialObject.δ_comp_σ_self_assoc, CategoryTheory.ShiftedHom.opEquiv'_symm_comp, CategoryTheory.Functor.unopComp_inv_app, AlgebraicTopology.NormalizedMooreComplex.objX_add_one, AlgebraicTopology.DoldKan.N₂_obj_X_X, CategoryTheory.Limits.multicospanIndexEnd_left, SimplicialObject.Split.Hom.comm_assoc, CategoryTheory.TwoSquare.instGuitartExactOppositeOp, SSet.HasDimensionLT.degenerate_eq_top, CategoryTheory.Sieve.functorInclusion_top_isIso, SSet.Truncated.HomotopyCategory.BinaryProduct.inverseCompFunctorIso_hom_app, CategoryTheory.Idempotents.DoldKan.isoN₁_hom_app_f, CategoryTheory.Limits.PullbackCone.unop_pt, CategoryTheory.Subpresheaf.sheafify_isSheaf, CategoryTheory.simplicialToCosimplicialAugmented_obj, AlgebraicTopology.DoldKan.N₁Γ₀_hom_app, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom_apply, SheafOfModules.toSheaf_map_hom, CategoryTheory.MonoidalCategory.MonoidalLeftAction.oppositeLeftAction_actionHom, CategoryTheory.Functor.PullbackObjObj.π_snd_assoc, InfiniteGalois.finGaloisGroupFunctor_map_proj_eq_proj, HomologicalComplex.isSupported_op_iff, CategoryTheory.Presheaf.instIsLocallySurjectiveHomToRangeSheafify, SSet.prodStdSimplex.nonDegenerate_iff_injective_objEquiv, CategoryTheory.MonoidalClosed.internalHom_obj, CategoryTheory.Functor.opComp_hom_app, CategoryTheory.ShortComplex.unop_g, CategoryTheory.ObjectProperty.IsConservativeFamilyOfPoints.jointlyReflectIsomorphisms, CategoryTheory.Limits.coconeRightOpOfConeEquiv_counitIso, CategoryTheory.essImage_yonedaAddMon, AlgebraicGeometry.Scheme.IdealSheafData.ideal_sup, CategoryTheory.cokernelOpUnop_hom, CategoryTheory.Equivalence.symmEquiv_inverse, AlgebraicGeometry.Scheme.evaluation_naturality_assoc, AlgebraicGeometry.AffineSpace.isoOfIsAffine_hom, CategoryTheory.Equivalence.rightOp_unitIso_inv_app, CategoryTheory.Coyoneda.coyoneda_full, CategoryTheory.Limits.reflectsFiniteColimits_of_rightOp, TopCat.toSSet_map_const, SSet.Quasicategory.hornFilling, AlgebraicGeometry.tilde.toOpen_res, Alexandrov.projSup_obj, AlgebraicGeometry.tilde.toOpen_res_assoc, SSet.prodStdSimplex.nonDegenerate_max_dim_iff, CategoryTheory.Presheaf.IsSheaf.amalgamateOfArrows_map_assoc, AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex_comp_inclusionOfMooreComplexMap_assoc, CategoryTheory.GrothendieckTopology.plusFunctor_preservesZeroMorphisms, CategoryTheory.Limits.hasZeroObject_op, SSet.stdSimplex.instFiniteObjSimplexCategory, CompHausLike.isIsoSigmaComparison, CategoryTheory.GrothendieckTopology.sheafificationWhiskerLeftIso_hom_app, AlgebraicGeometry.Scheme.Hom.ι_fromNormalization_assoc, CategoryTheory.Abelian.IsGrothendieckAbelian.OppositeModuleEmbedding.faithful_embedding, SSet.Subcomplex.prod_le_top_prod, CategoryTheory.Limits.IndObjectPresentation.ofCocone_F, CategoryTheory.Functor.FullyFaithful.homNatIso'_inv_app_down, HomologicalComplex.acyclic_op_iff, SSet.stdSimplex.objMk_apply, CategoryTheory.NatIso.op_hom, CategoryTheory.GrothendieckTopology.whiskerRight_toPlus_comp_plusCompIso_hom_assoc, CategoryTheory.GrothendieckTopology.ofArrows_mem_iff_isLocallySurjective_cofanIsColimitDesc_shrinkYoneda_map, AlgebraicGeometry.Scheme.emptyTo_c_app, TopCat.presheafToType_obj, SSet.horn₃₂.desc.multicofork_π_three, CategoryTheory.ObjectProperty.instIsClosedUnderColimitsOfShapeOppositeOpOfIsClosedUnderLimitsOfShape_1, CategoryTheory.Functor.IsRepresentedBy.iff_isIso_uliftYonedaEquiv, CategoryTheory.Limits.coyonedaOpColimitIsoLimitCoyoneda_hom_comp_π, SSet.Truncated.HomotopyCategory.homToNerveMk_app_edge, CategoryTheory.Limits.widePushoutShapeUnop_map, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.comp_hom_assoc, CategoryTheory.kernel.ι_op, CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheafObj_hom_ext_iff, CategoryTheory.Join.InclLeftCompRightOpOpEquivFunctor_inv_app, CategoryTheory.GrothendieckTopology.diagramNatTrans_app, SSet.Subcomplex.PairingCore.notMem₁, CategoryTheory.Sheaf.instHasColimitsOfSize, isZero_Ext_succ_of_projective, CategoryTheory.Meq.mk_apply, AlgebraicGeometry.affineAnd_apply, AlgebraicTopology.DoldKan.MorphComponents.postComp_φ, CategoryTheory.SimplicialObject.Augmented.rightOp_hom_app, CategoryTheory.Limits.preservesColimitsOfShape_leftOp, CategoryTheory.yonedaEquiv_naturality, CategoryTheory.prodOpEquiv_inverse_obj, AlgebraicGeometry.tilde.isUnit_algebraMap_end_basicOpen, CategoryTheory.full_linearYoneda, SimplicialObject.Splitting.πSummand_comp_cofan_inj_id_comp_PInfty_eq_PInfty_assoc, SSet.Truncated.StrictSegal.spineToSimplex_interval, SSet.Truncated.hom_ext_iff, HomologicalComplex.evalCompCoyonedaCorepresentableByDoubleId_homEquiv_symm_apply, CategoryTheory.unopUnop_map, SSet.N.mk_surjective, HomotopicalAlgebra.fibration_unop_iff, CategoryTheory.Functor.instFullOppositeOp, CategoryTheory.shrinkYonedaRepresentableBy_homEquiv, SheafOfModules.pushforwardNatTrans_app_val_app, CategoryTheory.Limits.IndObjectPresentation.ofCocone_ℐ, CategoryTheory.Presheaf.instIsLocallyPresentableFunctorOppositeOfHasPullbacks, Profinite.Extend.cocone_ι_app, CategoryTheory.Limits.coconeRightOpOfConeEquiv_functor_obj, CategoryTheory.GrothendieckTopology.W_iff, AlgebraicGeometry.HasRingHomProperty.iff_exists_appLE_locally, CategoryTheory.Limits.preservesColimit_op, CategoryTheory.StructuredArrow.functor_obj, CategoryTheory.Presheaf.isLocallySurjective_whisker_iff, CategoryTheory.SimplicialObject.Augmented.point_obj, SimplicialObject.Split.natTransCofanInj_app, CategoryTheory.MonoidalCategory.dayConvolutionInternalHomDiagramFunctor_obj_map_app_app, preservesColimit_coyoneda_of_finitePresentation, PresheafOfModules.Sheafify.one_smul, CategoryTheory.Limits.whiskeringLimYonedaIsoCones_hom_app_app_app, CategoryTheory.ULiftYoneda.instFaithfulFunctorOppositeTypeUliftYoneda, CategoryTheory.Pretriangulated.commShiftIso_opOp_inv_app_assoc, CategoryTheory.Limits.FormalCoproduct.cosimplicialObjectFunctor_obj_map, AlgebraicTopology.DoldKan.Γ₀.obj_map, CategoryTheory.Pretriangulated.commShiftIso_unopUnop_inv_app_assoc, AlgebraicGeometry.targetAffineLocally_affineAnd_iff', CategoryTheory.Limits.ProductsFromFiniteCofiltered.finiteSubproductsCone_π_app, AlgebraicGeometry.AffineSpace.reindex_appTop_coord, AlgebraicGeometry.Spec.algebraMap_stalkIso_inv_assoc, CategoryTheory.ComposableArrows.opEquivalence_inverse_map, HomologicalComplex.ExactAt.op, CategoryTheory.Pseudofunctor.IsStackFor.essSurj, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIsoApp_inv_hom, CategoryTheory.GrothendieckTopology.W_eq_inverseImage_isomorphisms_of_adjunction, AlgebraicGeometry.smoothOfRelativeDimension_iff, CategoryTheory.enrichedNatTransYoneda_obj, CondensedMod.epi_iff_locallySurjective_on_compHaus, CategoryTheory.LocalizerMorphism.LeftResolution.op_w, SSet.OneTruncation₂.ofNerve₂.natIso_inv_app_map, AlgebraicGeometry.HasRingHomProperty.iff_exists_appLE, CategoryTheory.Limits.PushoutCocone.op_pt, CategoryTheory.Equivalence.sheafCongrPreregular_unitIso_inv_app_hom_app, CategoryTheory.yonedaAddMonObj_obj_coe, CategoryTheory.GrothendieckTopology.pseudofunctorOver_toPrelaxFunctor_toPrelaxFunctorStruct_toPrefunctor_obj_α, CategoryTheory.SimplicialObject.δ_comp_δ_self', CategoryTheory.GrothendieckTopology.isoSheafify_inv, CategoryTheory.ObjectProperty.InheritedFromSource.op, AlgebraicGeometry.ΓSpec.adjunction_homEquiv_symm_apply, CategoryTheory.Comon.monoidal_leftUnitor_inv_hom, CategoryTheory.CosimplicialObject.Augmented.leftOp_left_obj, AlgebraicGeometry.Scheme.Hom.appLE_appIso_inv_apply, SSet.RelativeMorphism.Homotopy.h₀, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_presheafFiberMapObjIso_inv_assoc, CategoryTheory.Pseudofunctor.DescentData'.pullHom'_hom_comp, CategoryTheory.SimplicialObject.Augmented.rightOp_left, CategoryTheory.Limits.hasPullbacks_opposite, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.coalgebraEquivalence_functor_map_f, CategoryTheory.sheafOver_obj, CategoryTheory.GrothendieckTopology.toPlus_naturality_assoc, AlgebraicGeometry.Spec_Γ_naturality_assoc, AlgebraicGeometry.Scheme.IdealSheafData.ideal_mono, AlgebraicGeometry.Scheme.Hom.appLE_map'_assoc, AlgebraicGeometry.RingedSpace.isUnit_res_basicOpen, AlgebraicGeometry.Scheme.empty_presheaf, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_apply, CategoryTheory.CosimplicialObject.Augmented.leftOpRightOpIso_inv_right_app, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.section_app_op_mk_zero, TopCat.Presheaf.germ_res'_assoc, AlgebraicGeometry.Scheme.Opens.ι_app_self, CategoryTheory.Square.op_X₄, LightCondensed.epi_π_app_zero_of_epi, HomotopicalAlgebra.trivialCofibrations_eq_unop, CategoryTheory.Functor.mapConeOp_inv_hom, CategoryTheory.ShortComplex.LeftHomologyData.unop_Q, CategoryTheory.MonoidalCategory.MonoidalRightAction.oppositeRightAction_actionHomLeft, CategoryTheory.isDetecting_op_iff, CategoryTheory.Functor.WellOrderInductionData.Extension.compatibility, AlgebraicGeometry.ΓSpec.toSpecΓ_of, CategoryTheory.Functor.map_shift_unop_assoc, CategoryTheory.Functor.sheafPushforwardContinuousIso_hom, AlgebraicGeometry.Scheme.zeroLocus_span, CategoryTheory.Functor.IsCoverDense.Types.presheafHom_app, HomotopicalAlgebra.instIsStableUnderRetractsOppositeFibrationsOfCofibrations, CommRingCat.Opposite.regularEpiOfFaithfullyFlat, AlgebraicGeometry.StructureSheaf.toOpen_germ, TopCat.Presheaf.IsSheaf.isSheafPairwiseIntersections, CategoryTheory.GrothendieckTopology.yonedaULiftEquiv_naturality, HomologicalComplex.opEquivalence_functor, AlgebraicGeometry.Scheme.IdealSheafData.ideal_iSup, CategoryTheory.Square.isPullback_iff_map_coyoneda_isPullback, CategoryTheory.monoidalUnopUnop_μ, AlgebraicGeometry.Scheme.Hom.naturality, PresheafOfModules.instIsLocalizationSheafOfModulesSheafificationInverseImageFunctorOppositeAbWToPresheaf, CategoryTheory.Limits.reflectsFiniteLimits_of_rightOp, CategoryTheory.Comma.opFunctorCompSnd_inv_app, CategoryTheory.Pretriangulated.Opposite.UnopUnopCommShift.iso_hom_app, CategoryTheory.yonedaCommGrpGrp_map_app, CategoryTheory.regularTopology.equalizerCondition_w, SSet.Truncated.Path.arrow_tgt, CategoryTheory.Functor.sheafPushforwardContinuousComp_inv_app_hom_app, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_hom_app_assoc, CategoryTheory.RetractArrow.op_i_left, CategoryTheory.isRegularMono_op_iff_isRegularEpi, CategoryTheory.Functor.instIsEquivalenceOppositeLeftOp, CategoryTheory.ShortComplex.op_f, AlgebraicGeometry.exists_appTop_π_eq_of_isLimit, SSet.horn₃₁.ι₀_desc_assoc, SSet.mem_degenerate_iff, SSet.nonDegenerateEquivOfIso_apply_coe, SSet.S.le_iff_nonempty_hom, AlgebraicGeometry.Proj.stalkIso'_germ, TopCat.Presheaf.isGluing_iff_pairwise, Alexandrov.lowerCone_pt, SSet.modelCategoryQuillen.boundary_ι_mem_I, CategoryTheory.SimplicialObject.Truncated.sk_coreflective, SSet.Augmented.stdSimplex_map_left, TopCat.toSheafCompHausLike_obj_map, CategoryTheory.Functor.op_isTriangulated_iff, CategoryTheory.FunctorToTypes.rightAdj_obj_obj, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_inv_app_assoc, AlgebraicGeometry.IsAffine.affine, AlgebraicGeometry.IsAffineOpen.instAwayCarrierObjOppositeOpensCarrierCarrierCommRingCatSpecPresheafOpOpensBasicOpen, HomologicalComplex.op_X, CategoryTheory.Sheaf.cartesianMonoidalCategoryFst_val, CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinsetLimitCone_cone_π_app, AlgebraicGeometry.SheafedSpace.Γ_obj, CategoryTheory.Arrow.cechNerve_map, CategoryTheory.Limits.coyonedaOpColimitIsoLimitCoyoneda'_hom_comp_π_assoc, CategoryTheory.LocalizerMorphism.RightResolution.unopFunctor_map_f, CategoryTheory.isCardinalPresentable_iff_isCardinalAccessible_coyoneda_obj, SSet.Truncated.trunc_spine, CategoryTheory.Sieve.shrinkFunctorIsoFunctor_hom_app_coe, CategoryTheory.sheafification_obj, CategoryTheory.Limits.walkingParallelPairOpEquiv_unitIso_zero, SSet.stdSimplex.faceSingletonComplIso_hom_ι_assoc, AlgebraicGeometry.Scheme.restrict_presheaf_map, CategoryTheory.Comon.MonOpOpToComonObj_X, PresheafOfModules.instHasColimitModuleCatCarrierObjOppositeRingCatCompEvaluationRestrictScalarsHomMap, CategoryTheory.Sheaf.ΓHomEquiv_naturality_right, CategoryTheory.Sheaf.instIsLeftAdjointComposeAndSheafify, SSet.RelativeMorphism.Homotopy.precomp_h, HomologicalComplex.quasiIso_opFunctor_map_iff, CategoryTheory.GrothendieckTopology.Point.presheafFiber_map_shrinkYoneda_map_shrinkYonedaCompPresheafFiberIso_inv_app, CategoryTheory.Equivalence.sheafCongr.functor_map_hom_app, CategoryTheory.GrothendieckTopology.Point.skyscraperPresheafHomEquiv_app_π_assoc, CategoryTheory.ShortComplex.unopMap_id, CategoryTheory.NatIso.op_inv, PresheafOfModules.pushforward_assoc, CategoryTheory.MonoidalCategory.DayConvolutionUnit.leftUnitorCorepresentingIso_hom_app_app, SSet.Finite.exists_epi_from_isCardinalPresentable, CategoryTheory.map_yonedaEquiv, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.ofRestrict_invApp, AlgebraicGeometry.formallyUnramified_iff, CategoryTheory.GrothendieckTopology.Point.instIsRightAdjointSheafSkyscraperSheafFunctor, PresheafOfModules.map_id, AlgebraicGeometry.Spec.locallyRingedSpaceObj_presheaf_map', CategoryTheory.op_zsmul, CategoryTheory.Sheaf.isSeparator, CommGrpCat.coyonedaForget_inv_app_app, CategoryTheory.Sieve.natTransOfLe_app_coe, CategoryTheory.Idempotents.DoldKan.equivalence_functor, CategoryTheory.GrothendieckTopology.map_uliftYonedaEquiv, AlgebraicGeometry.Scheme.mem_basicOpen_top, CategoryTheory.Presheaf.coherentExtensiveEquivalence_unitIso_inv_app_hom_app, CommRingCat.coyonedaUnique_hom_app_hom_apply, Condensed.isoFinYoneda_hom_app, CategoryTheory.Pretriangulated.shift_unop_opShiftFunctorEquivalence_counitIso_hom_app, SSet.N.le_iff, CategoryTheory.ShortComplex.homologyMap_op, AlgebraicGeometry.IsAffineOpen.range_fromSpec, CategoryTheory.Limits.opProdIsoCoprod_inv_inr_assoc, AlgebraicTopology.NormalizedMooreComplex.objX_zero, CategoryTheory.Limits.hasColimit_rightOp_of_hasLimit, SSet.ι₀_fst, TopCat.Sheaf.eq_app_of_locally_eq, CategoryTheory.GrothendieckTopology.preservesLimits_diagramFunctor, CategoryTheory.Equivalence.sheafCongrPrecoherent_functor_map_hom_app, AlgebraicGeometry.Scheme.Opens.ι_image_basicOpen', CategoryTheory.Limits.coneOpEquiv_inverse_map, CategoryTheory.MonoidalCategory.DayConvolution.associatorCorepresentingIso_hom_app_app, CategoryTheory.GrothendieckTopology.Plus.inj_of_sep, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_naturality_assoc, CategoryTheory.Pseudofunctor.DescentData'.comm_assoc, CategoryTheory.SimplicialObject.Truncated.sk.faithful, CategoryTheory.shrinkYonedaIsoYoneda_inv_app_app, CategoryTheory.op_epi_of_mono, CategoryTheory.OverPresheafAux.OverArrows.map₁_val, AlgebraicTopology.DoldKan.N₂_obj_X_d, CategoryTheory.Limits.FormalCoproduct.evalOp_obj_map, CategoryTheory.Equivalence.sheafCongrPrecoherent_inverse_map_hom_app, CategoryTheory.Limits.reflectsLimit_of_unop, AlgebraicGeometry.PresheafedSpace.pushforwardDiagramToColimit_obj, CategoryTheory.Comon.monoidal_rightUnitor_hom_hom, Profinite.NobelingProof.spanFunctorIsoIndexFunctor_inv_app, lightDiagramToLightProfinite_map, CategoryTheory.MonoidalCategory.dayConvolutionInternalHomDiagramFunctor_obj_obj_obj_map, AlgebraicGeometry.LocallyRingedSpace.preimage_basicOpen, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.shortComplex_f, CategoryTheory.ObjectProperty.isClosedUnderColimitsOfShape_op_iff_op, CategoryTheory.Sheaf.isLocallyBijective_iff_isIso, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_toSpecΓ_assoc, AlgebraicGeometry.Scheme.isNilpotent_of_isNilpotent_cover, CategoryTheory.Join.InclRightCompRightOpOpEquivFunctor_inv_app, CategoryTheory.Subfunctor.toRangeSheafify_app_coe, CategoryTheory.Sieve.shrinkFunctor_obj, CategoryTheory.RanIsSheafOfIsCocontinuous.fac, CategoryTheory.Limits.preservesColimitsOfSize_rightOp, AlgebraicGeometry.AffineSpace.homOverEquiv_apply, CategoryTheory.SimplicialObject.Augmented.toArrow_obj_hom, SimplicialObject.opEquivalence_inverse, CategoryTheory.Limits.coconeUnopOfConeEquiv_inverse_obj, CategoryTheory.GrothendieckTopology.toPlus_naturality, AlgebraicGeometry.Scheme.Modules.instFullPresheafOfModulesToPresheafOfModules, SimpleGraph.componentComplFunctor_finite, CategoryTheory.Limits.preservesLimitsOfShape_of_leftOp, CategoryTheory.ObjectProperty.op_isoClosure, CategoryTheory.Pretriangulated.Opposite.contractibleTriangleIso_inv_hom₂, CategoryTheory.Limits.reflectsColimitsOfSize_of_unop, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_presheafFiberDesc_assoc, SSet.Truncated.liftOfStrictSegal_app_1, CategoryTheory.Functor.IsCoverDense.isoOver_inv_app, SheafOfModules.instReflectsIsomorphismsPresheafOfModulesObjFunctorOppositeRingCatIsSheafForget, CategoryTheory.Subfunctor.ofSection_le_iff, SSet.stdSimplex.δ_one_eq_const, AlgebraicGeometry.Scheme.basicOpen_res, CommMonCat.coyonedaForget_hom_app_app_hom, CategoryTheory.Limits.limitOpIsoOpColimit_inv_comp_π, AlgebraicGeometry.Scheme.Opens.fromSpecStalkOfMem_toSpecΓ_assoc, AlgebraicGeometry.StructureSheaf.toPushforwardStalk_comp, SSet.Truncated.liftOfStrictSegal.hδ'₀, TopCat.Sheaf.objSupIsoProdEqLocus_inv_fst, CategoryTheory.instIsConstantObjSheafSheafCompose, CategoryTheory.Pseudofunctor.DescentData'.pullHom'_self, smoothSheaf.ι_evalHom, CategoryTheory.OverPresheafAux.YonedaCollection.map₁_id, SSet.S.mk_surjective, CategoryTheory.linearYoneda_obj_obj_isModule, CategoryTheory.Limits.isColimitOfConeOfCoconeRightOp_desc, AlgebraicGeometry.Smooth.exists_isStandardSmooth, AlgebraicGeometry.SheafedSpace.Γ_def, SSet.associator_hom_app_apply, CategoryTheory.instIsIsoFunctorOppositeSheafToPresheafToSheafCompComposeAndSheafify, CategoryTheory.Abelian.DoldKan.equivalence_functor, AlgebraicGeometry.Scheme.Modules.pseudofunctor_mapComp_inv_τl, SSet.Truncated.cosk.full, CategoryTheory.SimplicialObject.Homotopy.h_zero_comp_δ_zero_assoc, CategoryTheory.Limits.preservesColimits_op, CategoryTheory.MorphismProperty.leftFractionRel_op_iff, AlgebraicGeometry.Scheme.comp_appTop, CategoryTheory.shrinkYonedaObjObjEquiv_obj_map, smoothSheafCommRing.ι_evalHom_assoc, CategoryTheory.Limits.Cone.unop_pt, CategoryTheory.SimplicialObject.σ_comp_σ_assoc, CategoryTheory.yonedaEvaluation_map_down, CategoryTheory.yoneda_preservesLimits, CategoryTheory.instIsDiscreteOpposite, AlgebraicGeometry.Scheme.IdealSheafData.radical_ideal, AlgebraicGeometry.isIso_pushoutSection_of_isQuasiSeparated_of_flat_left, CategoryTheory.Comma.opEquiv_inverse, CategoryTheory.MonoidalCategory.DayConvolutionUnit.corepresentableByLeft_homEquiv, CategoryTheory.PresheafHom.IsSheafFor.app_cond, CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheafObjObjIso.inv_restriction, TopCat.Presheaf.Pushforward.comp_hom_app, CategoryTheory.SimplicialObject.Homotopy.whiskerRight_h, SSet.Truncated.HomotopyCategory.BinaryProduct.associativity, CategoryTheory.Functor.mapCoconeOp_hom_hom, AlgebraicGeometry.StructureSheaf.instAwayObjOppositeOpensCarrierTopObjFunctorTypeIsSheafGrothendieckTopologyStructureSheafInTypeOpBasicOpen, CategoryTheory.sheafToPresheaf_ε, CategoryTheory.GrothendieckTopology.coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation_π_app, AugmentedSimplexCategory.equivAugmentedSimplicialObject_functor_obj_right, AlgebraicGeometry.IsClosedImmersion.isAffine_surjective_of_isAffine, AddCommGrpCat.coyoneda_map_app, CategoryTheory.Limits.pullbackIsoOpPushout_hom_inr, CategoryTheory.Pseudofunctor.DescentData.exists_equivalence_of_sieve_eq, CommRingCat.instIsRightAdjointOppositeObjFunctorTypeYoneda, AlgebraicGeometry.PresheafedSpace.restrictStalkIso_inv_eq_germ, CategoryTheory.Functor.instEssSurjOppositeRightOp, CategoryTheory.Presieve.piComparison_fac, SSet.OneTruncation₂.nerveEquiv_symm_apply_obj, AlgebraicGeometry.StructureSheaf.toPushforwardStalk_comp_assoc, AlgebraicGeometry.germ_eq_zero_of_pow_mul_eq_zero, AlgebraicGeometry.Spec.toPresheafedSpace_obj_op, ContinuousMap.yonedaPresheaf'_map, AlgebraicGeometry.StructureSheaf.toOpenₗ_eq_const, CategoryTheory.ParametrizedAdjunction.arrowHomEquiv_apply_right_fst, CategoryTheory.prodOpEquiv_functor_map, CategoryTheory.MorphismProperty.StableUnderInverse.op, CategoryTheory.NatTrans.op_whiskerLeft, AlgebraicGeometry.isIso_pushoutSection_iff, CategoryTheory.Comon.ComonToMonOpOpObj_mon_mul, CategoryTheory.GrothendieckTopology.uliftYonedaOpCompCoyoneda_inv_app_app, CategoryTheory.sheafificationAdjunction_counit_app_val, AugmentedSimplexCategory.equivAugmentedSimplicialObjectFunctorCompDropIso_inv_app_app, CategoryTheory.ShortComplex.rightHomologyFunctorOpNatIso_inv_app, CategoryTheory.preservesLimitsOfShape_presheafToSheaf, LightCondensed.lanPresheafExt_inv, CategoryTheory.ShortComplex.Homotopy.op_h₁, CategoryTheory.ObjectProperty.isClosedUnderLimitsOfShape_op_iff_op, CategoryTheory.isIso_iff_isIso_coyoneda_map, SSet.image_degenerate_le, AlgebraicGeometry.Scheme.fromSpecStalk_app, Condensed.lanPresheafExt_hom, CategoryTheory.Limits.colimitCoyonedaHomIsoLimit_π_apply, CategoryTheory.isFiltered_op_iff_isCofiltered, CategoryTheory.Enriched.FunctorCategory.diagram_obj_map, CategoryTheory.Limits.hasPushout_unop_iff_hasPullback, AlgebraicGeometry.IsAffineOpen.isLocalization_stalk, AlgebraicTopology.DoldKan.karoubi_PInfty_f, CategoryTheory.ParametrizedAdjunction.unit_whiskerRight_map, SSet.Subcomplex.liftPath_vertex_coe, SSet.modelCategoryQuillen.J_le_monomorphisms, AlgebraicGeometry.Smooth.smooth_appLE, AlgebraicGeometry.StructureSheaf.res_const, AlgebraicGeometry.PresheafedSpace.GlueData.ι_jointly_surjective, CategoryTheory.Limits.coneOfSectionCompCoyoneda_π, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_comp, AlgebraicGeometry.exists_basicOpen_le_appLE_of_appLE_of_isAffine, HomologicalComplex.evalCompCoyonedaCorepresentableBySingle_homEquiv_symm_apply, CategoryTheory.GrothendieckTopology.sheafification_map, CategoryTheory.GrothendieckTopology.sheafificationWhiskerLeftIso_inv_app, CategoryTheory.Functor.instFinalOppositeLeftOpOfInitial, CategoryTheory.Functor.op_comp_isSheaf, CategoryTheory.GrothendieckTopology.Point.instPreservesColimitsOfShapeOppositeElementsFiberObjFunctorCurriedTensor, AlgebraicGeometry.isField_of_universallyClosed, commBialgCatEquivComonCommAlgCat_inverse_obj, CategoryTheory.Functor.rightOpId_hom_app, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_base_apply, CategoryTheory.GrothendieckTopology.yonedaULiftEquiv_comp, CategoryTheory.Limits.instHasBinaryCoproductOppositeOp, CategoryTheory.hoFunctor.instIsIsoCatProdComparisonSSetHoFunctorNerve, CategoryTheory.Presieve.IsSheafFor.functorInclusion_comp_extend_assoc, AlgebraicGeometry.Scheme.IdealSheafData.ideal_inf, CategoryTheory.Presheaf.map_comp_uliftYonedaEquiv_down, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_eq_zero_iff_notMem_basicOpen, CategoryTheory.BraidedCategory.op_tensorμ, CategoryTheory.GrothendieckTopology.Point.skyscraperSheafAdjunction_homEquiv_apply_val, SSet.boundary_eq_iSup, AlgebraicGeometry.exists_of_res_zero_of_qcqs, CategoryTheory.NatIso.removeOp_hom, CategoryTheory.Subpresheaf.isSheaf_iff, CategoryTheory.MonoidalCategory.MonoidalLeftAction.leftActionOfOppositeLeftAction_actionHom, CategoryTheory.GrothendieckTopology.Cover.index_right, Opens.mayerVietorisSquare'_toSquare, AlgebraicGeometry.Scheme.LocalRepresentability.yonedaGluedToSheaf_app_comp, SSet.Subcomplex.unionProd.bicartSq, SheafOfModules.pushforwardPushforwardEquivalence_unit_app_val_app, AlgebraicGeometry.Scheme.Spec.algebraMap_residueFieldIso_inv_assoc, SSet.prod_σ_snd, CategoryTheory.Functor.WellOrderInductionData.map_lift, CategoryTheory.Pseudofunctor.DescentData.pullFunctorIdIso_inv_app_hom, SSet.RelativeMorphism.Homotopy.h₁, CategoryTheory.GrothendieckTopology.whiskerRight_toSheafify_sheafifyCompIso_hom, CategoryTheory.HasLiftingProperty.transfiniteComposition.sqFunctor_obj, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.instMonoSheafAddCommGrpCatFShortComplex, CategoryTheory.Limits.inr_opProdIsoCoprod_inv, smoothSheafCommRing.ι_forgetStalk_hom, CategoryTheory.Sieve.functorInclusion_is_mono, CategoryTheory.ShortComplex.HomologyData.unop_iso, SimplicialObject.Splitting.comp_PInfty_eq_zero_iff, CategoryTheory.hasColimitsEssentiallySmallSite, CategoryTheory.ShortComplex.SnakeInput.op_L₁, AlgebraicGeometry.exists_pow_mul_eq_zero_of_res_basicOpen_eq_zero_of_isAffineOpen, CategoryTheory.Limits.reflectsLimits_of_rightOp, CategoryTheory.Injective.injective_iff_preservesEpimorphisms_yoneda_obj, InfiniteGalois.mk_toAlgEquivAux, AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToSection_germ, SSet.stdSimplex.spineId_vertex, CategoryTheory.Limits.limitCompYonedaIsoCocone_hom_app, CategoryTheory.GrothendieckTopology.Cover.index_snd, CategoryTheory.ShortComplex.HomologyData.op_iso, AlgebraicGeometry.Scheme.IdealSheafData.subschemeι_app_surjective, CategoryTheory.Limits.fst_opProdIsoCoprod_hom, AlgebraicGeometry.Scheme.preimage_zeroLocus, SSet.OneTruncation₂.ofNerve₂.natIso_inv_app_obj_map, CategoryTheory.GrothendieckTopology.Point.presheafFiberMapIso_inv_app, CategoryTheory.Subfunctor.Subpresheaf.family_of_elements_compatible, CategoryTheory.Functor.LeibnizAdjunction.adj_unit_app_right, CategoryTheory.Limits.coconeOfConeLeftOp_ι_app, Opens.coe_mayerVietorisSquare_X₄, CategoryTheory.Limits.preservesLimitsOfShape_of_unop, CondensedMod.LocallyConstant.instFullSheafCompHausCoherentTopologyTypeConstantSheaf, AlgebraicGeometry.PresheafedSpace.congr_app, SSet.Subcomplex.range_tensorHom, CategoryTheory.Functor.IsRepresentedBy.iff_of_isoObj, CategoryTheory.OverPresheafAux.unitAuxAux_hom_app, CategoryTheory.simplicialCosimplicialEquiv_functor_obj_obj, CategoryTheory.Subfunctor.Subpresheaf.ofSection_le_iff, CategoryTheory.Limits.end_.map_comp_assoc, AlgebraicGeometry.Scheme.AffineZariskiSite.cocone_ι_app, SSet.stdSimplex.objMk₁_apply_eq_zero_iff, TopCat.Presheaf.presheafEquivOfIso_counitIso_hom_app_app, CategoryTheory.Pseudofunctor.LocallyDiscreteOpToCat.map_eq_pullHom_assoc, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_skyscraperPresheafHomEquiv_symm_assoc, CategoryTheory.Idempotents.DoldKan.Γ_map_app, CategoryTheory.PresheafOfGroups.OneCocycle.ev_refl, AlgebraicGeometry.IsAffineOpen.SpecMap_appLE_fromSpec_assoc, CategoryTheory.Limits.op_pullbackMap, CategoryTheory.Limits.Cocone.op_π, CategoryTheory.Limits.coneLeftOpOfCoconeEquiv_counitIso, SSet.Truncated.HomotopyCategory.BinaryProduct.right_unitality, CategoryTheory.ShortComplex.op_X₃, CategoryTheory.Limits.preservesFiniteLimits_of_unop, CategoryTheory.Functor.rightOp_additive, HomologicalComplex.evalCompCoyonedaCorepresentableBySingle_homEquiv_apply, SheafOfModules.pushforward_comp_id, CategoryTheory.Limits.coend.ι_map_assoc, CategoryTheory.Equivalence.sheafCongr_inverse, SSet.Truncated.cosk_reflective, CategoryTheory.Limits.coconeOfConeUnop_ι, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.section_comp_hom, CategoryTheory.HasLiftingProperty.transfiniteComposition.wellOrderInductionData.map_lift, AlgebraicGeometry.Scheme.IdealSheafData.coe_support_inter, CategoryTheory.Limits.colimitYonedaHomIsoLimitOp_π_apply, CategoryTheory.Pseudofunctor.DescentData.isEquivalence_toDescentData_iff_of_sieve_eq, CategoryTheory.coyoneda_preservesLimit, CategoryTheory.Pseudofunctor.CoGrothendieck.comp_const, AlgebraicTopology.DoldKan.Γ₂N₁.natTrans_app_f_app, CategoryTheory.SimplicialObject.id_right, CategoryTheory.hoFunctor.preservesBinaryProduct, HomologicalComplex.unopInverse_obj, SSet.degenerate_eq_iUnion_range_σ, AlgebraicGeometry.IsAffineOpen.fromSpec_top, CategoryTheory.nerve.functorOfNerveMap_nerveFunctor₂_map, CategoryTheory.WithInitial.opEquiv_functor_obj, CategoryTheory.ObjectProperty.isCodetecting_op_iff, LightCondensed.isColimitLocallyConstantPresheaf_desc_apply, SheafOfModules.relationsOfIsCokernelFree_s, CategoryTheory.Discrete.opposite_inverse_obj, CategoryTheory.SimplicialObject.instIsLeftKanExtensionOppositeTruncatedSimplexCategoryObjSkAppTruncatedUnitSkAdjTruncation, CategoryTheory.Injective.instEnoughInjectivesOppositeOfEnoughProjectives, CategoryTheory.ShiftedHom.opEquiv_symm_apply, HomologicalComplex.opEquivalence_inverse, CategoryTheory.prodOpEquiv_counitIso_hom_app, AlgebraicGeometry.preservesLimitsOfShape_discrete_of_isSheaf_zariskiTopology, SSet.RelativeMorphism.Homotopy.postcomp_h, CategoryTheory.Limits.Fork.op_ι_app_zero, CategoryTheory.Pseudofunctor.sheafHom_obj, CategoryTheory.prodOpEquiv_functor_obj, CategoryTheory.Limits.end_.map_π_assoc, CategoryTheory.instLocallySmallOpposite, CategoryTheory.Presheaf.restrictedULiftYonedaHomEquiv'_symm_app_naturality_left, CategoryTheory.simplicialToCosimplicialAugmented_map_right, CategoryTheory.Subfunctor.Subpresheaf.ofSection_image, CategoryTheory.Sieve.functor_map_coe, AlgebraicTopology.DoldKan.degeneracy_comp_PInfty, CategoryTheory.MonoidalCategory.MonoidalLeftAction.oppositeLeftAction_actionAssocIso_op, AlgebraicGeometry.Scheme.map_basicOpen_map, CategoryTheory.Limits.cospanOp_inv_app, CategoryTheory.Presheaf.freeYonedaHomEquiv_comp_assoc, CategoryTheory.GrothendieckTopology.overMapPullback_id_comp_assoc, SheafOfModules.forgetToSheafModuleCat_obj_obj, SSet.stdSimplex.σ_objMk₁_of_lt, CategoryTheory.kernelOpOp_inv, CategoryTheory.coyonedaPairing_map, CategoryTheory.MorphismProperty.instHasRightCalculusOfFractionsOppositeOpOfHasLeftCalculusOfFractions, AlgebraicGeometry.PresheafedSpace.id_c_app, CategoryTheory.Presheaf.tautologicalCocone_ι_app, CategoryTheory.balanced_opposite, CategoryTheory.Functor.PreOneHypercoverDenseData.multicospanIndex_left, CategoryTheory.NatTrans.removeUnop_id, CategoryTheory.Pseudofunctor.CoGrothendieck.map_id_eq, CategoryTheory.Limits.opProdIsoCoprod_inv_inl_assoc, CategoryTheory.Pseudofunctor.CoGrothendieck.compIso_inv_app, AlgebraicGeometry.Scheme.Hom.comp_appTop, CategoryTheory.CosimplicialObject.Augmented.leftOp_hom_app, CategoryTheory.Pretriangulated.instIsHomologicalAddCommGrpCatObjOppositeFunctorPreadditiveCoyoneda, CategoryTheory.ShiftedHom.opEquiv'_apply, SSet.S.equivElements_symm_apply_simplex, CategoryTheory.Presheaf.equalizerSieve_apply, SSet.stdSimplex.face_inter_face, CategoryTheory.LocalizerMorphism.LeftResolution.op_X₁, CategoryTheory.Limits.hasFiniteProducts_opposite, CategoryTheory.Idempotents.instIsIdempotentCompleteOpposite, CategoryTheory.Functor.RepresentableBy.equivUliftYonedaIso_apply, AlgebraicGeometry.IsAffineOpen.Spec_map_appLE_fromSpec, LightCondMod.LocallyConstant.instFullSheafLightProfiniteCoherentTopologyTypeConstantSheaf, CategoryTheory.isoOpEquiv_apply, CategoryTheory.Functor.leftOp_map, CategoryTheory.nerveFunctor_obj, CategoryTheory.simplicialCosimplicialAugmentedEquiv_inverse, AlgebraicGeometry.IsIntegralHom.hasAffineProperty, CategoryTheory.Equivalence.leftOp_counitIso_hom_app, CategoryTheory.GrothendieckTopology.pseudofunctorOver_toPrelaxFunctor_toPrelaxFunctorStruct_toPrefunctor_obj_str_id_hom_app, CategoryTheory.constantSheafAdj_counit_w, CategoryTheory.Limits.preservesColimits_of_leftOp, CategoryTheory.sheafCompose_id, SSet.stdSimplex.mem_nonDegenerate_iff_mono, CategoryTheory.OverPresheafAux.yonedaCollectionPresheafMap₁_app, SimplicialObject.Splitting.σ_comp_πSummand_id_eq_zero, CategoryTheory.Presheaf.tautologicalCocone'_ι_app, PresheafOfModules.pushforward_obj_map_apply', CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan.presheafHom_naturality_assoc, instPreservesFiniteCoproductsSheafTypeYonedaOfPreservesFiniteProductsOppositeObjFunctorIsSheaf, CategoryTheory.unop_hom_rightUnitor, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo_Spec, CategoryTheory.Limits.walkingSpanOpEquiv_functor_obj, CategoryTheory.Subfunctor.IsGeneratedBy.mem, CategoryTheory.Functor.PullbackObjObj.π_snd, CategoryTheory.Presheaf.coconeOfRepresentable_ι_app, SSet.horn₃₂.desc.multicofork_pt, CompHausLike.LocallyConstant.sigmaComparison_comp_sigmaIso, CategoryTheory.Limits.pushoutIsoOpPullback_inv_snd, SSet.prodStdSimplex.strictMono_orderHomOfSimplex, CompHausLike.LocallyConstant.unit_app, AlgebraicGeometry.Scheme.basicOpen_zero, InfiniteGalois.proj_of_le, AlgebraicGeometry.IsAffineOpen.isLocalization_stalk', CategoryTheory.PresheafIsGeneratedBy.of_epi, HomologicalComplex.cyclesOpNatIso_hom_app, CategoryTheory.instPreservesLimitsOfSizeFunctorOppositeTypeShrinkYoneda, CategoryTheory.Pseudofunctor.DescentData.pullFunctorIdIso_hom_app_hom, CategoryTheory.Limits.Cofork.op_π_app_one, SSet.StrictSegal.isPointwiseRightKanExtensionAt.fac_aux₁, AlgebraicGeometry.Scheme.IdealSheafData.ideal_ofIdeals_le, CategoryTheory.CategoryOfElements.costructuredArrowYonedaEquivalenceInverseπ_inv_app, CategoryTheory.typeEquiv_functor_obj_obj_map, CategoryTheory.Functor.PullbackObjObj.mapArrowRight_left, AlgebraicGeometry.LocallyOfFinitePresentation.finitePresentation_of_affine_subset, AlgebraicGeometry.Scheme.Hom.appLE_comp_appLE_assoc, SSet.N.nonDegenerate, AlgebraicGeometry.Scheme.SpecMap_presheaf_map_eqToHom, AlgebraicGeometry.IsAffineOpen.fromSpec_image_basicOpen, PresheafOfModules.Hom.naturality, AlgebraicGeometry.instHasAffinePropertyIsomorphismsSchemeAndIsAffineIsIsoCommRingCatAppTop, SSet.Subcomplex.mono_homOfLE, AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage, CategoryTheory.CompatiblePreserving.compatible, CategoryTheory.IsPushout.op, CategoryTheory.Subfunctor.IsGeneratedBy.image, SSet.Truncated.Path.mk₂_vertex, skyscraperPresheafCoconeOfSpecializes_ι_app, HomotopicalAlgebra.cofibration_unop_iff, SSet.horn₂₂.isPushout, LightProfinite.Extend.functor_initial, CategoryTheory.Under.opEquivOpOver_inverse_obj, CategoryTheory.Subpresheaf.eq_sheafify_iff, CategoryTheory.Limits.opProductIsoCoproduct'_inv_comp_lift, CategoryTheory.PresheafHom.IsSheafFor.exists_app, CategoryTheory.CostructuredArrow.overEquivPresheafCostructuredArrow_functor_map_toOverCompCoyoneda, CategoryTheory.Limits.pullbackIsoOpPushout_inv_snd_assoc, AlgebraicGeometry.IsAffineOpen.basicOpen_union_eq_self_iff, CategoryTheory.GrothendieckTopology.instIsIsoFunctorOppositeToPresheafFiberNatTransPointBotCoeEquivHomUnopOpObjTypeShrinkYonedaSymmShrinkYonedaObjObjEquivId, AlgebraicGeometry.Scheme.isZero_ellAdicSheaf_of_isEmpty, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_map_injective, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_hom_eq_germ_assoc, CategoryTheory.Iso.unop_inv_hom_id_app, AlgebraicGeometry.isFinite_iff, TopCat.Presheaf.presheafEquivOfIso_inverse_obj_obj, CategoryTheory.yoneda'_comp, AlgebraicTopology.DoldKan.N₁Γ₀_inv_app, SSet.prodStdSimplex.nonDegenerate_iff_strictMono_objEquiv, CategoryTheory.Coyoneda.fullyFaithful_preimage, SSet.yonedaEquiv_comp, CategoryTheory.GrothendieckTopology.overMapPullbackCongr_hom_app_hom_app, CategoryTheory.Functor.Elements.coconeπOpCompShrinkYonedaObj_ι_app, AlgebraicGeometry.etale_iff, AlgebraicGeometry.PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_inv_ι_app, SSet.Truncated.IsStrictSegal.spine_bijective, SSet.op_map, CategoryTheory.Limits.isColimitCoconeOfConeLeftOp_desc, CategoryTheory.GrothendieckTopology.toSheafify_naturality, CommGrpCat.coyonedaType_obj_map, CategoryTheory.Functor.mapConeOp_hom_hom, CategoryTheory.HasLiftingProperty.transfiniteComposition.sqFunctor_map, CategoryTheory.Subpresheaf.le_sheafify, CategoryTheory.coherentTopology.isLocallySurjective_π_app_zero_of_isLocallySurjective_map, CategoryTheory.GrothendieckTopology.yonedaULiftEquiv_symm_app_apply, LightCondSet.topCatAdjunctionUnit_hom_app, HomologicalComplex.fromOpcycles_op_cyclesOpIso_inv, CategoryTheory.MorphismProperty.IsCardinalForSmallObjectArgument.preservesColimit, SSet.stdSimplex.spineId_arrow_apply_one, PresheafOfModules.toPresheaf_preservesColimit, LightCondensed.instFinalNatCostructuredArrowOppositeFintypeCatLightProfiniteOpToLightProfiniteOpPtAsLimitConeFunctorOp, AlgebraicGeometry.Scheme.germ_residue_assoc, CategoryTheory.unop_hom_braiding, CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheafObjObjIso_inv_naturality_assoc, AlgebraicTopology.DoldKan.HigherFacesVanish.comp_Hσ_eq, CategoryTheory.simplicialCosimplicialAugmentedEquiv_functor, SSet.Quasicategory.hornFilling', CategoryTheory.Limits.piFunctor_map_app, CategoryTheory.GrothendieckTopology.ι_plusCompIso_hom, CategoryTheory.RetractArrow.unop_i_left, AlgebraicGeometry.IsAffineOpen.Spec_map_appLE_fromSpec_assoc, CategoryTheory.Limits.reflectsFiniteColimits_of_leftOp, AlgebraicGeometry.Scheme.AffineZariskiSite.PreservesLocalization.opensRange_map, CategoryTheory.SimplicialObject.Homotopy.h_succ_comp_δ_castSucc_succ_assoc, CategoryTheory.OverPresheafAux.MakesOverArrow.of_arrow, CategoryTheory.cosimplicialToSimplicialAugmented_map, AlgebraicGeometry.Scheme.kerFunctor_obj, CategoryTheory.typeEquiv_functor_obj_obj_obj, PresheafOfModules.freeAdjunction_unit_app, SSet.OneTruncation₂.nerveHomEquiv_apply, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.coassoc, TopCat.Presheaf.Γgerm_res_apply, AlgebraicGeometry.Spec.toPresheafedSpace_map, CategoryTheory.Equivalence.unop_inverse, CategoryTheory.GrothendieckTopology.pseudofunctorOver_mapComp_hom_toNatTrans_app_hom_app, CategoryTheory.Limits.coneUnopOfCoconeEquiv_inverse_map, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_jointly_surjective₂, CategoryTheory.isCardinalPresentable_iff_isCardinalAccessible_uliftCoyoneda_obj, AlgebraicGeometry.StructureSheaf.comap_basicOpen, CategoryTheory.Limits.π_comp_colimitRightOpIsoUnopLimit_inv_assoc, Condensed.underlying_obj, AlgebraicGeometry.mono_pushoutSection_of_isCompact_of_flat_left, CategoryTheory.Functor.instIsEquivalenceOppositeRightOp, AlgebraicGeometry.Scheme.Modules.pseudofunctor_mapId_hom_τl, AlgebraicGeometry.exists_appTop_map_eq_zero_of_isLimit, CategoryTheory.Sheaf.sheafifyCocone_ι_app_val_assoc, PresheafOfModules.HasDifferentials.exists_universal_derivation, commBialgCatEquivComonCommAlgCat_inverse_map_unop_hom, CategoryTheory.LocalizerMorphism.LeftResolution.opEquivalence_counitIso, CategoryTheory.unop_zsmul, SheafOfModules.pushforwardPushforwardAdj_counit_app_val_app, SheafOfModules.conjugateEquiv_pullbackId_hom, CategoryTheory.Sheaf.Hom.add_app, AlgebraicGeometry.Scheme.mem_basicOpen'', CategoryTheory.PresheafIsGeneratedBy.range, SSet.Subcomplex.instEpiToRange, CategoryTheory.Presieve.extension_iff_amalgamation, HomologicalComplex.quasiIso_unopFunctor_map_iff, SSet.StrictSegalCore.map_mkOfSucc_zero_spineToSimplex, SimpleGraph.componentComplFunctor_nonempty_of_infinite, CategoryTheory.Sheaf.hasLimitsOfSize, SSet.instFiniteSigmaObjOfFinite, CategoryTheory.GrothendieckTopology.Point.instPreservesColimitsOfShapeOppositeElementsFiberForget, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.SheafCondition.bijective_toPullbackObj, AlgebraicGeometry.Scheme.AffineZariskiSite.cocone_pt, CategoryTheory.ShortComplex.RightHomologyData.op_i, AugmentedSimplexCategory.equivAugmentedSimplicialObject_functor_obj_left_map, CategoryTheory.Limits.coconeOpEquiv_counitIso, CategoryTheory.GrothendieckTopology.diagramNatTrans_id, SSet.RelativeMorphism.Homotopy.rel, AlgebraicGeometry.Scheme.Spec_obj, AlgebraicGeometry.tilde.toOpen_map_app_assoc, CategoryTheory.Limits.isLimitOfCoconeUnopOfCone_lift, SSet.horn.primitiveEdge_coe_down, CategoryTheory.CategoryOfElements.fromCostructuredArrow_obj_mk, CategoryTheory.GrothendieckTopology.Point.shrinkYonedaCompPresheafFiberIso_inv_app_toPresheafFiber, CategoryTheory.Functor.PullbackObjObj.mapArrowLeft_right, CategoryTheory.Abelian.DoldKan.comparisonN_hom_app_f, AlgebraicGeometry.exists_app_map_eq_zero_of_isLimit, CategoryTheory.constantCommuteCompose_hom_app_hom, CategoryTheory.Equivalence.preregular_isSheaf_iff_of_essentiallySmall, CategoryTheory.ObjectProperty.isClosedUnderColimitsOfShape_iff_unop, CategoryTheory.Under.opEquivOpOver_counitIso, CategoryTheory.Functor.IsDenseSubsite.map_eq_of_eq, AlgebraicGeometry.basicOpen_eq_of_affine, AlgebraicGeometry.LocallyRingedSpace.Γ_map_op, CategoryTheory.Functor.op_comp_isSheaf_of_preservesOneHypercovers, SSet.Truncated.HomotopyCategory.BinaryProduct.associativityIso_hom_app, CategoryTheory.Functor.PreOneHypercoverDenseData.multicospanIndex_snd, CategoryTheory.GrothendieckTopology.plusCompIso_whiskerLeft, CategoryTheory.PreGaloisCategory.PointedGaloisObject.instHasColimitOppositeFunctorTypeCompOpInclCoyoneda, CategoryTheory.MonoidalCategory.DayConvolutionUnit.rightUnitorCorepresentingIso_inv_app_app, CategoryTheory.Limits.IndObjectPresentation.cocone_pt, CategoryTheory.Functor.IsCoverDense.sheaf_eq_amalgamation, localCohomology.hasColimitDiagram, CategoryTheory.Limits.preservesLimits_of_rightOp, AlgebraicGeometry.Scheme.Opens.ι_appLE, CategoryTheory.Yoneda.fullyFaithful_preimage, AlgebraicGeometry.AffineScheme.forgetToScheme_obj, CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheafMap_π, CategoryTheory.MonoidalCategory.MonoidalRightAction.oppositeRightAction_actionHom_op, SSet.stdSimplex.faceSingletonIso_one_hom_comp_ι_eq_δ_assoc, AlgebraicGeometry.Scheme.ΓSpecIso_naturality_assoc, SSet.Truncated.Edge.CompStruct.exists_of_simplex, SSet.StrictSegal.spineToSimplex_arrow, CategoryTheory.SimplicialObject.Augmented.drop_map, CategoryTheory.coyonedaEquiv_symm_map, AlgebraicGeometry.IsAffineOpen.fromSpec_toSpecΓ_assoc, CategoryTheory.Limits.instHasColimitOppositeDiscreteOpFunctor, CategoryTheory.Functor.opHom_map_app, CategoryTheory.Limits.opCoproductIsoProduct_hom_comp_π_assoc, CategoryTheory.Comon.monoidal_associator_inv_hom, CategoryTheory.Presieve.FamilyOfElements.isCompatible_map_smul_aux, SSet.Subcomplex.range_eq_top_iff, CategoryTheory.Limits.proj_comp_opProductIsoCoproduct'_hom, CategoryTheory.Comon.Comon_EquivMon_OpOp_inverse, CategoryTheory.ShortComplex.unopMap_τ₃, AlgebraicGeometry.IsAffineOpen.fromSpec_image_zeroLocus, AlgebraicGeometry.Spec.toLocallyRingedSpace_map, AlgebraicGeometry.Scheme.appLE_comp_appLE, CategoryTheory.instFaithfulIndFunctorOppositeTypeInclusion, CategoryTheory.Functor.map_opShiftFunctorEquivalence_counitIso_inv_app_unop, CategoryTheory.GrothendieckTopology.yonedaULiftEquiv_symm_map, AlgebraicGeometry.Scheme.IdealSheafData.mem_supportSet_iff_of_mem, AlgebraicGeometry.LocallyRingedSpace.isUnit_res_toΓSpecMapBasicOpen, AlgebraicGeometry.Scheme.affineBasisCover_map_range, CategoryTheory.Pseudofunctor.LocallyDiscreteOpToCat.map_eq_pullHom, CategoryTheory.regularTopology.equalizerCondition_iff_of_equivalence, CategoryTheory.ObjectProperty.isSeparating_op_iff, SimplicialObject.Split.toKaroubiNondegComplexFunctorIsoN₁_inv_app_f_f, CategoryTheory.Limits.PushoutCocone.unop_pt, CategoryTheory.Sheaf.adjunction_counit_app_hom, CategoryTheory.ObjectProperty.colimitsOfShape_eq_unop_limitsOfShape, CategoryTheory.sheafToPresheaf_isRightAdjoint, LightCondensed.id_val, InfiniteGalois.algEquivToLimit_continuous, TopCat.Presheaf.germ_stalkSpecializes, LightCondensed.ofSheafForgetLightProfinite_obj, CategoryTheory.Limits.SequentialProduct.cone_π_app, PresheafOfModules.freeYonedaEquiv_comp, CategoryTheory.sheafToPresheafCompCoyonedaCompWhiskeringLeftSheafToPresheaf_hom_app_app_hom, AlgebraicGeometry.ΓSpec_adjunction_homEquiv_eq, CategoryTheory.PresheafOfGroups.OneCochain.inv_ev, CategoryTheory.instFullSheafFunctorOppositeCompSheafComposeSheafToPresheafOfFaithful, HomotopicalAlgebra.instIsStableUnderRetractsOppositeCofibrationsOfFibrations, AddCommMonCat.coyoneda_obj_map, CategoryTheory.Pretriangulated.Opposite.instAdditiveOppositeShiftFunctorInt, CategoryTheory.NatIso.unop_refl, PresheafOfModules.Sheafify.zero_smul, AlgebraicGeometry.Scheme.Opens.ι_app, CategoryTheory.toSheafify_naturality_assoc, SSet.StrictSegal.spine_spineToSimplex, CategoryTheory.RetractArrow.unop_i_right, AlgebraicGeometry.Scheme.Modules.pushforwardCongr_inv_app_app, CategoryTheory.Pseudofunctor.DescentData.id_hom, CategoryTheory.Limits.limitRightOpIsoOpColimit_inv_comp_π_assoc, AlgebraicGeometry.Scheme.IdealSheafData.supportSet_inter, AlgebraicGeometry.locallyOfFinitePresentation_iff, AlgebraicGeometry.exists_basicOpen_le_affine_inter, CategoryTheory.Presheaf.restrictedULiftYonedaHomEquiv'_symm_naturality_right_assoc, SSet.Truncated.HomotopyCategory.homToNerveMk_app_zero, AlgebraicGeometry.IsAffineOpen.isOpenImmersion_fromSpec, SSet.Truncated.liftOfStrictSegal.hσ'₁, CategoryTheory.cones_map_app_app, CategoryTheory.Limits.preservesLimits_of_unop, CategoryTheory.Limits.limitLeftOpIsoUnopColimit_hom_comp_ι_assoc, SSet.Truncated.Edge.CompStruct.d₀, CommGrpCat.coyoneda_obj_map, HomotopicalAlgebra.weakEquivalences_eq_unop, CategoryTheory.GrothendieckTopology.Point.sheafFiberComapIso_inv_app, CompHausLike.LocallyConstant.incl_comap, SSet.Truncated.Edge.map_whiskerRight, CategoryTheory.Sheaf.hom_ext_iff, CategoryTheory.isDetecting_unop_iff, CategoryTheory.Presieve.FamilyOfElements.compPresheafMap_id, SSet.Truncated.comp_app, CategoryTheory.Limits.hasPushout_op_iff_hasPullback, LightProfinite.Extend.functor_obj, LightCondensed.isoFinYoneda_hom_app, CategoryTheory.Pretriangulated.shiftFunctorCompIsoId_op_hom_app_assoc, AlgebraicTopology.DoldKan.Γ₀.Obj.mapMono_on_summand_id_assoc, CategoryTheory.coyonedaPairingExt_iff, CategoryTheory.Pseudofunctor.DescentData.pullFunctorCompIso_hom_app_hom, CategoryTheory.Pretriangulated.preadditiveYoneda_shiftMap_apply, AlgebraicGeometry.SheafedSpace.congr_hom_app, CategoryTheory.GrothendieckTopology.uliftYonedaEquiv_symm_naturality_right, AlgebraicGeometry.Scheme.Hom.instIsIsoCommRingCatApp, CategoryTheory.Presieve.IsSheafFor.valid_glue, CategoryTheory.MonoidalCategory.MonoidalRightAction.oppositeRightAction_actionHom, CategoryTheory.Limits.coconeLeftOpOfConeEquiv_counitIso, CategoryTheory.Pseudofunctor.DescentData.isoMk_hom_hom, AlgebraicGeometry.Scheme.Spec_map, CategoryTheory.shrinkYonedaMon_obj_map_shrinkYonedaMonObjObjEquiv_symm, CategoryTheory.RelCat.opFunctor_comp_unopFunctor_eq, CategoryTheory.evalEquiv_symm_apply, CategoryTheory.Functor.map_opShiftFunctorEquivalence_unitIso_inv_app_unop_assoc, CategoryTheory.instPreservesFiniteProductsOppositeObjFunctorIsSheafExtensiveTopology, SSet.Subcomplex.preimage_iSup, CategoryTheory.NatTrans.Equifibered.op, CategoryTheory.Limits.Cone.op_pt, SSet.Subcomplex.fromPreimage_ι, CategoryTheory.GrothendieckTopology.Point.presheafFiberCocone_ι_app, CategoryTheory.Equivalence.unop_unitIso, CategoryTheory.instFullFunctorOppositeTypeShrinkYoneda, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.isoMk_inv_hom, CategoryTheory.cosimplicialSimplicialEquiv_inverse_map, CategoryTheory.Presheaf.classifier_Ω₀, PresheafOfModules.sheafificationAdjunction_homEquiv_apply, Condensed.comp_hom, CategoryTheory.Adjunction.representableBy_homEquiv, CategoryTheory.Functor.sheafPushforwardContinuousComp'_inv_app_hom_app, CategoryTheory.Functor.Elements.initialOfRepresentableBy_snd, CategoryTheory.Equivalence.inverseFunctorObjIso_hom, AlgebraicGeometry.PresheafedSpace.restrict_presheaf, CategoryTheory.Limits.preservesColimits_of_rightOp, TopCat.Presheaf.germ_eq, CategoryTheory.ParametrizedAdjunction.homEquiv_naturality_one, SheafOfModules.instPreservesFiniteLimitsSheafAddCommGrpCatToSheaf, CategoryTheory.SimplicialObject.δ_comp_σ_of_le_assoc, CategoryTheory.SimplicialObject.δ_comp_δ, HomologicalComplex.instIsStrictlySupportedOppositeOpOp, SSet.Path.map_vertex, CategoryTheory.Limits.preservesLimitsOfSize_of_leftOp, CategoryTheory.shrinkYonedaIsoYoneda_hom_app_app, SSet.Truncated.Edge.CompStruct.map_simplex, CategoryTheory.Pseudofunctor.DescentData.pullFunctorObjHom_eq, CategoryTheory.GrothendieckTopology.W_eq_isLocal_range_sheafToPresheaf_obj, AlgebraicGeometry.Scheme.germ_residue, CategoryTheory.GrothendieckTopology.instIsIsoSheafAppFunctorOppositeSheafComposeNatTransPlusPlusAdjunction, CategoryTheory.yonedaAddMon_naturality, CategoryTheory.MonoidalCategory.MonoidalLeftAction.oppositeLeftAction_actionHomLeft, HomologicalComplex.unopEquivalence_inverse, CategoryTheory.Limits.coconeRightOpOfCone_pt, CategoryTheory.Limits.walkingSpanOpEquiv_unitIso_hom_app, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.Hom.comm_assoc, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.app_inv_app', AlgebraicGeometry.PresheafedSpace.GlueData.ιInvApp_π, ContinuousMap.comp_yonedaPresheaf', CategoryTheory.GrothendieckTopology.yonedaEquiv_symm_naturality_left, CategoryTheory.Presieve.FamilyOfElements.IsAmalgamation.map, TopCat.Presheaf.presheafEquivOfIso_functor_map_app, SSet.stdSimplex.face_le_face_iff, CategoryTheory.Equivalence.inverseFunctorObj'_inv_app, CategoryTheory.Enriched.FunctorCategory.enrichedComp_π, LightProfinite.proj_surjective, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.shortComplex_X₂, SSet.Finite.instIsFinitelyPresentable, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_map_surjective, CategoryTheory.eqToHom_comp_homOfLE_op_assoc, CategoryTheory.extensiveTopology.surjective_of_isLocallySurjective_sheaf_of_types, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_naturality, AlgebraicTopology.DoldKan.Γ₂_obj_X_obj, SSet.RelativeMorphism.Homotopy.refl_h, AlgebraicGeometry.Scheme.Hom.ι_fromNormalization, CategoryTheory.Limits.reflectsLimitsOfSize_of_rightOp, AlgebraicGeometry.LocallyRingedSpace.toΓSpec_preimage_zeroLocus_eq, SSet.horn₃₁.ι₃_desc, SSet.truncation_spine, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_inv_eq_germ, CategoryTheory.GrothendieckTopology.liftToPlusObjLimitObj_fac, PresheafOfModules.toPresheaf_preservesColimitsOfSize, AlgebraicGeometry.LocallyRingedSpace.SpecΓIdentity_hom_app, CategoryTheory.ParametrizedAdjunction.homEquiv_eq, CategoryTheory.Equivalence.symmEquivInverse_obj_unitIso_inv, CategoryTheory.Pretriangulated.TriangleOpEquivalence.functor_map_hom₁, AlgebraicGeometry.instIsQuasicoherentOpensCarrierCarrierCommRingCatSpecTilde, CategoryTheory.Pseudofunctor.DescentData.hom_self, CategoryTheory.toSheafify_plusPlusIsoSheafify_hom_assoc, CategoryTheory.Functor.IsDenseSubsite.sheafifyHomEquivOfIsEquivalence_naturality_left_assoc, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_hom, CategoryTheory.Pseudofunctor.DescentData.comp_hom_assoc, CategoryTheory.Functor.sheafPushforwardContinuousId'_inv_app_hom_app, AlgebraicGeometry.Scheme.Hom.map_appLE, TopCat.Presheaf.pullbackPushforwardAdjunction_unit_pullback_map_germToPullbackStalk_assoc, AlgebraicGeometry.Scheme.LocalRepresentability.glueData_t, SSet.PtSimplex.RelStruct.δ_succ_map, CategoryTheory.Functor.PullbackObjObj.ofHasPullback_pt, SheafOfModules.forgetPreservesLimitsOfShape, CategoryTheory.Limits.op_zero, CategoryTheory.Iso.unop_inv_hom_id_app_assoc, CategoryTheory.GrothendieckTopology.yoneda_map_hom, CategoryTheory.MonoidalCategory.MonoidalRightAction.oppositeRightAction_actionHomLeft_op, CategoryTheory.ShortComplex.opcyclesOpIso_hom_toCycles_op, AlgebraicGeometry.Scheme.Γevaluation_naturality, CategoryTheory.Presheaf.isLimit_iff_isSheafFor, SheafOfModules.Presentation.IsFinite.finite_relations, LightProfinite.proj_comp_transitionMapLE, LightProfinite.instEpiAppOppositeNatπAsLimitCone, SSet.OneTruncation₂.map_obj, CategoryTheory.Limits.reflectsFiniteLimits_of_unop, CategoryTheory.Limits.π_comp_colimitUnopIsoOpLimit_inv_assoc, AlgebraicGeometry.basicOpen_eq_bot_iff, SSet.stdSimplex.δ_objMk₁_of_lt, nonempty_sections_of_finite_inverse_system, SSet.σ_naturality_apply, CategoryTheory.Sheaf.instIsRightAdjointSheafComposeOfHasWeakSheafify, AlgebraicGeometry.Scheme.IdealSheafData.ker_glueDataObjι_appTop, SheafOfModules.pushforwardNatTrans_app_val_app_apply, AlgebraicGeometry.FormallyUnramified.formallyUnramified_appLE, CategoryTheory.Sieve.uliftFunctorInclusion_top_isIso, CategoryTheory.Limits.walkingParallelPairOp_one, AlgebraicGeometry.Spec.germ_stalkMapIso_hom, CategoryTheory.GrothendieckTopology.Point.instIsIsoηFunctorOppositePresheafFiber, PresheafOfModules.colimitCocone_ι_app_app, CategoryTheory.Equivalence.unop_functor, CategoryTheory.GrothendieckTopology.Point.toPresheafFiberNatTrans_app, AlgebraicTopology.DoldKan.Γ₂_map_f_app, AlgebraicGeometry.Scheme.Hom.appIso_inv_app_apply, CategoryTheory.Pretriangulated.commShiftIso_unopUnop_hom_app, CategoryTheory.Limits.reflectsColimits_rightOp, CategoryTheory.whiskering_linearYoneda₂, SSet.Truncated.HomotopyCategory.homToNerveMk_comp, SSet.Truncated.HomotopyCategory.BinaryProduct.functor_map, CategoryTheory.SimplicialObject.Homotopy.h_zero_comp_δ_zero, CategoryTheory.Pseudofunctor.CoGrothendieck.categoryStruct_comp_fiber, CategoryTheory.op_neg, PresheafOfModules.colimitPresheafOfModules_obj, CategoryTheory.isCoseparator_op_iff, AddCommMonCat.coyonedaForget_hom_app_app_hom, CategoryTheory.Limits.IndizationClosedUnderFilteredColimitsAux.isFiltered, CategoryTheory.Pseudofunctor.DescentData'.comp_pullHom''_assoc, AlgebraicTopology.DoldKan.decomposition_Q, PresheafOfModules.Finite.evaluation_preservesFiniteColimits, CategoryTheory.ShortComplex.RightHomologyData.unop_π, CategoryTheory.ShortComplex.leftHomologyFunctorOpNatIso_inv_app, CategoryTheory.Presheaf.preservesColimitsOfSize_leftKanExtension, LightCondMod.LocallyConstant.instFaithfulSheafLightProfiniteCoherentTopologyTypeConstantSheaf, CategoryTheory.Presheaf.restrictedULiftYonedaHomEquiv'_symm_app_naturality_left_assoc, CategoryTheory.LocalizerMorphism.RightResolution.op_w, CategoryTheory.Functor.op_commShiftIso_hom_app, CategoryTheory.Limits.reflectsFiniteColimits_of_unop, SheafOfModules.comp_val, CategoryTheory.instSmallHomFunctorOppositeTypeColimitCompYoneda, AlgebraicTopology.DoldKan.σ_comp_P_eq_zero, CategoryTheory.Equivalence.op_unitIso, CategoryTheory.Groupoid.invEquivalence_unitIso, CategoryTheory.GrothendieckTopology.Point.toPresheafFiberOfIsCofiltered_w, AlgebraicGeometry.Spec.fromSpecStalk_eq, skyscraperPresheafCoconeOfSpecializes_pt, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_over, CategoryTheory.instPreservesColimitFunctorOppositeTypeObjCoyonedaOpYoneda, PresheafOfModules.instReflectsFiniteLimitsSheafOfModulesSheafAddCommGrpCatToSheaf, CategoryTheory.Over.opEquivOpUnder_functor_map, CategoryTheory.op_sub, CategoryTheory.Subfunctor.to_sheafifyLift, CategoryTheory.GrothendieckTopology.Point.presheafFiberMap_hom_ext_iff, SSet.S.mk_map_le, CategoryTheory.ShortComplex.Homotopy.unop_h₀, SSet.S.IsUniquelyCodimOneFace.existsUnique_δ_cast_simplex, AlgebraicGeometry.Spec.locallyRingedSpaceObj_presheaf, CategoryTheory.Equivalence.sheafCongrPreregular_functor_obj_obj_map, CategoryTheory.SimplicialObject.isCoskeletal_iff_isIso, CategoryTheory.instIsRegularEpiCategorySheafTypeOfHasSheafify, AlgebraicGeometry.PresheafedSpace.GlueData.π_ιInvApp_π, CategoryTheory.Presheaf.instIsLeftKanExtensionFunctorOppositeTypeIdCompUliftYonedaOfPreservesColimitsOfSizeOfHasPointwiseLeftKanExtension, TopCat.Presheaf.stalkFunctor_map_germ, CategoryTheory.PresheafHom.isAmalgamation_iff, CategoryTheory.Limits.SequentialProduct.functorMap_commSq_succ, LightCondensed.instHasColimitsOfShapeCostructuredArrowOppositeFintypeCatLightProfiniteOpToLightProfiniteType, AlgebraicGeometry.Scheme.IdealSheafData.ideal_biInf, CategoryTheory.Limits.reflectsLimitsOfSize_of_leftOp, CategoryTheory.Pretriangulated.Opposite.OpOpCommShift.iso_hom_app, CategoryTheory.Comon.monoidal_tensorUnit_comon_counit, PresheafOfModules.sections_ext_iff, CategoryTheory.Functor.IsContinuous.op_comp_isSheaf_of_types, CategoryTheory.Equivalence.op_inverse, CategoryTheory.Presheaf.pullback_imageSieve, AlgebraicTopology.DoldKan.Γ₀.obj_obj, CategoryTheory.Limits.isColimitOfConeOfCoconeUnop_desc, PresheafOfModules.toPresheaf_map_sheafificationAdjunction_unit_app, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_naturality, CategoryTheory.Functor.PullbackObjObj.π_fst_assoc, CategoryTheory.ShortComplex.opEquiv_unitIso, CategoryTheory.cokernelUnopOp_inv, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_basicOpen', PresheafOfModules.limitPresheafOfModules_obj, CondensedSet.toTopCatMap_hom_apply, CategoryTheory.Functor.IsRepresentedBy.of_isoObj, CategoryTheory.Pseudofunctor.DescentData'.pullHom'₁₂_eq_pullHom_of_chosenPullback₃_assoc, CategoryTheory.Square.IsPushout.op, CategoryTheory.Pretriangulated.TriangleOpEquivalence.counitIso_hom_app_hom₃, AlgebraicGeometry.structurePresheafInCommRingCat_obj_carrier, CategoryTheory.nerveMap_app, CategoryTheory.Presieve.isSheaf_yoneda', TopCat.Presheaf.germToPullbackStalk_stalkPullbackHom_assoc, CategoryTheory.Limits.opProductIsoCoproduct_inv_comp_lift, CategoryTheory.Limits.Cocone.unop_pt, CategoryTheory.Limits.hasFiniteLimits_opposite, HomologicalComplex.unop_d, CategoryTheory.SimplicialThickening.functor_id, CategoryTheory.SimplicialObject.augmentedCechNerve_obj_right, SSet.Subcomplex.top_prod_le_unionProd, CategoryTheory.Presheaf.classifier_χ₀, instSecondCountableTopologyCarrierToTopTotallyDisconnectedSpacePtOppositeNatProfiniteCone, CategoryTheory.Limits.ι_comp_colimitLeftOpIsoUnopLimit_hom_assoc, AlgebraicGeometry.IsAffineOpen.app_basicOpen_eq_away_map, AlgebraicGeometry.LocallyRingedSpace.toΓSpecMapBasicOpen_eq, CategoryTheory.Pretriangulated.opShiftFunctorEquivalenceSymmHomEquiv_left_inv_assoc, CategoryTheory.sheafCompose_obj_obj, CategoryTheory.uliftYonedaMap_app_apply, SheafOfModules.pushforward_id_comp, CategoryTheory.Functor.WellOrderInductionData.Extension.map_limit, SSet.modelCategoryQuillen.fibration_iff, InfiniteGalois.limitToAlgEquiv_symm_apply, CategoryTheory.OverPresheafAux.counitForward_counitBackward, CategoryTheory.MonoidalCategory.dayConvolutionInternalHomDiagramFunctor_obj_obj_map_app, CategoryTheory.MonoidalCategory.MonoidalRightAction.oppositeRightAction_actionAssocIso, CategoryTheory.Limits.Wedge.condition_assoc, CategoryTheory.Functor.IsCoverDense.Types.appIso_hom, CategoryTheory.Functor.IsCoverDense.restrictHomEquivHom_naturality_right_symm_assoc, CategoryTheory.NatIso.unop_inv, AlgebraicGeometry.Scheme.preimage_basicOpen, CategoryTheory.OverPresheafAux.costructuredArrowPresheafToOver_obj, HomologicalComplex.opFunctor_additive, CategoryTheory.Limits.hasColimits_op_of_hasLimits, CategoryTheory.GrothendieckTopology.sheafificationWhiskerRightIso_inv_app, CategoryTheory.GrothendieckTopology.Point.ofIsCofiltered.instHasColimitsOfShapeOppositeType, CategoryTheory.GrothendieckTopology.Point.presheafFiber_hom_ext_iff, AlgebraicGeometry.Scheme.Opens.instIsIsoCommRingCatAppLEιTopToScheme, CategoryTheory.Limits.pullbackIsoOpPushout_hom_inl_assoc, CategoryTheory.orderDualEquivalence_inverse_obj, CategoryTheory.Sieve.equalizer_eq_equalizerSieve, SSet.Truncated.rightExtensionInclusion_hom_app, CategoryTheory.Equivalence.leftOp_inverse_obj, LightCondensed.comp_val, CategoryTheory.Limits.multispanIndexCoend_fst, TopCat.Presheaf.toPushforwardOfIso_app, CategoryTheory.Limits.snd_opProdIsoCoprod_hom_assoc, LightCondensed.id_hom, AlgebraicGeometry.IsAffineOpen.ΓSpecIso_hom_fromSpec_app, CategoryTheory.sheafBotEquivalence_counitIso, CategoryTheory.Equivalence.sheafCongrPrecoherent_unitIso_hom_app_hom_app, CategoryTheory.kernelOpOp_hom, CategoryTheory.Limits.coconeUnopOfCone_ι, AlgebraicGeometry.Scheme.exists_le_and_germ_injective, AlgebraicGeometry.Scheme.Modules.pseudofunctor_map_r, CategoryTheory.cones_obj_obj, AlgebraicGeometry.Scheme.instIsGrothendieckAbelianSheafProEtTopologyAb, CategoryTheory.Functor.FullyFaithful.homNatIso_hom_app_down, AlgebraicGeometry.Scheme.Hom.flat_appLE, CategoryTheory.SimplicialObject.augmentedCechNerve_map_left_app, CategoryTheory.Limits.IndizationClosedUnderFilteredColimitsAux.exists_nonempty_limit_obj_of_colimit, AlgebraicTopology.AlternatingFaceMapComplex.obj_X, PresheafOfModules.ofPresheaf_map, PresheafOfModules.instIsLocallyInjectiveToSheafify, CategoryTheory.shrinkYoneda_map_app_shrinkYonedaObjObjEquiv_symm, CategoryTheory.Pseudofunctor.CoGrothendieck.instFaithfulαCategoryObjLocallyDiscreteOppositeCatMkOpFiberForgetInducedFunctor, CategoryTheory.nerve.functorOfNerveMap_obj, CategoryTheory.Square.unop_X₄, SSet.N.iSup_subcomplex_eq_top, CategoryTheory.GrothendieckTopology.W_eq_W_range_sheafToPresheaf_obj, AlgebraicGeometry.Scheme.affineOpenCover_X, CategoryTheory.CostructuredArrow.overEquivPresheafCostructuredArrow_inverse_map_toOverCompYoneda, AlgebraicGeometry.Scheme.comp_app_assoc, AlgebraicTopology.DoldKan.Γ₂N₁_inv, CategoryTheory.MorphismProperty.instIsStableUnderRetractsOppositeOp, CategoryTheory.ShortComplex.cyclesOpIso_inv_op_iCycles, TopCat.Presheaf.restrict_sum, CategoryTheory.Limits.opCoproductIsoProduct'_hom_comp_proj_assoc, SSet.Truncated.Path₁.arrow_src, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_invApp_assoc, AlgebraicGeometry.instQuasiCompactToSpecΓOfCompactSpaceCarrierCarrierCommRingCat, SkyscraperPresheafFunctor.map'_app, CategoryTheory.Subobject.functor_map, AlgebraicGeometry.LocallyRingedSpace.evaluation_naturality_assoc, AlgebraicGeometry.IsAffineOpen.SpecMap_appLE_fromSpec, CategoryTheory.Limits.walkingParallelPairOpEquiv_unitIso_hom_app_one, CategoryTheory.sheafComposeIso_hom_fac_assoc, AlgebraicGeometry.morphismRestrict_app, CategoryTheory.Functor.PullbackObjObj.hom_ext_iff, CategoryTheory.SimplicialObject.augmentedCechNerve_obj_left_map, AlgebraicGeometry.AffineSpace.comp_homOfVector_assoc, CategoryTheory.Functor.leftOpRightOpEquiv_functor_map_app, CategoryTheory.sheafification_reflective, CategoryTheory.yonedaGrp_map_app, AlgebraicTopology.DoldKan.instReflectsIsomorphismsKaroubiSimplicialObjectChainComplexNatN₂, AlgebraicGeometry.Scheme.Hom.appLE_comp_appLE, CategoryTheory.ShortComplex.HomologyData.unop_right, PresheafOfModules.germ_ringCat_smul, SSet.hasDimensionLE_prod, AlgebraicTopology.DoldKan.Q_f_naturality, SSet.ι₁_fst_assoc, CategoryTheory.sheafComposeIso_inv_fac, CategoryTheory.Equalizer.FirstObj.ext_iff, AlgebraicGeometry.LocallyRingedSpace.toStalk_stalkMap_toΓSpec, CategoryTheory.Limits.walkingSpanOpEquiv_counitIso_inv_app, CategoryTheory.Functor.SmallCategories.instPreservesFiniteLimitsSheafSheafPullbackOfRepresentablyFlat, CategoryTheory.Pretriangulated.commShiftIso_unopUnop_hom_app_assoc, CategoryTheory.Limits.preservesLimit_of_rightOp, CategoryTheory.nerve.nerveFunctor₂_map_functorOfNerveMap, CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan.coconeApp_naturality, CategoryTheory.Under.opEquivOpOver_unitIso, AlgebraicGeometry.Scheme.Modules.instIsRightAdjointPresheafOfModulesToPresheafOfModules, AlgebraicGeometry.Scheme.Hom.toImage_app, CategoryTheory.sheafHomSectionsEquiv_symm_apply_coe_apply, AlgebraicGeometry.Scheme.Modules.restrictFunctorId_hom_app_app, TopCat.Sheaf.eq_of_locally_eq_iff, AlgebraicGeometry.exists_lift_of_germInjective_aux, CategoryTheory.MorphismProperty.presheaf_monomorphisms_le_monomorphisms, CategoryTheory.Limits.isIndObject_limit_comp_yoneda_comp_colim, CategoryTheory.PreOneHypercover.Hom.mapMultiforkOfIsLimit_ι_assoc, CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheafObjObjIso.hom_map, AlgebraicGeometry.isIso_pushoutSection_of_isQuasiSeparated_of_flat_right, CategoryTheory.Limits.limitLeftOpIsoUnopColimit_inv_comp_π_assoc, CategoryTheory.Functor.isEquivalence_of_isOneHypercoverDense, LightCondensed.underlying_map, PresheafOfModules.naturality_apply, CategoryTheory.ShortComplex.RightHomologyData.unop_H, AlgebraicGeometry.IsAffineOpen.isoSpec_hom, PresheafOfModules.toPresheaf_map_sheafificationHomEquiv_def, SimplicialObject.opEquivalence_functor, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_presheafFiberMapObjIso_inv, AlgebraicGeometry.Scheme.Modules.Hom.app_smul, AlgebraicTopology.DoldKan.Γ₀.Obj.map_epi_on_summand_id, SSet.Truncated.Edge.CompStruct.d₁, SSet.Subcomplex.N.mk'_surjective, CategoryTheory.unop_hom_associator, CategoryTheory.regularTopology.EqualizerCondition.bijective_mapToEqualizer_pullback, CategoryTheory.Limits.PreservesFiniteLimitsOfIsFilteredCostructuredArrowYonedaAux.flipFunctorToInterchange_hom_app_app, CategoryTheory.Limits.Cofork.unop_ι, CategoryTheory.NatTrans.unop_whiskerRight_assoc, AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_app_spec, AlgebraicTopology.DoldKan.identity_N₂_objectwise, AlgebraicGeometry.Scheme.Γ_obj_op, CategoryTheory.Join.opEquiv_functor_map_op_inclLeft, SSet.stdSimplex.objMk₁_apply_eq_one_iff, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.inv_invApp, CategoryTheory.coyonedaFunctor_preservesLimits, CategoryTheory.map_shrinkYonedaEquiv, CategoryTheory.GrothendieckTopology.Point.sheafFiberCompIso_hom_app, CategoryTheory.ObjectProperty.small_unop_iff, SimpleGraph.componentComplFunctor_map, Condensed.underlying_map, CategoryTheory.OverPresheafAux.YonedaCollection.map₁_snd, CategoryTheory.Limits.reflectsColimitsOfSize_leftOp, CategoryTheory.Functor.sieves_map, TopCat.Presheaf.IsSheaf.isSheafOpensLeCover, SSet.Truncated.HomotopyCategory.BinaryProduct.id_prod_mapHomotopyCategory_comp_inverse, CategoryTheory.GrothendieckTopology.ofArrows_mem_iff_isLocallySurjective_sigmaDesc_uliftYoneda_map, CategoryTheory.WithInitial.opEquiv_inverse_map, CategoryTheory.Presieve.FamilyOfElements.map_comp, CategoryTheory.Under.opEquivOpOver_functor_map, CategoryTheory.Functor.IsRepresentedBy.iff_natIso, TopCat.Presheaf.IsSheaf.isSheafUniqueGluing, CategoryTheory.Limits.preservesColimitsOfShape_op, CategoryTheory.Adjunction.sheafPushforwardContinuous_counit_app_hom_app, SSet.Subcomplex.PairingCore.notMem₂, PresheafOfModules.instFullRestrictScalarsIdFunctorOppositeRingCat, SSet.Subcomplex.topIso_hom, SSet.OneTruncation₂.id_edge, CategoryTheory.Limits.coneRightOpOfCoconeEquiv_unitIso, AlgebraicTopology.inclusionOfMooreComplexMap_f, CategoryTheory.yonedaEquiv_symm_map, CategoryTheory.Pretriangulated.Opposite.instIsTriangulatedOpposite, AlgebraicGeometry.Scheme.image_zeroLocus, AlgebraicGeometry.Scheme.Hom.appLE_map, CategoryTheory.HasCodetector.hasDetector_op, CategoryTheory.Idempotents.DoldKan.hη, AlgebraicGeometry.Scheme.basicOpen_eq_bot_iff_forall_evaluation_eq_zero, CategoryTheory.GrothendieckTopology.yoneda_obj_obj, AlgebraicGeometry.HasRingHomProperty.iff_of_source_openCover, AlgebraicGeometry.LocallyRingedSpace.toΓSpecCBasicOpens_app, CategoryTheory.ShortComplex.cyclesOpIso_inv_naturality_assoc, SSet.stdSimplex.toSSetObj_app_const_one, AlgebraicGeometry.LocallyRingedSpace.evaluation_eq_zero_iff_notMem_basicOpen, AlgebraicGeometry.isIso_toSpecΓ, CategoryTheory.SimplicialObject.augmentedCechNerve_obj_left_obj, CategoryTheory.Limits.pullbackIsoUnopPushout_inv_snd_assoc, SSet.Edge.toTruncated_id, SSet.horn.ι_ι_assoc, CategoryTheory.Functor.IsDenseSubsite.isIso_ranCounit_app_of_isDenseSubsite, CategoryTheory.Pseudofunctor.CoGrothendieck.map_id_map, CategoryTheory.Limits.opParallelPairIso_inv_app_zero, AlgebraicGeometry.Proj.res_apply, CategoryTheory.CostructuredArrow.projectQuotient_factors, TopCat.Presheaf.app_injective_iff_stalkFunctor_map_injective, AlgebraicGeometry.StructureSheaf.comapₗ_eq_localRingHom, CategoryTheory.Sieve.toFunctor_app_coe, CategoryTheory.isFinitelyPresentable_iff_preservesFilteredColimits, CategoryTheory.constantPresheafAdj_counit_app_app, SheafOfModules.pushforwardSections_coe, AlgebraicGeometry.Scheme.Opens.topIso_hom, AlgebraicTopology.DoldKan.N₁Γ₀_inv_app_f_f, CategoryTheory.Limits.ι_comp_colimitLeftOpIsoUnopLimit_hom, CategoryTheory.GrothendieckTopology.map_yonedaEquiv, AlgebraicGeometry.Scheme.Modules.pseudofunctor_mapComp_inv_τr, AlgebraicTopology.alternatingFaceMapComplex_obj_X, PresheafOfModules.Sheafify.smul_zero, PresheafOfModules.fromFreeYonedaCoproduct_app_mk, CategoryTheory.Functor.reprW_hom_app, CategoryTheory.Sheaf.isLocallySurjective_iff_isIso, PresheafOfModules.Sheafify.map_smul, CategoryTheory.ComposableArrows.opEquivalence_counitIso_hom_app_app, CategoryTheory.uliftCoyonedaEquiv_symm_map, CategoryTheory.GrothendieckTopology.Point.presheafFiberMapIso_hom_app, CategoryTheory.GrothendieckTopology.overMapPullback_assoc, AlgebraicGeometry.StructureSheaf.toOpen_res, CategoryTheory.Presheaf.isSheaf_coherent_of_hasPullbacks_comp, CategoryTheory.Limits.opParallelPairIso_inv_app_one, AlgebraicGeometry.PresheafedSpace.restrictStalkIso_hom_eq_germ_assoc, CategoryTheory.Limits.colimitHomIsoLimitYoneda_inv_comp_π, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_ε, CategoryTheory.presheafIsGeneratedBy_of_isFinite, CategoryTheory.coherentTopology.isLocallySurjective_iff, SSet.Truncated.sk.faithful, AlgebraicGeometry.Scheme.ofRestrict_app, AlgebraicGeometry.structurePresheafInModuleCat_obj_carrier, SSet.opEquivalence_counitIso, AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_app_spec_assoc, AlgebraicGeometry.SheafedSpace.congr_app, CategoryTheory.op_associator, CategoryTheory.yonedaYonedaColimit_app_inv, CategoryTheory.Sheaf.ΓObjEquivSections_naturality, CategoryTheory.Limits.Cone.extensions_app, CategoryTheory.yonedaEquiv_symm_naturality_left, CategoryTheory.Limits.inl_opProdIsoCoprod_inv, PresheafOfModules.germ_smul, CategoryTheory.Limits.preservesLimitsOfShape_rightOp, AlgebraicGeometry.Scheme.zeroLocus_univ, CategoryTheory.Pseudofunctor.presheafHomObjHomEquiv_symm_apply, CategoryTheory.Limits.snd_opProdIsoCoprod_hom, CategoryTheory.Equalizer.Presieve.Arrows.SecondObj.ext_iff, CategoryTheory.imageUnopOp_inv_comp_op_factorThruImage, CategoryTheory.Presheaf.comp_isLocallyInjective_iff, SSet.leftUnitor_hom_app_apply, AlgebraicGeometry.LocallyRingedSpace.toΓSpecCApp_spec, CategoryTheory.SmallObject.preservesColimit, CategoryTheory.Limits.walkingParallelPairOpEquiv_counitIso_one, SimplicialObject.Splitting.toKaroubiNondegComplexIsoN₁_hom_f_f, AlgebraicTopology.DoldKan.Γ₀.Obj.map_on_summand', TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv_functor, CategoryTheory.Limits.coneUnopOfCocone_π, CategoryTheory.ShortComplex.Splitting.unop_s, TopCat.Presheaf.pullbackPushforwardAdjunction_unit_app_app_germToPullbackStalk_assoc, CategoryTheory.ObjectProperty.IsCodetecting.isIso_iff_of_epi, CategoryTheory.NatIso.unop_whiskerLeft, CategoryTheory.Pseudofunctor.DescentData'.pullHom'₂₃_eq_pullHom_of_chosenPullback₃_assoc, CategoryTheory.Pretriangulated.Opposite.complete_distinguished_triangle_morphism, HomotopicalAlgebra.instHasTwoOutOfThreePropertyOppositeWeakEquivalences, CategoryTheory.Limits.isColimitCoconeOfConeUnop_desc, LightCondSet.toTopCatMap_hom_apply, CategoryTheory.Presieve.isSheaf_comp_uliftFunctor_iff, CategoryTheory.Limits.PreservesFiniteLimitsOfIsFilteredCostructuredArrowYonedaAux.isIso_post, SimplicialObject.Split.id_F, CategoryTheory.Limits.limitLeftOpIsoUnopColimit_hom_comp_ι, AlgebraicGeometry.Scheme.Hom.comp_appTop_assoc, CategoryTheory.Functor.map_opShiftFunctorEquivalence_counitIso_hom_app_unop_assoc, CategoryTheory.GrothendieckTopology.diagramNatTrans_zero, CategoryTheory.Equivalence.leftOp_functor_map, CategoryTheory.NatIso.unop_hom, AlgebraicGeometry.IsAffineOpen.fromSpec_app_of_le, CondensedSet.epi_iff_locallySurjective_on_compHaus, CategoryTheory.Hom.mulEquivCongrRight_apply, AlgebraicGeometry.SheafedSpace.instEpiTopCatBaseHomPresheafedSpaceπ, AlgebraicGeometry.StructureSheaf.const_mul, CategoryTheory.GrothendieckTopology.toPlus_plusLift, AlgebraicGeometry.AffineSpace.map_appTop_coord, CategoryTheory.GrothendieckTopology.ofArrows_mem_iff_isLocallySurjective_cofanIsColimitDesc_uliftYoneda_map, CategoryTheory.Comma.unopFunctorCompSnd_hom_app, CategoryTheory.Pretriangulated.preadditiveYoneda_map_distinguished, CategoryTheory.coyonedaFunctor_reflectsLimits, AlgebraicGeometry.Scheme.toSpecΓ_preimage_basicOpen, CategoryTheory.Presheaf.instIsLocallyInjectiveHomιOpposite, CategoryTheory.Functor.sheafPushforwardContinuousComp'_hom_app_hom_app, LightCondensed.lanPresheafExt_hom, CategoryTheory.yonedaEquiv_symm_naturality_right, CategoryTheory.PreGaloisCategory.PointedGaloisObject.cocone_app, CategoryTheory.Limits.FormalCoproduct.cech_obj, CategoryTheory.Limits.reflectsFiniteProducts_leftOp, CategoryTheory.AddMonObj.ofRepresentableBy_zero, CategoryTheory.constantSheafAdj_counit_app, PresheafOfModules.toSheafify_app_apply, CategoryTheory.Equalizer.Sieve.SecondObj.ext_iff, CategoryTheory.Functor.leibnizPullback_map_app, CategoryTheory.isDetector_op_iff, CategoryTheory.unop_tensorObj, CategoryTheory.NatTrans.rightOp_id, TopCat.Presheaf.isIso_iff_stalkFunctor_map_iso, AlgebraicGeometry.Scheme.Hom.app_appIso_inv, AlgebraicGeometry.isIso_fromTildeΓ_iff, CategoryTheory.isCodetecting_unop_iff, TopCat.Sheaf.interUnionPullbackCone_pt, CategoryTheory.Pseudofunctor.DescentData'.pullHom_pullHom'_assoc, CategoryTheory.NatTrans.removeRightOp_id, CategoryTheory.GrothendieckTopology.uliftYoneda_map_hom_app_down, HomologicalComplex.unopFunctor_map_f, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv_unitIso, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_functor, CategoryTheory.sheafSectionsNatIsoEvaluation_hom_app, HomologicalComplex.homologyOp_hom_naturality, CategoryTheory.Enriched.FunctorCategory.enrichedId_π, CategoryTheory.StructuredArrow.toCostructuredArrow'_map, CategoryTheory.Sieve.shrinkFunctorIsoFunctor_inv_app_coe, CategoryTheory.Square.opFunctor_map_τ₄, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_presheafFiberCompIso_hom_app_assoc, CategoryTheory.GrothendieckTopology.preservesLimit_diagramFunctor, CategoryTheory.MonoidalCategory.MonoidalRightAction.oppositeRightAction_actionObj_op, CategoryTheory.Adjunction.op_counit, SSet.RelativeMorphism.comp_map, CategoryTheory.Presheaf.isSheaf_iff_isSheaf_forget, SSet.prod_map_snd, CategoryTheory.ShortComplex.fromOpcycles_op_cyclesOpIso_inv, SheafOfModules.instIsRightAdjointPushforwardIdSheafRingCat, SSet.horn₃₂.desc.multicofork_π_zero, CategoryTheory.Limits.FormalCoproduct.cosimplicialObjectFunctor_map_app, CategoryTheory.GrothendieckTopology.overMapPullback_id_comp, AlgebraicGeometry.Scheme.Hom.normalizationObjIso_hom_val, AlgebraicGeometry.Scheme.openCoverBasicOpenTop_f, CategoryTheory.Functor.unopId_hom_app, AlgebraicGeometry.Scheme.zero_of_zero_cover, CategoryTheory.Limits.IndObjectPresentation.yoneda_I, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.sheafCondition_iff_comp_coyoneda, SSet.stdSimplex.objEquiv_symm_apply, AlgebraicTopology.DoldKan.Γ₀_obj_termwise_mapMono_comp_PInfty_assoc, SSet.StrictSegal.instIsStrictSegalObjTruncatedHAddNatOfNatTruncationOfIsStrictSegal, CategoryTheory.MorphismProperty.LeftFractionRel.op, AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_hom_c_app, SSet.horn₃₂.desc.multicofork_π_three_assoc, CategoryTheory.OverPresheafAux.OverArrows.map_val, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverseObj_pt, CategoryTheory.Adjunction.Quadruple.op_adj₂, instIsCommMonObjOppositeCommAlgCatOpOfOfIsCocomm, CategoryTheory.RetractArrow.unop_r_left, CommRingCat.moduleCatRestrictScalarsPseudofunctor_obj, CategoryTheory.Functor.sheafAdjunctionCocontinuous_homEquiv_apply_val, CategoryTheory.hasCodetector_op_iff, SheafOfModules.instFullPresheafOfModulesObjFunctorOppositeRingCatIsSheafForget, CondensedSet.topCatAdjunctionUnit_hom_app, CategoryTheory.PreOneHypercover.forkOfIsColimit_ι_map_inj, CategoryTheory.Equivalence.symmEquivInverse_obj_inverse, CategoryTheory.ParametrizedAdjunction.homEquiv_naturality_one_assoc, PresheafOfModules.pushforward₀_obj_obj_isModule, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.invApp_app_assoc, CategoryTheory.CostructuredArrow.unop_left_comp_underlyingIso_hom_unop, CategoryTheory.GrothendieckTopology.yonedaULiftEquiv_yonedaULift_map, CategoryTheory.Limits.walkingCospanOpEquiv_unitIso_hom_app, AlgebraicTopology.DoldKan.map_hσ', CategoryTheory.ShortComplex.LeftHomologyData.unop_g', CategoryTheory.Limits.reflectsLimits_of_leftOp, AlgebraicTopology.DoldKan.N₂_map_f_f, CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinset_obj_obj, CategoryTheory.Subfunctor.family_of_elements_compatible, SSet.Subcomplex.PairingCore.nonDegenerate₁, SSet.horn₃₂.desc.multicofork_π_one_assoc, CategoryTheory.Presheaf.coherentExtensiveEquivalence_counitIso_inv_app_hom_app, CategoryTheory.Injective.injective_iff_preservesEpimorphisms_preadditiveYoneda_obj, CategoryTheory.uliftYonedaIsoYoneda_inv_app_app_down, AlgebraicGeometry.isField_of_isIntegral_of_subsingleton, CategoryTheory.NatTrans.leftOp_comp, SSet.KanComplex.hornFilling, CategoryTheory.uliftYonedaEquiv_symm_map_assoc, CategoryTheory.StructuredArrow.toCostructuredArrow'_obj, commBialgCatEquivComonCommAlgCat_counitIso_hom_app, AlgebraicGeometry.exists_of_res_eq_of_qcqs, AlgebraicGeometry.PresheafedSpace.map_comp_c_app, SSet.PtSimplex.RelStruct.δ_map_of_gt, TopCat.Sheaf.isLocallySurjective_iff_epi, CategoryTheory.Limits.pullbackIsoUnopPushout_hom_inr, HomologicalComplex.opcyclesOpIso_hom_toCycles_op_assoc, CategoryTheory.Functor.instEssSurjOppositeLeftOp, CategoryTheory.Limits.opSpan_inv_app, CategoryTheory.SimplicialObject.instHasLimits, CategoryTheory.unop_inv_rightUnitor, CategoryTheory.Limits.SequentialProduct.cone_π_app_comp_Pi_π_neg, AlgebraicGeometry.ProjectiveSpectrum.Proj.toOpen_toSpec_val_c_app, CategoryTheory.Abelian.IsGrothendieckAbelian.OppositeModuleEmbedding.preservesFiniteColimits_embedding, CategoryTheory.Limits.coendFunctor_map, CategoryTheory.Limits.coyonedaCompLimIsoCones_hom_app_app, HomotopicalAlgebra.trivialFibrations_op, TopCat.subpresheafToTypes_obj, TopCat.Presheaf.submonoidPresheafOfStalk_obj, Condensed.isoFinYoneda_inv_app, HomologicalComplex.quasiIsoAt_opFunctor_map_iff, SSet.Truncated.StrictSegal.spine_δ_vertex_lt, CategoryTheory.Limits.FormalCoproduct.evalOp_obj_obj, CategoryTheory.Presheaf.isLocallyInjective_toPlus, SSet.nondegenerate_zero, CategoryTheory.RelCat.instIsEquivalenceOppositeOpFunctor, CategoryTheory.Functor.OneHypercoverDenseData.essSurj.restriction_map, CategoryTheory.isPreconnected_op, CategoryTheory.PreOneHypercover.Hom.mapMultiforkOfIsLimit_ι, CategoryTheory.Functor.leftAdjointObjIsDefined_iff, CategoryTheory.GrothendieckTopology.Point.skyscraperPresheafFunctor_obj_obj, TopCat.Presheaf.stalkFunctor_map_germ_apply', AlgebraicGeometry.StructureSheaf.algebraMap_pushforward_stalk, CondensedMod.LocallyConstant.instFaithfulModuleCatSheafCompHausCoherentTopologyConstantSheaf, CategoryTheory.Functor.PullbackObjObj.π_iso_of_iso_left_hom, CategoryTheory.instAdhesiveSheafOfHasPullbacksOfHasPushoutsOfHasSheafify, AlgebraicTopology.DoldKan.P_f_naturality, fintypeToFinBoolAlgOp_obj, PresheafOfModules.freeAdjunction_homEquiv, CategoryTheory.Pretriangulated.Opposite.rotate_distinguished_triangle, CategoryTheory.GrothendieckTopology.Point.skyscraperPresheafHomEquiv_app_π, SSet.Truncated.HomotopyCategory.descOfTruncation_obj_mk, SSet.S.IsUniquelyCodimOneFace.iff, CategoryTheory.Functor.leftOpRightOpEquiv_inverse_obj, CategoryTheory.sheafifyMap_sheafifyLift_assoc, CategoryTheory.Equivalence.op_counitIso, CategoryTheory.CostructuredArrow.well_copowered_costructuredArrow, AlgebraicTopology.DoldKan.MorphComponents.id_b, CategoryTheory.Presheaf.isLocallyInjective_forget, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.inv_invApp, CategoryTheory.SimplicialObject.Homotopy.postcomp_h, AlgebraicGeometry.isLocalization_basicOpen_of_qcqs, CategoryTheory.unop_sub, AlgebraicGeometry.Scheme.IdealSheafData.strictMono_ideal, SimplicialObject.Split.cofan_inj_naturality_symm, CommGrpCat.coyoneda_map_app, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.SheafCondition.map_f₃₄_op_glue, CategoryTheory.cocones_obj_obj, CategoryTheory.Comon.monoidal_tensorHom_hom, commBialgCatEquivComonCommAlgCat_functor_map_unop_hom, CategoryTheory.Limits.colimitYonedaHomIsoLimitRightOp_π_apply, CategoryTheory.WithInitial.opEquiv_unitIso_hom_app, CategoryTheory.Abelian.subobjectIsoSubobjectOp_symm_apply, CategoryTheory.GrothendieckTopology.plusMap_zero, CategoryTheory.Functor.RepresentableBy.ofIsoObj_homEquiv, CategoryTheory.Functor.op_additive, AlgebraicGeometry.Scheme.Opens.eq_presheaf_map_eqToHom, CategoryTheory.unop_hom_leftUnitor, CategoryTheory.LocalizerMorphism.hasRightResolutions_iff_op, CategoryTheory.Comma.unopFunctorCompFst_hom_app, CategoryTheory.isArtinianObject_iff_isEventuallyConstant, CategoryTheory.Sheaf.isConstant_iff_isIso_counit_app, CategoryTheory.Presheaf.tautologicalCocone_pt, CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence_functor, CategoryTheory.Pretriangulated.TriangleOpEquivalence.inverse_map, CategoryTheory.Functor.CorepresentableBy.coyoneda_homEquiv, AlgebraicGeometry.LocallyRingedSpace.evaluation_naturality_apply, AlgebraicTopology.DoldKan.factors_normalizedMooreComplex_PInfty, SimplexCategory.II_obj, CategoryTheory.Limits.reflectsColimits_of_unop, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_appLE, AlgebraicGeometry.IsAffineOpen.fromSpec_toSpecΓ, SSet.associator_inv_app_apply, AlgebraicGeometry.map_injective_of_isIntegral, CategoryTheory.Limits.end_.lift_π_assoc, CategoryTheory.GrothendieckTopology.pseudofunctorOver_toPrelaxFunctor_toPrelaxFunctorStruct_toPrefunctor_map_toFunctor_obj_obj_map, CategoryTheory.instIsLeftAdjointFunctorOppositeSheafPresheafToSheaf, CategoryTheory.Functor.map_distinguished_op_exact, AlgebraicGeometry.exists_eq_pow_mul_of_isCompact_of_isQuasiSeparated, AlgebraicGeometry.AffineSpace.homOfVector_appTop_coord, SSet.stdSimplex.nonDegenerateEquiv_symm_apply_coe, AlgebraicTopology.DoldKan.homotopyEquivNormalizedMooreComplexAlternatingFaceMapComplex_hom, CategoryTheory.GrothendieckTopology.isClosed_χ_app_apply_of_isSheaf_of_isSeparated, AlgebraicGeometry.sourceAffineLocally_morphismRestrict, PresheafOfModules.evaluation_obj, CategoryTheory.RanIsSheafOfIsCocontinuous.liftAux_map, CategoryTheory.OverPresheafAux.map_mkPrecomp_eqToHom, CategoryTheory.SimplicialObject.Homotopy.map_homology_eq, CategoryTheory.Limits.end_.condition_assoc, CategoryTheory.Presheaf.subsingleton_iff_isSeparatedFor, CategoryTheory.Sheaf.cartesianMonoidalCategoryWhiskerLeft_val, AlgebraicGeometry.Scheme.IsQuasiAffine.isBasis_basicOpen, AlgebraicGeometry.Proj.stalkIso'_symm_mk, SSet.horn.ι_ι, CategoryTheory.Sheaf.ΓRes_naturality, CategoryTheory.Presheaf.coherentExtensiveEquivalence_counitIso_hom_app_hom_app, SimplicialObject.Splitting.cofan_inj_comp_app_assoc, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_hom_app_eq, CategoryTheory.Subfunctor.Subpresheaf.IsGeneratedBy.ofSection_le, CategoryTheory.Limits.preservesFiniteColimits_of_rightOp, SheafOfModules.pushforwardNatIso_hom, CategoryTheory.Limits.preservesFiniteCoproducts_leftOp, AlgebraicGeometry.structureSheafInType.smul_apply, CategoryTheory.prodOpEquiv_counitIso_inv_app, CategoryTheory.unop_associator, CategoryTheory.GrothendieckTopology.diagramNatTrans_comp, CategoryTheory.uliftYonedaFunctor_preservesLimits, CategoryTheory.Functor.sheafPushforwardContinuousComp_hom_app_hom_app, AlgebraicTopology.DoldKan.N₁Γ₀_app, AlgebraicGeometry.Scheme.isoSpec_image_zeroLocus, CategoryTheory.Equalizer.Presieve.isSheafFor_singleton_iff_of_hasPullback, CategoryTheory.SimplicialObject.Homotopy.h_castSucc_comp_δ_succ_of_lt, SSet.Subcomplex.prod_le_prod_top, CategoryTheory.Limits.reflectsLimit_op, CategoryTheory.Limits.cokernelOrderHom_coe, CategoryTheory.Functor.IsRepresentedBy.iff_exists_representableBy, AlgebraicGeometry.PresheafedSpace.GlueData.π_ιInvApp_eq_id, CategoryTheory.Subfunctor.sieveOfSection_apply, CategoryTheory.Limits.preservesColimit_of_rightOp, CategoryTheory.ObjectProperty.essentiallySmall_op_iff, CategoryTheory.Equivalence.sheafCongr_unitIso, CategoryTheory.CategoryOfElements.costructuredArrowULiftYonedaEquivalence_inverse_map, CategoryTheory.hoFunctor.isIso_prodComparison, CategoryTheory.Functor.rightOp_map_unop, CategoryTheory.sheafifyLift_id_toSheafify_assoc, AlgebraicGeometry.Scheme.IdealSheafData.ideal_map_of_isAffineHom, CategoryTheory.Comon.MonOpOpToComonObj_comon_counit, AlgebraicGeometry.isCompact_and_isOpen_iff_finite_and_eq_biUnion_basicOpen, CategoryTheory.Pseudofunctor.DescentData'.descentDataEquivalence_counitIso, SSet.horn₃₁.ι₀_desc, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.app_inv_app'_assoc, LightProfinite.proj_comp_transitionMap', CategoryTheory.monoidalUnopUnop_η, CategoryTheory.preadditiveCoyoneda_obj, PresheafOfModules.pushforward_id_comp, CategoryTheory.cosimplicialSimplicialEquiv_functor_obj_map, CategoryTheory.DinatTrans.dinaturality, AlgebraicGeometry.IsAffineOpen.toSpecΓ_isoSpec_inv_assoc, CategoryTheory.GrothendieckTopology.Point.instPreservesFiniteColimitsSheafSheafFiber, AlgebraicGeometry.LocallyRingedSpace.SpecΓIdentity_inv_app, CategoryTheory.SimplicialThickening.functor_obj_as, CategoryTheory.GrothendieckTopology.OneHypercover.multiforkLift_map_assoc, CategoryTheory.Limits.piFunctor_obj_map, SSet.instIsRegularEpiCategory, CategoryTheory.GrothendieckTopology.toSheafify_sheafifyLift, Condensed.isSheafProfinite, CategoryTheory.classifier_isSheaf, CategoryTheory.representablyCoflat_op_iff, CategoryTheory.Functor.leftOpRightOpIso_inv_app, CategoryTheory.SimplicialObject.Homotopy.ToChainHomotopy.hom_eq, HomologicalComplex.unopFunctor_additive, CategoryTheory.Meq.condition, AlgebraicGeometry.Scheme.AffineZariskiSite.presieveOfSections_eq_ofArrows, CategoryTheory.Sheaf.sheafifyCocone_ι_app_val, CategoryTheory.Idempotents.DoldKan.N_map, CategoryTheory.Functor.instIsRepresentableObjOppositeTypeUliftYoneda, AlgebraicGeometry.Spec.toTop_obj_carrier, CategoryTheory.SimplicialObject.δ_comp_σ_succ_assoc, AlgebraicGeometry.Scheme.Hom.inv_app, CategoryTheory.NonemptyParallelPairPresentationAux.hg, SSet.RelativeMorphism.map_coe, PresheafOfModules.ofPresheaf_obj_isAddCommGroup, CategoryTheory.CommSq.LiftStruct.opEquiv_apply, AlgebraicGeometry.PresheafedSpace.GlueData.snd_invApp_t_app', CategoryTheory.Adjunction.unop_unit, CategoryTheory.Pretriangulated.shift_unop_opShiftFunctorEquivalence_counitIso_inv_app, CategoryTheory.Pretriangulated.shift_opShiftFunctorEquivalenceSymmHomEquiv_unop, CategoryTheory.WithTerminal.opEquiv_unitIso_inv_app, CategoryTheory.Functor.OneHypercoverDenseData.essSurj.restriction_eq_of_fac, CategoryTheory.Pseudofunctor.DescentData'.instIsIsoαCategoryObjLocallyDiscreteOppositeCatMkOpPullHom'Hom, CategoryTheory.Functor.pushforwardContinuousSheafificationCompatibility_hom_app_val, AlgebraicGeometry.RingedSpace.mem_basicOpen, CategoryTheory.Functor.IsLocalization.op, CategoryTheory.Limits.PushoutCocone.unop_snd, CategoryTheory.OverPresheafAux.unitForward_naturality₁, CategoryTheory.Square.op_f₃₄, CategoryTheory.isCoseparator_unop_iff, SSet.Truncated.HomotopyCategory.BinaryProduct.inverse_obj, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.coalgebraEquivalence_functor_obj_A, AlgebraicGeometry.Scheme.Hom.app_injective, CategoryTheory.equivYoneda'_inv_hom, CategoryTheory.Functor.mapPresheaf_obj_presheaf, CategoryTheory.Sheaf.isSeparated, CategoryTheory.ShortComplex.opcyclesOpIso_inv_naturality_assoc, SheafOfModules.pushforwardPushforwardEquivalence_counit_app_val_app, CategoryTheory.OverPresheafAux.counitAuxAux_hom, CategoryTheory.Pseudofunctor.DescentData'.comp_pullHom'_assoc, CategoryTheory.shrinkYonedaGrpObjObjEquiv_symm_comp, CategoryTheory.NatTrans.removeOp_app, CategoryTheory.GrothendieckTopology.mem_iff_isSheafFor_closedSieves, CategoryTheory.MorphismProperty.RightFraction.unop_Y', HomologicalComplex.unopEquivalence_counitIso, SSet.Subcomplex.N.notMem, AlgebraicGeometry.Scheme.IdealSheafData.subschemeFunctor_map, CategoryTheory.Limits.colimitCoyonedaHomIsoLimitLeftOp_π_apply, CategoryTheory.Functor.IsDenseSubsite.full_sheafPushforwardContinuous, CategoryTheory.Functor.IsCoverDense.restrictHomEquivHom_naturality_right, CategoryTheory.SimplicialObject.Homotopy.h_succ_comp_δ_castSucc_succ, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_inv_app, Condensed.id_val, TopCat.Presheaf.mono_iff_stalk_mono, AlgebraicGeometry.Scheme.Modules.inv_app, CategoryTheory.instIsIsoSSetProdComparisonCatCompNerveFunctorHoFunctorOf, SSet.PtSimplex.MulStruct.δ_succ_castSucc_map_assoc, AugmentedSimplexCategory.equivAugmentedSimplicialObject_inverse_obj_map, CategoryTheory.Functor.Elements.initialOfRepresentableBy_fst, CategoryTheory.Limits.colimitHomIsoLimitYoneda_inv_comp_π_assoc, Condensed.locallyConstantIsoFinYoneda_hom_app, AlgebraicTopology.DoldKan.hσ'_naturality, TopCat.Presheaf.ext_iff, AlgebraicGeometry.Scheme.Opens.toSpecΓ_preimage_zeroLocus, CategoryTheory.ParametrizedAdjunction.homEquiv_naturality_two_assoc, CategoryTheory.Functor.leibnizPullback_obj_obj, AlgebraicTopology.DoldKan.P_add_Q_f, PresheafOfModules.evaluation_map, CategoryTheory.linearCoyoneda_obj_obj_isModule, CategoryTheory.ShortComplex.opFunctor_map, SSet.horn.const_val_apply, CategoryTheory.yoneda_obj_isGeneratedBy, CategoryTheory.Limits.coend.ι_desc_assoc, Condensed.instPreservesFiniteProductsOppositeCompHausObjFunctorIsSheafCoherentTopology, CategoryTheory.Limits.coconeLeftOpOfConeEquiv_inverse_map, SSet.horn₃₁.desc.multicofork_π_three, CategoryTheory.MonoidalCategory.MonoidalLeftAction.oppositeLeftAction_actionObj_op, CategoryTheory.Pretriangulated.instIsTriangulatedOppositeUnopUnop, AlgebraicGeometry.AffineScheme.Spec_full, CategoryTheory.Functor.OneHypercoverDenseData.isSheaf_iff.lift_map, AlgebraicGeometry.nonempty_isColimit_Γ_mapCocone, CategoryTheory.nerveAdjunction.isIso_counit, AlgebraicTopology.DoldKan.hσ'_eq', CategoryTheory.GrothendieckTopology.map_yonedaULiftEquiv, HomologicalComplex.opInverse_map, CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheafObjObjIso.inv_restriction_assoc, CategoryTheory.MorphismProperty.RightFraction.op_s, SSet.StrictSegal.isRightKanExtension, CategoryTheory.GrothendieckTopology.Plus.toPlus_apply, CategoryTheory.Limits.limitUnopIsoUnopColimit_inv_comp_π, CategoryTheory.GrothendieckTopology.pointBot_fiber, CategoryTheory.Subfunctor.Subpresheaf.ofSection_eq_range, CategoryTheory.Presheaf.isLocallyInjective_iff_injective_of_separated, CommRingCat.preservesFilteredColimits_coyoneda, CategoryTheory.GrothendieckTopology.uliftYonedaOpCompCoyoneda_inv_app_app_hom_app, CategoryTheory.Limits.preservesColimitsOfShape_of_rightOp, SSet.horn₃₂.ι₃_desc_assoc, CategoryTheory.Limits.walkingParallelPairOpEquiv_counitIso_inv_app_op_one, CategoryTheory.Subfunctor.Subpresheaf.IsGeneratedBy.iSup_eq, SSet.Truncated.IsStrictSegal.segal, CategoryTheory.unop_whiskerLeft, AlgebraicTopology.DoldKan.HigherFacesVanish.induction, CategoryTheory.Limits.opHomCompWhiskeringLimYonedaIsoCocones_inv_app_app, AlgebraicTopology.DoldKan.Γ₀_obj_termwise_mapMono_comp_PInfty, CategoryTheory.GrothendieckTopology.plusMap_comp, PresheafOfModules.freeObjDesc_app, CategoryTheory.Presheaf.freeYonedaHomEquiv_symm_comp_assoc, AlgebraicGeometry.LocallyRingedSpace.comp_c_app, CategoryTheory.sheafSectionsNatIsoEvaluation_inv_app, CompHausLike.LocallyConstantModule.functorToPresheaves_obj_obj_isAddCommGroup, CategoryTheory.nerveMap_app_mk₀, CategoryTheory.OverPresheafAux.YonedaCollection.map₂_fst, CategoryTheory.Sheaf.IsConstant.mem_essImage, CategoryTheory.Functor.sheafPushforwardContinuousId_inv_app_hom_app, CategoryTheory.Presheaf.instIsIsoFunctorLeftKanExtensionUnitOppositeTypeUliftYoneda, TopCat.Presheaf.presheafEquivOfIso_functor_obj_map, CategoryTheory.NatTrans.unop_app, SSet.horn₂₁.sq, AlgebraicGeometry.LocallyRingedSpace.toΓSpec_preimage_basicOpen_eq, CategoryTheory.Subpresheaf.to_sheafifyLift, CategoryTheory.Limits.preservesLimits_leftOp, AlgebraicGeometry.StructureSheaf.algebraMap_germ_apply, CategoryTheory.SimplicialThickening.SimplicialCategory.id_comp, CategoryTheory.Functor.sheafAdjunctionCocontinuous_unit_app_val, CategoryTheory.unop_sum, SSet.Subcomplex.image_iSup, CategoryTheory.Functor.map_opShiftFunctorEquivalence_unitIso_hom_app_unop, CategoryTheory.Sieve.toUliftFunctor_app_down_coe, CategoryTheory.Functor.homObjFunctor_obj, CategoryTheory.Functor.op_obj, CategoryTheory.Limits.walkingParallelPairOpEquiv_counitIso_hom_app_op_zero, AlgebraicGeometry.Scheme.stalkMap_germ, Condensed.instAB4CondensedMod, CategoryTheory.Functor.OneHypercoverDenseData.isSheaf_iff.lift_map_assoc, CategoryTheory.yoneda'_obj_obj, CategoryTheory.MonoidalCategory.MonoidalLeftAction.oppositeLeftAction_actionUnitIso, CompHausLike.LocallyConstantModule.functor_map_hom_app_hom_apply_apply, CategoryTheory.Presieve.isSheafFor_ofArrows_iff_bijective_toCompabible, SheafOfModules.pushforward_obj_val, AlgebraicGeometry.Scheme.AffineZariskiSite.mem_grothendieckTopology_iff_sectionsOfPresieve, CategoryTheory.Equivalence.symmEquivInverse_obj_unitIso_hom, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_ι, CategoryTheory.Limits.coconeOpEquiv_functor_map_hom, CommRingCat.Opposite.effectiveEpi_of_faithfullyFlat, CategoryTheory.Functor.instInitialOppositeRightOpOfFinal, CategoryTheory.Under.opEquivOpOver_inverse_map, AlgebraicGeometry.Proj.fromOfGlobalSections_preimage_basicOpen, CategoryTheory.Limits.reflectsFiniteLimits_rightOp, Condensed.equalizerCondition_profinite, CategoryTheory.GrothendieckTopology.isIso_toPlus_of_isSheaf, AlgebraicGeometry.Scheme.LocalRepresentability.yonedaGluedToSheaf_app_toGlued, SimplicialObject.Splitting.cofan_inj_eq_assoc, CategoryTheory.SimplicialObject.whiskering_obj_obj_δ, PresheafOfModules.Sheafify.mul_smul, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_appTop, CategoryTheory.GrothendieckTopology.Point.skyscraperSheafAdjunction_homEquiv_apply_hom, CategoryTheory.ObjectProperty.isClosedUnderLimitsOfShape_iff_unop, AlgebraicGeometry.PresheafedSpace.restrictStalkIso_inv_eq_germ_assoc, SSet.instFinitePullback, CategoryTheory.GrothendieckTopology.Point.sheafFiberCompIso_inv_app, CategoryTheory.Limits.FormalCoproduct.evalOpCompInlIsoId_inv_app_app, CategoryTheory.Abelian.DoldKan.comparisonN_inv_app_f, SimplexCategory.II_δ, CategoryTheory.Limits.Cofork.op_ι, AlgebraicGeometry.Scheme.IdealSheafData.map_ideal_basicOpen, CommGrpCat.coyonedaType_obj_obj_coe, CategoryTheory.MorphismProperty.MapFactorizationData.op_p, CategoryTheory.isRegularMono_unop_iff_isRegularEpi, SSet.Truncated.HomotopyCategory.homToNerveMk_comp_assoc, Condensed.equalizerCondition, TopCat.Presheaf.presheafEquivOfIso_inverse_map_app, CategoryTheory.SimplicialThickening.functor_comp, AlgebraicGeometry.instIsIsoModulesSpecOfCarrierFromTildeΓFreeOpensCarrierCarrierCommRingCat, AlgebraicGeometry.RingedSpace.zeroLocus_singleton, TopCat.Presheaf.stalkFunctor_map_germ_assoc, CategoryTheory.Limits.Cocone.extensions_app, CategoryTheory.Limits.IndObjectPresentation.yoneda_F, CommRingCat.preservesColimit_coyoneda_of_finitePresentation, CategoryTheory.Functor.PullbackObjObj.ofHasPullback_fst, instPreservesColimitsOfShapeSheafTypeUliftYonedaOfPreservesLimitsOfShapeOppositeObjFunctorIsSheaf, CategoryTheory.Functor.IsRepresentedBy.representableBy_homEquiv_apply, PresheafOfModules.instAdditiveFunctorOppositeAbToPresheaf, CategoryTheory.Sheaf.tensorUnit_isSheaf, CategoryTheory.Limits.Types.surjective_π_app_zero_of_surjective_map_aux, CategoryTheory.LocalizerMorphism.LeftResolution.opEquivalence_functor, CategoryTheory.Subfunctor.eq_sheafify_iff, CategoryTheory.Adjunction.Quadruple.op_adj₃, CategoryTheory.MonoidalCategory.MonoidalLeftAction.leftActionOfOppositeLeftAction_actionHomLeft, CategoryTheory.Retract.op_r, CategoryTheory.Pseudofunctor.DescentData'.pullHom'_hom_comp_assoc, CategoryTheory.orderDualEquivalence_inverse_map, AlgebraicGeometry.essImage_Spec, CategoryTheory.Limits.widePullbackShapeOpEquiv_counitIso, TopCat.Presheaf.pullbackPushforwardAdjunction_unit_app_app_germToPullbackStalk, SSet.horn.primitiveTriangle_coe, CategoryTheory.Limits.endFunctor_map, CompHausLike.LocallyConstant.functorToPresheaves_obj_map, AlgebraicGeometry.PresheafedSpace.colimitCocone_ι_app_c, CategoryTheory.Limits.walkingParallelPairOp_right, CategoryTheory.WithTerminal.opEquiv_functor_map, CategoryTheory.Join.InclRightCompRightOpOpEquivFunctor_hom_app, PresheafOfModules.zero_app, AlgebraicGeometry.Scheme.Modules.pseudofunctor_mapId_inv_τl, instPreservesLimitsOfShapeSheafTypeYoneda, CategoryTheory.Limits.opCoproductIsoProduct_inv_comp_ι, CategoryTheory.CostructuredArrow.projectQuotient_mk, CategoryTheory.Pretriangulated.TriangleOpEquivalence.counitIso_inv_app_hom₁, AlgebraicGeometry.Scheme.comp_appLE_assoc, CategoryTheory.hasSeparator_op_iff, CategoryTheory.HasSubobjectClassifier.reflectsIsomorphismsOp, AlgebraicGeometry.Scheme.Hom.fromNormalization_app, CategoryTheory.Functor.mapTriangleOpCompTriangleOpEquivalenceFunctorApp_hom_hom₂, AlgebraicGeometry.Proj.awayToSection_comp_appLE_assoc, CategoryTheory.Limits.coneOfCoconeUnop_pt, CategoryTheory.Presheaf.restrictedULiftYonedaHomEquiv'_symm_naturality_right, CategoryTheory.nerve.instFaithfulCatTruncatedOfNatNatNerveFunctor₂, AlgebraicTopology.DoldKan.Γ₀.Obj.map_on_summand'_assoc, CategoryTheory.Sheaf.instHasFiniteProducts, PresheafOfModules.map_comp_assoc, CategoryTheory.Equalizer.Presieve.Arrows.w, AlgebraicGeometry.Scheme.Hom.eqToHom_app, CategoryTheory.CostructuredArrow.toStructuredArrow_map, CategoryTheory.yonedaMon_obj, TopCat.presheafToTypes_obj, CategoryTheory.Functor.unopComp_hom_app, PresheafOfModules.Sheafify.SMulCandidate.h, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_presheafFiberCompIso_hom_app, CategoryTheory.Limits.preservesLimits_rightOp, TopCat.Presheaf.isSheaf_iff_isTerminal_of_indiscrete, CategoryTheory.OverPresheafAux.counitForward_val_fst, CategoryTheory.isRegularEpi_op_iff_isRegularMono, CategoryTheory.Limits.hasPullback_op_iff_hasPushout, CategoryTheory.Limits.instHasCoproductOppositeOp, CategoryTheory.MonoidalCategory.MonoidalRightAction.rightActionOfOppositeRightAction_actionHomLeft_unop, CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily_apply, CategoryTheory.GrothendieckTopology.yonedaEquiv_yoneda_map, CategoryTheory.Pseudofunctor.DescentData'.id_hom, commBialgCatEquivComonCommAlgCat_counitIso_inv_app, CategoryTheory.Pseudofunctor.CoGrothendieck.compIso_hom_app, AlgebraicGeometry.Scheme.id_app, CategoryTheory.Pseudofunctor.DescentData.hom_comp, CategoryTheory.GrothendieckTopology.pseudofunctorOver_toPrelaxFunctor_toPrelaxFunctorStruct_toPrefunctor_map_toFunctor_map_hom_app, CategoryTheory.Limits.coneUnopOfCoconeEquiv_unitIso, SSet.stdSimplex.ι₀_whiskerLeft_toSSetObjI_μ, AlgebraicGeometry.StructureSheaf.instIsLocalizedModuleObjOppositeOpensCarrierTopObjFunctorTypeIsSheafGrothendieckTopologyStructureSheafInTypeOpBasicOpenPowersToOpenₗ, CategoryTheory.Pseudofunctor.IsPrestackFor.nonempty_fullyFaithful, PresheafOfModules.evaluation_preservesColimit, Alexandrov.projSup_map, CategoryTheory.LocalizerMorphism.instHasRightResolutionsOppositeOpOpOfHasLeftResolutions, CategoryTheory.GrothendieckTopology.plusCompIso_inv_eq_plusLift, CategoryTheory.Presheaf.FamilyOfElementsOnObjects.IsCompatible.existsUnique_section, AlgebraicGeometry.flat_iff, CategoryTheory.SimplicialObject.equivalenceLeftToRight_left_app, CategoryTheory.Limits.Fork.unop_π, CategoryTheory.MorphismProperty.RightFraction.op_Y', CategoryTheory.MonoidalCategory.DayConvolutionUnit.corepresentableByRight_homEquiv, TopCat.Presheaf.SheafConditionEqualizerProducts.res_π_apply, AlgebraicGeometry.Scheme.zeroLocus_map_of_eq, HomologicalComplex.opSymm_X, PresheafOfModules.forgetToPresheafModuleCat_map, AlgebraicGeometry.IsAffineOpen.toSpecΓ_fromSpec, CategoryTheory.Limits.coconeLeftOpOfConeEquiv_functor_obj, LightProfinite.Extend.functorOp_final, PresheafOfModules.toPresheaf_map_sheafificationHomEquiv, AlgebraicGeometry.Scheme.Modules.pseudofunctor_mapId_inv_τr, AlgebraicGeometry.Scheme.AffineZariskiSite.PreservesLocalization.isOpenImmersion, CategoryTheory.presheafHom_map_app_op_mk_id, CategoryTheory.Limits.preservesColimit_of_unop, Condensed.lanPresheafIso_hom, SSet.degenerate_iff_of_isIso, HomologicalComplex.instHasHomologyObjOppositeSymmUnopFunctorOp, Condensed.discrete_obj, AlgebraicGeometry.IsAffineOpen.fromSpec_primeIdealOf, CategoryTheory.GrothendieckTopology.uliftYonedaEquiv_comp, CategoryTheory.ShortComplex.RightHomologyMapData.unop_φH, CategoryTheory.Limits.IndObjectPresentation.yoneda_ℐ, CategoryTheory.cokernelOpOp_inv, SSet.Truncated.liftOfStrictSegal.spineEquiv_f₂_arrow_one, SSet.stdSimplex.face_singleton_compl, TopCat.Presheaf.sections_exact_of_exact, PresheafOfModules.instIsRightAdjointPushforwardIdFunctorOppositeRingCat, CategoryTheory.CategoryOfElements.fromCostructuredArrow_obj_fst, SSet.ι₀_app_fst, SSet.horn₃₂.ι₀_desc_assoc, CategoryTheory.Functor.IsRightAdjoint.rightOp, SSet.Truncated.spine_injective, CategoryTheory.Functor.pushforwardContinuousSheafificationCompatibility_hom_app_hom, CategoryTheory.Pseudofunctor.CoGrothendieck.Hom.ext_iff, CategoryTheory.Abelian.wellPowered_opposite, CategoryTheory.Limits.PushoutCocone.unop_fst, CategoryTheory.isIso_op_iff, AlgebraicGeometry.IsOpenImmersion.lift_app, CategoryTheory.RelCat.opEquivalence_unitIso, CategoryTheory.sheafComposeNatTrans_fac, CategoryTheory.toSheafify_naturality, CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence.functor_obj_left_left_as, CategoryTheory.Limits.limitOpIsoOpColimit_hom_comp_ι, CategoryTheory.Limits.widePushoutShapeOpEquiv_unitIso, CategoryTheory.ObjectProperty.IsConservativeFamilyOfPoints.jointlyReflectEpimorphisms, CategoryTheory.GrothendieckTopology.yonedaEquiv_apply, CategoryTheory.GrothendieckTopology.Point.toPresheafFiberMap_naturality, CategoryTheory.SimplicialObject.Truncated.whiskering_obj_map_app, AugmentedSimplexCategory.equivAugmentedSimplicialObject_functor_obj_left_obj, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_ι_assoc, germ_skyscraperPresheafStalkOfSpecializes_hom, AlgebraicGeometry.HasRingHomProperty.iff_of_isAffine, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.toBiprod_apply, CategoryTheory.MonoidalCategory.dayConvolutionInternalHomDiagramFunctor_obj_obj_obj_obj, SheafOfModules.Finite.evaluationPreservesFiniteLimits, CategoryTheory.shrinkYonedaEquiv_symm_map, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_η_assoc, CategoryTheory.SimplicialObject.augmentedCechNerve_obj_hom_app, SimplicialObject.Splitting.cofan_inj_epi_naturality, CategoryTheory.map_coyonedaEquiv, CategoryTheory.CommSq.HasLift.iff_unop, CategoryTheory.Limits.PushoutCocone.op_fst, CategoryTheory.Presieve.FamilyOfElements.map_id, CategoryTheory.ShortComplex.LeftHomologyData.op_ι, CategoryTheory.ShortComplex.RightHomologyData.op_π, AlgebraicGeometry.Scheme.IdealSheafData.le_def, AddCommGrpCat.coyonedaType_map_app, CategoryTheory.Over.opEquivOpUnder_functor_obj, SSet.Edge.toTruncated_edge, CategoryTheory.ObjectProperty.small_op_iff, CategoryTheory.instIsCardinalAccessibleObjOppositeFunctorTypeUliftCoyonedaOpOfIsCardinalPresentable, SSet.Truncated.Edge.CompStruct.idCompId_simplex, CategoryTheory.SimplicialObject.augmentedCechNerve_map_right, CategoryTheory.Yoneda.obj_map_id, CategoryTheory.Arrow.mapAugmentedCechNerve_right, TopCat.Presheaf.isSheaf_of_isOpenEmbedding, AlgebraicGeometry.Scheme.AffineZariskiSite.restrictIsoSpec_inv_app, CategoryTheory.Groupoid.invFunctor_obj, CategoryTheory.GrothendieckTopology.Point.skyscraperPresheafHomEquiv_naturality_left, AlgebraicGeometry.ΓSpec.toSpecΓ_unop, AlgebraicTopology.DoldKan.Γ₂N₂_inv, CategoryTheory.Sheaf.instIsMonoidalFunctorOppositeIsSheaf, CategoryTheory.Limits.widePullbackShapeOp_map, CategoryTheory.map_yonedaEquiv', CategoryTheory.opOpEquivalence_functor, SSet.stdSimplex.ι₁_whiskerLeft_toSSetObjI_μ_assoc, CategoryTheory.ShortComplex.quasiIso_opMap_iff, CategoryTheory.Limits.Cowedge.condition, CommMonCat.coyonedaType_map_app, AlgebraicGeometry.isIso_pushoutSection_of_isAffineOpen, SSet.range_eq_iSup_of_isColimit, CategoryTheory.CategoryOfElements.toCostructuredArrow_map, CategoryTheory.yoneda'_map_hom, CategoryTheory.MonoidalCategory.MonoidalRightAction.oppositeRightAction_actionUnitIso, CategoryTheory.shrinkYonedaGrp_map_app_shrinkYonedaObjObjEquiv_symm, CategoryTheory.Equivalence.sheafCongr.inverse_map_hom_app, CategoryTheory.ObjectProperty.isClosedUnderLimitsOfShape_iff_op, CategoryTheory.instIsRepresentableObjFunctorOppositeTypeShrinkYoneda, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.counit_assoc, PresheafOfModules.pushforward_comp_id, CategoryTheory.Pretriangulated.triangleOpEquivalence_inverse, CategoryTheory.CategoryOfElements.costructuredArrowYonedaEquivalenceFunctorProj_inv_app, CategoryTheory.Pretriangulated.shift_opShiftFunctorEquivalenceSymmHomEquiv_unop_assoc, CategoryTheory.Limits.walkingParallelPairOpEquiv_inverse, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_inv_naturality, CategoryTheory.MorphismProperty.isomorphisms_op, CategoryTheory.Pretriangulated.Opposite.isomorphic_distinguished, AlgebraicGeometry.AffineScheme.ΓIsEquiv, CategoryTheory.Idempotents.DoldKan.η_hom_app_f, AlgebraicTopology.normalizedMooreComplex_obj, CategoryTheory.GrothendieckTopology.yonedaEquiv_symm_naturality_right, CategoryTheory.Limits.π_comp_opProductIsoCoproduct_hom, CategoryTheory.Limits.walkingCospanOpEquiv_inverse_map, SSet.Subcomplex.preimage_iInf, SSet.Subcomplex.instIsIsoToRangeOfMono, CompHausLike.LocallyConstant.adjunction_unit, CategoryTheory.simplicialCosimplicialEquiv_functor_obj_map, CategoryTheory.instFullIndFunctorOppositeTypeInclusion, AlgebraicGeometry.SheafedSpace.Γ_map_op, CategoryTheory.Pretriangulated.preadditiveCoyoneda_homologySequenceδ_apply, SSet.S.le_def, CategoryTheory.SimplicialObject.Homotopy.h_succ_comp_δ_castSucc_of_lt_assoc, AlgebraicGeometry.tilde.instIsLocalizedModuleCarrierCarrierObjOppositeOpensCarrierCarrierCommRingCatSpecModuleCatPresheafModulesSheafModulesSpecToSheafOpBasicOpenPowersHomToOpen, CategoryTheory.StructuredArrow.toCostructuredArrow_obj, CategoryTheory.Functor.OneHypercoverDenseData.isSheaf_iff.liftAux_fac, AlgebraicGeometry.Scheme.isNilpotent_iff_basicOpen_eq_bot_of_isCompact, CategoryTheory.Limits.coneRightOpOfCoconeEquiv_counitIso, classifyingSpaceUniversalCover_obj, AlgebraicGeometry.SheafedSpace.GlueData.ι_jointly_surjective, SSet.ι₁_comp_assoc, CategoryTheory.Comon.ComonToMonOpOpObj_X, CategoryTheory.Presheaf.isLocallyInjective_comp, AlgebraicTopology.DoldKan.natTransQ_app, CategoryTheory.Limits.reflectsFiniteCoproducts_rightOp, CategoryTheory.Limits.hasInitial_op_of_hasTerminal, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.instIsIsoInvApp, CategoryTheory.Limits.preservesColimitsOfShape_of_unop, SSet.skeletonOfMono_succ, CategoryTheory.SimplicialObject.δ_comp_δ_self'_assoc, CategoryTheory.Square.unop_f₃₄, LightCondensed.instIsMonoidalFunctorOppositeLightProfiniteModuleCatWCoherentTopology, skyscraperPresheaf_map, CategoryTheory.Limits.preservesLimit_rightOp, AlgebraicGeometry.Scheme.IdealSheafData.supportSet_eq_iInter_zeroLocus, CategoryTheory.isSeparator_unop_iff, CategoryTheory.Functor.IsCoverDense.restrictHomEquivHom_naturality_left_symm, AlgebraicGeometry.IsAffineOpen.appLE_eq_away_map, AlgebraicTopology.DoldKan.Γ₂N₂.natTrans_app_f_app, CategoryTheory.ParametrizedAdjunction.homEquiv_naturality_three_assoc, StalkSkyscraperPresheafAdjunctionAuxs.toSkyscraperPresheaf_app, SSet.PtSimplex.RelStruct.δ_succ_map_assoc, AlgebraicGeometry.Scheme.Hom.appIso_inv_appLE, CategoryTheory.isCofiltered_op_iff_isFiltered, CategoryTheory.Pretriangulated.opShiftFunctorEquivalenceSymmHomEquiv_apply_assoc, AlgebraicGeometry.Scheme.Modules.map_smul, CategoryTheory.SimplicialObject.instHasColimits, CategoryTheory.PresheafOfGroups.Cochain₀.mul_apply, AlgebraicGeometry.RingedSpace.mem_basicOpen', CategoryTheory.ShortComplex.op_X₂, CategoryTheory.Functor.IsDenseSubsite.faithful_sheafPushforwardContinuous, CategoryTheory.SimplicialObject.equivalenceLeftToRight_right, SSet.opEquivalence_functor, CategoryTheory.Presheaf.isLocallySurjective_presheafToSheaf_map_iff, CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan.coconeApp_naturality_assoc, SheafOfModules.pushforwardNatTrans_comp, CategoryTheory.Limits.coconeLeftOpOfCone_pt, CategoryTheory.ShortComplex.leftHomologyMap'_op, CategoryTheory.shrinkYonedaEquiv_shrinkYoneda_map, CategoryTheory.ShortComplex.instHasRightHomologyOppositeOpOfHasLeftHomology, AlgebraicGeometry.Scheme.mem_basicOpen, CategoryTheory.enrichedNatTransYoneda_map_app, CategoryTheory.yonedaGrp_naturality, CategoryTheory.Idempotents.DoldKan.equivalence_unitIso, CategoryTheory.Sheaf.instEpiAppArrowILocallySurjectiveLocallyInjectiveFunctorialLocallySurjectiveInjectiveFactorization, CategoryTheory.WithTerminal.opEquiv_inverse_map, CategoryTheory.Functor.PullbackObjObj.isPullback, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.isoMk_hom_hom, Subobject.presheaf_map, CategoryTheory.constantPresheafAdj_unit_app, CategoryTheory.GrothendieckTopology.pseudofunctorOver_mapId_inv_toNatTrans_app_hom_app, CategoryTheory.Limits.limitRightOpIsoOpColimit_inv_comp_π, CategoryTheory.Limits.preservesColimitsOfSize_of_leftOp, CategoryTheory.Presheaf.freeYoneda_obj, CategoryTheory.Equivalence.symmEquivInverse_map_app, SSet.StrictSegal.spineToSimplex_vertex, AlgebraicGeometry.SheafedSpace.comp_c_app', CategoryTheory.Pretriangulated.shiftFunctorCompIsoId_op_inv_app_assoc, CategoryTheory.ShiftedHom.opEquiv'_add_symm, CategoryTheory.Functor.functorHomEquiv_symm_apply_app_app, SSet.Truncated.Edge.tgt_eq, SSet.stdSimplex.isoNerve_inv_app_apply, CategoryTheory.colimitYonedaHomEquiv_π_apply, CategoryTheory.Enriched.FunctorCategory.diagram_obj_obj, CategoryTheory.Sheaf.cartesianMonoidalCategoryWhiskerRight_hom, CategoryTheory.Presheaf.instIsLeftKanExtensionOppositeObjFunctorTypeUliftYonedaUliftYonedaMap, AlgebraicGeometry.SpecMap_ΓSpecIso_inv_toSpecΓ_assoc, CategoryTheory.GrothendieckTopology.ofArrows_mem_iff_isLocallySurjective_sigmaDesc_shrinkYoneda_map, CategoryTheory.ParametrizedAdjunction.whiskerLeft_map_counit_assoc, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv_apply, AlgebraicGeometry.PresheafedSpace.ofRestrict_c_app, SSet.Truncated.liftOfStrictSegal.spineEquiv_f₂_arrow_zero, PresheafOfModules.instPreservesFiniteLimitsSheafAddCommGrpCatCompSheafOfModulesSheafificationToSheaf, CategoryTheory.Limits.SequentialProduct.functorMap_commSq, CategoryTheory.instMonoSheafTypeImageι, CategoryTheory.Sheaf.isSheaf_yoneda_obj, AlgebraicGeometry.LocallyRingedSpace.toΓSpecCApp_iff, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.ofRestrict_invApp_apply, CategoryTheory.Sheaf.isTerminalTerminal_from_hom, CategoryTheory.Functor.IsCoverDense.full_sheafPushforwardContinuous, CategoryTheory.Functor.OneHypercoverDenseData.isSheaf_iff.fac, CategoryTheory.Comon.monoidal_tensorObj_comon_comul, CategoryTheory.uliftCoyonedaEquiv_comp, CategoryTheory.Functor.IsCoverDense.Types.naturality_assoc, CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinsetObj_map, CompHausLike.LocallyConstant.functorToPresheaves_obj_obj, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit, CategoryTheory.Limits.Cone.op_ι, CategoryTheory.Limits.hasColimit_op_of_hasLimit, CategoryTheory.yonedaEquiv_symm_app_apply, SSet.instFiniteInitial, AlgebraicGeometry.LocallyRingedSpace.Γ_obj, AlgebraicGeometry.Spec.sheafedSpaceObj_presheaf, SSet.Truncated.StrictSegal.spineToSimplex_map, CategoryTheory.MorphismProperty.ContainsIdentities.of_unop, AlgebraicGeometry.Spec.map_app, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.shortComplex_exact, CategoryTheory.preservesColimitsOfShape_of_isCardinalPresentable, SSet.PtSimplex.MulStruct.δ_castSucc_castSucc_map, CategoryTheory.instIsClosedUnderColimitsOfShapeFunctorOppositeTypeIsIndObjectOfIsFiltered, ContinuousMap.instPreservesFiniteProductsOppositeTopCatYonedaPresheaf', CategoryTheory.Limits.ι_comp_colimitUnopIsoOpLimit_hom, AlgebraicGeometry.exists_lift_of_germInjective, AlgebraicGeometry.Scheme.Spec_fromSpecStalk, CategoryTheory.Functor.op_map, CategoryTheory.Equivalence.sheafCongr.counitIso_hom_app_hom_app, PresheafOfModules.pushforward₀_obj_map, SSet.RelativeMorphism.Homotopy.rel_assoc, CategoryTheory.yonedaAddMon_map_app, CategoryTheory.NatTrans.unop_comp, SheafOfModules.unitHomEquiv_apply_coe, CategoryTheory.Abelian.IsGrothendieckAbelian.OppositeModuleEmbedding.preservesFiniteLimits_embedding, CategoryTheory.Limits.widePullbackShapeOpEquiv_unitIso, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.comp_hom, CategoryTheory.preadditiveYonedaObj_map, CategoryTheory.GrothendieckTopology.W.whiskerRight, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.coalgebraEquivalence_counitIso_inv_app_f, AlgebraicGeometry.IsAffineOpen.ideal_ext_iff, CategoryTheory.Presheaf.isSheaf_iff_isLimit_pretopology, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_hom_app_shift, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_inv_app_assoc, CategoryTheory.Limits.colimitHomIsoLimitYoneda'_inv_comp_π, CategoryTheory.sheafCompose_comp, CategoryTheory.whiskering_preadditiveCoyoneda, CategoryTheory.ParametrizedAdjunction.arrowHomEquiv_apply_right_snd_assoc, typeToBoolAlgOp_obj, CategoryTheory.ShiftedHom.opEquiv_symm_comp, CategoryTheory.Coyoneda.ULiftCoyoneda.instFaithfulOppositeFunctorTypeUliftCoyoneda, CategoryTheory.GrothendieckTopology.Point.presheafFiberCocone_pt, CategoryTheory.ShortComplex.cyclesOpIso_hom_naturality_assoc, CategoryTheory.Square.isPushout_iff_op_map_yoneda_isPullback, CategoryTheory.MonoidalCategory.dayConvolutionInternalHomDiagramFunctor_map_app_app_app, CategoryTheory.Pseudofunctor.DescentData.Hom.comm, CategoryTheory.Coyoneda.coyoneda_faithful, AlgebraicGeometry.tilde.toOpen_map_app, CategoryTheory.Limits.compYonedaSectionsEquiv_symm_apply_coe, CategoryTheory.Functor.PreOneHypercoverDenseData.multicospanMapIso_hom, CategoryTheory.yonedaEquiv_comp, CategoryTheory.OverPresheafAux.counitForward_naturality₂, CategoryTheory.ShortComplex.HomologyMapData.unop_left, AddCommGrpCat.coyoneda_obj_map, CategoryTheory.sheafifyLift_comp, SSet.skeleton_zero, CategoryTheory.Comon.ComonToMonOpOpObj_mon_one, AlgebraicGeometry.IsAffineOpen.primeIdealOf_genericPoint, ChainComplex.linearYonedaObj_X, CategoryTheory.Limits.parallelPairOpIso_hom_app_zero, CategoryTheory.Functor.IsCoverDense.Types.naturality, CategoryTheory.LocalizerMorphism.isLeftDerivabilityStructure_iff_op, AlgebraicGeometry.LocallyRingedSpace.stalkMap_germ, AlgebraicGeometry.IsAffineOpen.mem_ideal_iff, AlgebraicGeometry.Scheme.toSpecΓ_base, AlgebraicGeometry.Smooth.iff_forall_exists_isStandardSmooth

CategoryTheory.CategoryStruct

Definitions

Name	Category	Theorems
opposite 📖	CompOp	291 math math: CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_inv_app_eq, CategoryTheory.Pseudofunctor.DescentData.ofObj_hom, CategoryTheory.Pretriangulated.shiftFunctorCompIsoId_op_hom_app, CategoryTheory.Pretriangulated.shift_opShiftFunctorEquivalence_counitIso_inv_app, CategoryTheory.Pseudofunctor.DescentData'.comm, CategoryTheory.Pseudofunctor.isEquivalence_toDescentDataAsCoalgebra_iff_isEquivalence_comonadComparison, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_add_unitIso_hom_app_eq, CategoryTheory.ShortComplex.homologyOpIso_hom_naturality, CategoryTheory.Pseudofunctor.DescentData.pullFunctorCompIso_inv_app_hom, CategoryTheory.Pseudofunctor.DescentData'.comp_pullHom', CategoryTheory.op_comp, CategoryTheory.eqToHom_comp_homOfLE_op, CategoryTheory.WithInitial.opEquiv_unitIso_inv_app, CategoryTheory.Pseudofunctor.DescentData'.pullHom'_eq_pullHom_assoc, AlgebraicGeometry.Scheme.restrictFunctorΓ_inv_app, CategoryTheory.Pseudofunctor.DescentData'.instIsIsoαCategoryObjLocallyDiscreteOppositeCatMkOpPullbackHom, CategoryTheory.Limits.parallelPairOpIso_inv_app_zero, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.counit, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.coalgebraEquivalence_inverse_obj_hom, CategoryTheory.ObjectProperty.unop_ofObj, CategoryTheory.Pretriangulated.shiftFunctorZero_op_inv_app, CategoryTheory.Join.inclRightCompOpEquivInverse_inv_app_op, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.invApp_app_assoc, CategoryTheory.Limits.walkingParallelPairOpEquiv_counitIso_hom_app_op_one, CategoryTheory.Pretriangulated.shiftFunctorCompIsoId_op_inv_app, CategoryTheory.Pseudofunctor.DescentData.pullFunctor_map_hom, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.invApp_app_apply, CategoryTheory.Limits.walkingParallelPairOpEquiv_counitIso_inv_app_op_zero, CategoryTheory.Limits.desc_op_comp_opCoproductIsoProduct'_hom, TopCat.Presheaf.pushforwardToOfIso_app, CategoryTheory.Pseudofunctor.CoGrothendieck.map_map_fiber, CategoryTheory.Pseudofunctor.DescentData'.pullHom'_self', CategoryTheory.WithInitial.opEquiv_counitIso_hom_app, CategoryTheory.Pretriangulated.shiftFunctorAdd'_op_hom_app, CategoryTheory.Pseudofunctor.toDescentDataAsCoalgebraCompCoalgebraEquivalenceFunctorIso_inv_app_f, CategoryTheory.Pretriangulated.Opposite.OpOpCommShift.iso_inv_app, CategoryTheory.monoidalOpOp_δ, CategoryTheory.Pretriangulated.TriangleOpEquivalence.functor_map_hom₂, HomologicalComplex.opcyclesOpIso_inv_naturality, HomologicalComplex.extend.XOpIso_hom_d_op, CategoryTheory.Limits.parallelPairOpIso_inv_app_one, AlgebraicGeometry.PresheafedSpace.componentwiseDiagram_map, CategoryTheory.ShortComplex.op_pOpcycles_opcyclesOpIso_hom, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.invApp_app, CategoryTheory.WithTerminal.opEquiv_counitIso_inv_app, CategoryTheory.Pretriangulated.commShiftIso_opOp_inv_app, CategoryTheory.Limits.pushoutIsoUnopPullback_inv_snd, CategoryTheory.image_ι_op_comp_imageUnopOp_hom, CategoryTheory.Limits.pushoutIsoUnopPullback_inv_fst, CategoryTheory.Limits.opCoproductIsoProduct'_comp_self, PresheafOfModules.map_comp_apply, CategoryTheory.ShortComplex.leftHomologyMap_op, CategoryTheory.Pseudofunctor.toDescentDataAsCoalgebra_obj_obj, CategoryTheory.Limits.opCoproductIsoProduct'_hom_comp_proj, CategoryTheory.Functor.shift_map_op_assoc, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_hom_naturality, CategoryTheory.ShortComplex.homologyOpIso_inv_naturality, HomologicalComplex.cyclesOpIso_inv_naturality_assoc, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.coalgebraEquivalence_unitIso_inv_app_hom, CategoryTheory.Pretriangulated.commShiftIso_opOp_hom_app_assoc, AlgebraicGeometry.PresheafedSpace.GlueData.snd_invApp_t_app_assoc, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_inv_naturality, CategoryTheory.Pretriangulated.shiftFunctorAdd'_op_inv_app, CategoryTheory.ShortComplex.cyclesOpIso_hom_naturality, CategoryTheory.Pseudofunctor.DescentData.iso_hom, CategoryTheory.ObjectProperty.op_monotone_iff, CategoryTheory.Limits.pullbackIsoUnopPushout_hom_inl, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.coassoc_assoc, CategoryTheory.WithTerminal.opEquiv_unitIso_hom_app, CategoryTheory.WithTerminal.opEquiv_counitIso_hom_app, CategoryTheory.Pretriangulated.TriangleOpEquivalence.functor_map_hom₃, CategoryTheory.homOfLE_op_comp_eqToHom_assoc, CategoryTheory.Pseudofunctor.DescentData.iso_inv, CategoryTheory.Limits.parallelPairOpIso_hom_app_one, TopCat.Presheaf.pushforwardEq_hom_app, CategoryTheory.Limits.opCoproductIsoProduct_hom_comp_π, CategoryTheory.OverPresheafAux.YonedaCollection.map₂_snd, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.invApp_app_apply, CategoryTheory.ShortComplex.homologyMap'_op, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_hom_naturality_assoc, CategoryTheory.Limits.opParallelPairIso_hom_app_zero, CategoryTheory.ObjectProperty.op_singleton, CategoryTheory.Pseudofunctor.DescentData.instIsIsoαCategoryObjLocallyDiscreteOppositeCatMkOpHom, CategoryTheory.Pseudofunctor.toDescentDataAsCoalgebra_map_hom, CategoryTheory.OverPresheafAux.counitForward_val_snd, CategoryTheory.Limits.opParallelPairIso_hom_app_one, HomologicalComplex.opcyclesOpIso_hom_naturality_assoc, CategoryTheory.Limits.opCoproductIsoProduct'_inv_comp_inj, CategoryTheory.Pseudofunctor.DescentData'.comp_pullHom'', CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_hom_app, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.Hom.comm, CategoryTheory.cokernel.π_unop, HomologicalComplex.opcyclesOpIso_hom_toCycles_op, CategoryTheory.Pseudofunctor.DescentData.Hom.comm_assoc, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_zero_unitIso_inv_app, TopCat.Presheaf.presheafEquivOfIso_unitIso_hom_app_app, AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap_app, CategoryTheory.factorThruImage_comp_imageUnopOp_inv, CategoryTheory.Limits.limitRightOpIsoOpColimit_hom_comp_ι, CategoryTheory.ShortComplex.opcyclesOpIso_hom_naturality, CategoryTheory.unop_comp, CategoryTheory.Pseudofunctor.CoGrothendieck.Hom.congr, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.inv_invApp, finGaloisGroupMap.map_id, TopCat.Presheaf.whiskerIsoMapGenerateCocone_inv_hom, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_hom_app, CategoryTheory.op_id_unop, AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap_app', CategoryTheory.Limits.opProductIsoCoproduct'_comp_self, CategoryTheory.Pretriangulated.Opposite.OpOpCommShift.iso_hom_app_assoc, CategoryTheory.Pseudofunctor.toDescentDataAsCoalgebra_obj_hom, CategoryTheory.Pseudofunctor.DescentData.ofObj_obj, CategoryTheory.op_comp_assoc, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_zero_unitIso_hom_app, HomologicalComplex.homologyOp_hom_naturality_assoc, CategoryTheory.Pseudofunctor.DescentData.hom_comp_assoc, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.invApp_app, CategoryTheory.ShortComplex.cyclesOpIso_inv_naturality, CategoryTheory.kernel.ι_unop, CategoryTheory.ObjectProperty.op_ofObj, CategoryTheory.monoidalOpOp_μ, HomologicalComplex.cyclesOpIso_inv_naturality, AlgebraicGeometry.Scheme.restrictFunctorΓ_hom_app, HomologicalComplex.opcyclesOpIso_hom_naturality, CategoryTheory.Pseudofunctor.CoGrothendieck.ι_map_fiber, CategoryTheory.Pretriangulated.shiftFunctorZero_op_hom_app, CategoryTheory.Pretriangulated.opShiftFunctorEquivalenceSymmHomEquiv_apply, CategoryTheory.Pseudofunctor.DescentData'.pullHom'₁₃_eq_pullHom_of_chosenPullback₃_assoc, CategoryTheory.Limits.pullbackIsoOpPushout_inv_fst, AlgebraicGeometry.PresheafedSpace.GlueData.f_invApp_f_app_assoc, CategoryTheory.monoidalOpOp_η, CategoryTheory.Limits.pullbackIsoOpPushout_inv_snd, HomologicalComplex.extend_op_d, CategoryTheory.Pseudofunctor.DescentData.pullFunctorObj_obj, CategoryTheory.Limits.π_comp_colimitOpIsoOpLimit_inv, CochainComplex.homotopyOp_hom_eq, HomologicalComplex.cyclesOpIso_hom_naturality_assoc, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.coalgebraEquivalence_functor_obj_a, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_inv_naturality_assoc, CategoryTheory.Join.inclRightCompOpEquivInverse_hom_app_op, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.coalgebraEquivalence_inverse_map_hom, CategoryTheory.Functor.op_commShiftIso_inv_app, PresheafOfModules.map_comp, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.coalgebraEquivalence_counitIso_hom_app_f, AlgebraicGeometry.PresheafedSpace.GlueData.f_invApp_f_app, AugmentedSimplexCategory.equivAugmentedSimplicialObject_functor_obj_hom_app, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_hom_naturality_assoc, CategoryTheory.Functor.const.opObjOp_hom_app, CategoryTheory.ShortComplex.opcyclesOpIso_inv_naturality, CategoryTheory.Pretriangulated.Opposite.OpOpCommShift.iso_inv_app_assoc, Alexandrov.lowerCone_π_app, CategoryTheory.Limits.ι_comp_colimitRightOpIsoUnopLimit_hom, CategoryTheory.WithInitial.opEquiv_counitIso_inv_app, CategoryTheory.Pretriangulated.TriangleOpEquivalence.functor_obj, CategoryTheory.Join.inclLeftCompOpEquivInverse_inv_app_op, AlgebraicGeometry.IsAffineOpen.fromSpec_app_self_apply, HomologicalComplex.cyclesOpIso_hom_naturality, CategoryTheory.eqToHom_op, TopCat.Presheaf.presheafEquivOfIso_unitIso_inv_app_app, CategoryTheory.Functor.const.opObjOp_inv_app, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_hom_naturality, CategoryTheory.Pseudofunctor.toDescentDataAsCoalgebraCompCoalgebraEquivalenceFunctorIso_hom_app_f, TopCat.Presheaf.whiskerIsoMapGenerateCocone_hom_hom, CategoryTheory.monoidalOpOp_ε, CategoryTheory.ShortComplex.rightHomologyMap_op, ContinuousMap.piComparison_fac, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.invApp_app_apply, CategoryTheory.ObjectProperty.op_monotone, AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap_app_assoc, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_hom_app_assoc, CategoryTheory.yonedaPairing_map, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_inv_app, CategoryTheory.Pseudofunctor.DescentData'.Hom.comm, AlgebraicGeometry.PresheafedSpace.GlueData.snd_invApp_t_app, CategoryTheory.ObjectProperty.unop_monotone_iff, CategoryTheory.OverPresheafAux.YonedaCollection.mk_snd, CategoryTheory.Join.inclLeftCompOpEquivInverse_hom_app_op, CategoryTheory.unop_comp_assoc, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.coalgebraEquivalence_unitIso_hom_app_hom, CategoryTheory.ShiftedHom.opEquiv'_symm_op_opShiftFunctorEquivalence_counitIso_inv_app_op_shift, CategoryTheory.Functor.shift_map_op, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.coalgebraEquivalence_inverse_obj_obj, CategoryTheory.Limits.ι_comp_colimitOpIsoOpLimit_hom, CategoryTheory.Pretriangulated.shiftFunctor_op_map, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.invApp_app, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_add_unitIso_inv_app_eq, CategoryTheory.Pretriangulated.commShiftIso_opOp_hom_app, CategoryTheory.unop_id_op, CategoryTheory.Limits.pushout.op_codiagonal, CategoryTheory.imageUnopOp_hom_comp_image_ι, CategoryTheory.Limits.desc_op_comp_opCoproductIsoProduct_hom, CategoryTheory.Pseudofunctor.toDescentData_map_hom, CategoryTheory.homOfLE_op_comp_eqToHom, CategoryTheory.Limits.π_comp_colimitRightOpIsoUnopLimit_inv, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.invApp_app_assoc, CategoryTheory.Pseudofunctor.DescentData'.pullHom'_hom_self, CategoryTheory.Pseudofunctor.DescentData.pullFunctorObjHom_eq_assoc, TopCat.Presheaf.presheafEquivOfIso_counitIso_inv_app_app, CategoryTheory.Pseudofunctor.DescentData'.Hom.comm_assoc, CategoryTheory.ShiftedHom.opEquiv'_symm_comp, CategoryTheory.Pretriangulated.commShiftIso_opOp_inv_app_assoc, CategoryTheory.Pseudofunctor.DescentData'.pullHom'_hom_comp, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.coalgebraEquivalence_functor_map_f, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_hom_app_assoc, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_inv_app_assoc, CategoryTheory.MorphismProperty.instRespectsLeftOppositeOpOfRespectsRight, PresheafOfModules.map_id, CategoryTheory.ShortComplex.homologyMap_op, CategoryTheory.Pseudofunctor.DescentData'.comm_assoc, CategoryTheory.Limits.limitOpIsoOpColimit_inv_comp_π, CategoryTheory.Pseudofunctor.DescentData'.pullHom'_self, CategoryTheory.yonedaEvaluation_map_down, CategoryTheory.Limits.fst_opProdIsoCoprod_hom, CategoryTheory.MorphismProperty.instRespectsRightOppositeOpOfRespectsLeft, AlgebraicGeometry.PresheafedSpace.congr_app, TopCat.Presheaf.presheafEquivOfIso_counitIso_hom_app_app, CategoryTheory.Pseudofunctor.LocallyDiscreteOpToCat.map_eq_pullHom_assoc, AlgebraicGeometry.PresheafedSpace.id_c_app, CategoryTheory.Limits.opProductIsoCoproduct'_inv_comp_lift, HomologicalComplex.fromOpcycles_op_cyclesOpIso_inv, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.coassoc, AugmentedSimplexCategory.equivAugmentedSimplicialObject_functor_obj_left_map, CategoryTheory.Limits.proj_comp_opProductIsoCoproduct'_hom, CategoryTheory.Pseudofunctor.LocallyDiscreteOpToCat.map_eq_pullHom, CompHausLike.LocallyConstant.incl_comap, CategoryTheory.Pseudofunctor.DescentData.pullFunctorCompIso_hom_app_hom, AlgebraicGeometry.SheafedSpace.congr_hom_app, CategoryTheory.Pseudofunctor.DescentData.pullFunctorObjHom_eq, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.Hom.comm_assoc, AlgebraicGeometry.PresheafedSpace.GlueData.ιInvApp_π, CategoryTheory.eqToHom_comp_homOfLE_op_assoc, CategoryTheory.Pretriangulated.TriangleOpEquivalence.functor_map_hom₁, CategoryTheory.Pseudofunctor.DescentData.hom_self, CategoryTheory.ShortComplex.opcyclesOpIso_hom_toCycles_op, CategoryTheory.Pseudofunctor.CoGrothendieck.categoryStruct_comp_fiber, CategoryTheory.Pseudofunctor.DescentData'.comp_pullHom''_assoc, CategoryTheory.Functor.op_commShiftIso_hom_app, CategoryTheory.Pretriangulated.Opposite.OpOpCommShift.iso_hom_app, CategoryTheory.Pseudofunctor.DescentData'.pullHom'₁₂_eq_pullHom_of_chosenPullback₃_assoc, CategoryTheory.Limits.opProductIsoCoproduct_inv_comp_lift, TopCat.Presheaf.toPushforwardOfIso_app, CategoryTheory.ShortComplex.cyclesOpIso_inv_op_iCycles, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.inv_invApp, CategoryTheory.OverPresheafAux.YonedaCollection.map₁_snd, CategoryTheory.WithInitial.opEquiv_inverse_map, CategoryTheory.Limits.opParallelPairIso_inv_app_zero, CategoryTheory.Limits.opParallelPairIso_inv_app_one, AlgebraicGeometry.SheafedSpace.congr_app, CategoryTheory.Limits.snd_opProdIsoCoprod_hom, CategoryTheory.imageUnopOp_inv_comp_op_factorThruImage, CategoryTheory.Pseudofunctor.DescentData'.pullHom'₂₃_eq_pullHom_of_chosenPullback₃_assoc, CategoryTheory.Pretriangulated.Opposite.complete_distinguished_triangle_morphism, CategoryTheory.Pseudofunctor.DescentData'.pullHom_pullHom'_assoc, HomologicalComplex.homologyOp_hom_naturality, CategoryTheory.ShortComplex.fromOpcycles_op_cyclesOpIso_inv, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.invApp_app_assoc, CategoryTheory.Limits.pullbackIsoUnopPushout_hom_inr, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.inv_invApp, finGaloisGroupMap.map_comp, CategoryTheory.WithInitial.opEquiv_unitIso_hom_app, CategoryTheory.OverPresheafAux.map_mkPrecomp_eqToHom, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_hom_app_eq, AlgebraicGeometry.PresheafedSpace.GlueData.snd_invApp_t_app', CategoryTheory.WithTerminal.opEquiv_unitIso_inv_app, CategoryTheory.Pseudofunctor.DescentData'.instIsIsoαCategoryObjLocallyDiscreteOppositeCatMkOpPullHom'Hom, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.coalgebraEquivalence_functor_obj_A, CategoryTheory.Pseudofunctor.DescentData'.comp_pullHom'_assoc, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_inv_app, CategoryTheory.Limits.walkingParallelPairOpEquiv_counitIso_inv_app_op_one, CategoryTheory.eqToHom_unop, CategoryTheory.Limits.walkingParallelPairOpEquiv_counitIso_hom_app_op_zero, CategoryTheory.Pseudofunctor.DescentData'.pullHom'_hom_comp_assoc, CategoryTheory.op_comp_unop, CategoryTheory.Limits.opCoproductIsoProduct_inv_comp_ι, PresheafOfModules.map_comp_assoc, CategoryTheory.Pseudofunctor.DescentData.hom_comp, CategoryTheory.Pseudofunctor.CoGrothendieck.Hom.ext_iff, CategoryTheory.Limits.limitOpIsoOpColimit_hom_comp_ι, CategoryTheory.unop_id, CategoryTheory.op_id, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.counit_assoc, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_inv_naturality, CategoryTheory.Limits.π_comp_opProductIsoCoproduct_hom, CategoryTheory.ObjectProperty.unop_singleton, CategoryTheory.WithTerminal.opEquiv_inverse_map, CategoryTheory.Limits.limitRightOpIsoOpColimit_inv_comp_π, CategoryTheory.ShiftedHom.opEquiv'_add_symm, CategoryTheory.Pseudofunctor.DescentDataAsCoalgebra.coalgebraEquivalence_counitIso_inv_app_f, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_inv_app_assoc, CategoryTheory.Pseudofunctor.DescentData.Hom.comm, CategoryTheory.Limits.parallelPairOpIso_hom_app_zero

CategoryTheory.Equivalence

Definitions

Name	Category	Theorems
leftOp 📖	CompOp	8 math math: leftOp_unitIso_hom_app, leftOp_unitIso_inv_app, leftOp_functor_obj, leftOp_counitIso_inv_app, leftOp_inverse_map, leftOp_counitIso_hom_app, leftOp_inverse_obj, leftOp_functor_map
op 📖	CompOp	26 math math: leftOp_unitIso_hom_app, CategoryTheory.ComposableArrows.opEquivalence_counitIso_inv_app_app, rightOp_counitIso_inv_app, leftOp_unitIso_inv_app, leftOp_counitIso_inv_app, CategoryTheory.ComposableArrows.opEquivalence_unitIso_inv_app, rightOp_unitIso_hom_app, TopCat.Presheaf.presheafEquivOfIso_unitIso_hom_app_app, TopCat.Presheaf.whiskerIsoMapGenerateCocone_inv_hom, CategoryTheory.ComposableArrows.opEquivalence_functor_map_app, CategoryTheory.ComposableArrows.opEquivalence_unitIso_hom_app, TopCat.Presheaf.presheafEquivOfIso_unitIso_inv_app_app, op_functor, TopCat.Presheaf.whiskerIsoMapGenerateCocone_hom_hom, rightOp_counitIso_hom_app, TopCat.Presheaf.presheafEquivOfIso_counitIso_inv_app_app, rightOp_unitIso_inv_app, TopCat.Presheaf.presheafEquivOfIso_counitIso_hom_app_app, leftOp_counitIso_hom_app, CategoryTheory.regularTopology.equalizerCondition_iff_of_equivalence, TopCat.Presheaf.presheafEquivOfIso_functor_map_app, op_unitIso, op_inverse, CategoryTheory.ComposableArrows.opEquivalence_counitIso_hom_app_app, op_counitIso, TopCat.Presheaf.presheafEquivOfIso_inverse_map_app
rightOp 📖	CompOp	8 math math: rightOp_counitIso_inv_app, rightOp_functor_map, rightOp_inverse_map, rightOp_unitIso_hom_app, rightOp_inverse_obj, rightOp_functor_obj, rightOp_counitIso_hom_app, rightOp_unitIso_inv_app
unop 📖	CompOp	4 math math: unop_counitIso, unop_inverse, unop_unitIso, unop_functor

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
leftOp_counitIso_hom_app 📖	mathematical	—	CategoryTheory.NatTrans.app CategoryTheory.Functor.comp Opposite CategoryTheory.Category.opposite inverse CategoryTheory.opOpEquivalence op functor CategoryTheory.Functor.id CategoryTheory.Iso.hom CategoryTheory.Functor CategoryTheory.Functor.category counitIso leftOp Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct Opposite.op CategoryTheory.Functor.obj CategoryTheory.Iso.inv	—	CategoryTheory.Category.comp_id CategoryTheory.Category.id_comp
leftOp_counitIso_inv_app 📖	mathematical	—	CategoryTheory.NatTrans.app CategoryTheory.Functor.id CategoryTheory.Functor.comp Opposite CategoryTheory.Category.opposite inverse CategoryTheory.opOpEquivalence op functor CategoryTheory.Iso.inv CategoryTheory.Functor CategoryTheory.Functor.category counitIso leftOp Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.obj Opposite.op CategoryTheory.Iso.hom	—	CategoryTheory.Category.id_comp CategoryTheory.Category.comp_id
leftOp_functor_map 📖	mathematical	—	CategoryTheory.Functor.map Opposite CategoryTheory.Category.opposite functor leftOp Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.obj Opposite.unop	—	—
leftOp_functor_obj 📖	mathematical	—	CategoryTheory.Functor.obj Opposite CategoryTheory.Category.opposite functor leftOp Opposite.unop	—	—
leftOp_inverse_map 📖	mathematical	—	CategoryTheory.Functor.map Opposite CategoryTheory.Category.opposite inverse leftOp Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.obj Opposite.op	—	—
leftOp_inverse_obj 📖	mathematical	—	CategoryTheory.Functor.obj Opposite CategoryTheory.Category.opposite inverse leftOp Opposite.op	—	—
leftOp_unitIso_hom_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.id CategoryTheory.Functor.comp functor op CategoryTheory.opOpEquivalence inverse CategoryTheory.Iso.hom CategoryTheory.Functor CategoryTheory.Functor.category unitIso leftOp Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.obj Opposite.unop CategoryTheory.Iso.inv	—	CategoryTheory.Category.comp_id CategoryTheory.Functor.map_id
leftOp_unitIso_inv_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.comp functor op CategoryTheory.opOpEquivalence inverse CategoryTheory.Functor.id CategoryTheory.Iso.inv CategoryTheory.Functor CategoryTheory.Functor.category unitIso leftOp Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct Opposite.unop CategoryTheory.Functor.obj CategoryTheory.Iso.hom	—	CategoryTheory.Category.comp_id CategoryTheory.Functor.map_id CategoryTheory.Category.id_comp
op_counitIso 📖	mathematical	—	counitIso Opposite CategoryTheory.Category.opposite op CategoryTheory.Iso.symm CategoryTheory.Functor CategoryTheory.Functor.category CategoryTheory.Functor.op CategoryTheory.Functor.id CategoryTheory.Functor.comp inverse functor CategoryTheory.NatIso.op	—	—
op_functor 📖	mathematical	—	functor Opposite CategoryTheory.Category.opposite op CategoryTheory.Functor.op	—	—
op_inverse 📖	mathematical	—	inverse Opposite CategoryTheory.Category.opposite op CategoryTheory.Functor.op	—	—
op_unitIso 📖	mathematical	—	unitIso Opposite CategoryTheory.Category.opposite op CategoryTheory.Iso.symm CategoryTheory.Functor CategoryTheory.Functor.category CategoryTheory.Functor.op CategoryTheory.Functor.comp functor inverse CategoryTheory.Functor.id CategoryTheory.NatIso.op	—	—
rightOp_counitIso_hom_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.comp inverse op symm CategoryTheory.opOpEquivalence functor CategoryTheory.Functor.id CategoryTheory.Iso.hom CategoryTheory.Functor CategoryTheory.Functor.category counitIso rightOp Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct Opposite.unop CategoryTheory.Functor.obj CategoryTheory.Iso.inv	—	CategoryTheory.Category.comp_id CategoryTheory.Functor.map_id CategoryTheory.Category.id_comp
rightOp_counitIso_inv_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.id CategoryTheory.Functor.comp inverse op symm CategoryTheory.opOpEquivalence functor CategoryTheory.Iso.inv CategoryTheory.Functor CategoryTheory.Functor.category counitIso rightOp Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.obj Opposite.unop CategoryTheory.Iso.hom	—	CategoryTheory.Category.comp_id CategoryTheory.Functor.map_id
rightOp_functor_map 📖	mathematical	—	CategoryTheory.Functor.map Opposite CategoryTheory.Category.opposite functor rightOp Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.obj Opposite.op	—	—
rightOp_functor_obj 📖	mathematical	—	CategoryTheory.Functor.obj Opposite CategoryTheory.Category.opposite functor rightOp Opposite.op	—	—
rightOp_inverse_map 📖	mathematical	—	CategoryTheory.Functor.map Opposite CategoryTheory.Category.opposite inverse rightOp Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.obj Opposite.unop	—	—
rightOp_inverse_obj 📖	mathematical	—	CategoryTheory.Functor.obj Opposite CategoryTheory.Category.opposite inverse rightOp Opposite.unop	—	—
rightOp_unitIso_hom_app 📖	mathematical	—	CategoryTheory.NatTrans.app CategoryTheory.Functor.id CategoryTheory.Functor.comp Opposite CategoryTheory.Category.opposite functor symm CategoryTheory.opOpEquivalence op inverse CategoryTheory.Iso.hom CategoryTheory.Functor CategoryTheory.Functor.category unitIso rightOp Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.obj Opposite.op CategoryTheory.Iso.inv	—	CategoryTheory.Category.comp_id CategoryTheory.Category.id_comp
rightOp_unitIso_inv_app 📖	mathematical	—	CategoryTheory.NatTrans.app CategoryTheory.Functor.comp Opposite CategoryTheory.Category.opposite functor symm CategoryTheory.opOpEquivalence op inverse CategoryTheory.Functor.id CategoryTheory.Iso.inv CategoryTheory.Functor CategoryTheory.Functor.category unitIso rightOp Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct Opposite.op CategoryTheory.Functor.obj CategoryTheory.Iso.hom	—	CategoryTheory.Category.id_comp CategoryTheory.Category.comp_id
unop_counitIso 📖	mathematical	—	counitIso unop CategoryTheory.Iso.symm CategoryTheory.Functor CategoryTheory.Functor.category CategoryTheory.Functor.unop CategoryTheory.Functor.id Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.comp inverse functor CategoryTheory.NatIso.unop	—	—
unop_functor 📖	mathematical	—	functor unop CategoryTheory.Functor.unop Opposite CategoryTheory.Category.opposite	—	—
unop_inverse 📖	mathematical	—	inverse unop CategoryTheory.Functor.unop Opposite CategoryTheory.Category.opposite	—	—
unop_unitIso 📖	mathematical	—	unitIso unop CategoryTheory.Iso.symm CategoryTheory.Functor CategoryTheory.Functor.category CategoryTheory.Functor.unop CategoryTheory.Functor.comp Opposite CategoryTheory.Category.opposite functor inverse CategoryTheory.Functor.id CategoryTheory.NatIso.unop	—	—

CategoryTheory.Functor

Definitions

Name	Category	Theorems
leftOp 📖	CompOp	136 math math: CategoryTheory.Adjunction.leftOp_unit, CategoryTheory.Limits.reflectsLimitsOfShape_leftOp, CategoryTheory.CategoryOfElements.costructuredArrowYonedaEquivalenceInverseπ_hom_app, CategoryTheory.Limits.coconeLeftOpOfConeEquiv_functor_map_hom, CategoryTheory.Limits.cospanUnop_hom_app, CategoryTheory.NatTrans.removeLeftOp_id, leftOpRightOpEquiv_counitIso_inv_app_app, CategoryTheory.simplicialCosimplicialEquiv_counitIso_inv_app_app, CategoryTheory.simplicialCosimplicialEquiv_inverse_map, CategoryTheory.Limits.preservesLimitsOfShape_leftOp, CategoryTheory.Limits.reflectsColimitsOfShape_leftOp, CategoryTheory.Limits.hasLimit_leftOp_of_hasColimit, CategoryTheory.NatTrans.leftOp_app, CategoryTheory.Comma.opEquiv_counitIso, CategoryTheory.Limits.isColimitCoconeLeftOpOfCone_desc, CategoryTheory.Limits.coconeOfConeLeftOp_pt, CategoryTheory.Limits.reflectsColimits_leftOp, CategoryTheory.Limits.colimitYonedaHomIsoLimit'_π_apply, CategoryTheory.Limits.π_comp_colimitLeftOpIsoUnopLimit_inv, CategoryTheory.Limits.reflectsFiniteCoproducts_leftOp, leftOp_obj, rightOpLeftOpIso_hom_app, CategoryTheory.Limits.reflectsFiniteColimits_leftOp, CategoryTheory.Limits.isLimitOfCoconeOfConeLeftOp_lift, CategoryTheory.Limits.spanUnop_inv_app, leftOp_faithful, CategoryTheory.Limits.coconeLeftOpOfConeEquiv_inverse_obj, const.opObjUnop_hom_app, instInitialOppositeLeftOpOfFinal, leftOpId_hom_app, CategoryTheory.Limits.preservesColimit_leftOp, leftOpRightOpEquiv_unitIso_inv_app, CategoryTheory.ComposableArrows.opEquivalence_unitIso_inv_app, CategoryTheory.NatTrans.removeLeftOp_app, CategoryTheory.Limits.preservesLimit_leftOp, CategoryTheory.NatTrans.leftOp_id, CategoryTheory.Limits.reflectsLimits_leftOp, CategoryTheory.Limits.reflectsLimit_leftOp, CategoryTheory.Limits.cospanUnop_inv_app, CategoryTheory.simplicialCosimplicialEquiv_unitIso_hom_app, CategoryTheory.Limits.preservesLimitsOfSize_leftOp, CategoryTheory.Limits.coneOfCoconeLeftOp_π_app, CategoryTheory.Limits.isColimitOfConeOfCoconeLeftOp_desc, leftOpRightOpIso_hom_app, CategoryTheory.Limits.spanUnop_hom_app, CategoryTheory.simplicialCosimplicialEquiv_counitIso_hom_app_app, CategoryTheory.Limits.π_comp_colimitLeftOpIsoUnopLimit_inv_assoc, CategoryTheory.Limits.preservesFiniteLimits_leftOp, AlgebraicGeometry.PresheafedSpace.ColimitCoconeIsColimit.desc_c_naturality, CategoryTheory.Limits.isLimitOfCoconeLeftOpOfCone_lift, CategoryTheory.Limits.preservesFiniteColimits_leftOp, CategoryTheory.ComposableArrows.opEquivalence_inverse_obj, CategoryTheory.Limits.hasLimit_leftOp_iff_hasColimit, CategoryTheory.Limits.preservesFiniteProducts_leftOp, CategoryTheory.Limits.coneLeftOpOfCoconeEquiv_inverse_obj, CategoryTheory.Limits.coneOfCoconeLeftOp_pt, CategoryTheory.Limits.reflectsFiniteLimits_leftOp, rightOpLeftOpIso_inv_app, CategoryTheory.Limits.hasColimit_leftOp_iff_hasLimit, rightOp_leftOp_eq, CategoryTheory.ComposableArrows.opEquivalence_unitIso_hom_app, CategoryTheory.MorphismProperty.IsInvertedBy.leftOp, leftOpId_inv_app, CategoryTheory.Limits.preservesColimits_leftOp, leftOpRightOpEquiv_inverse_map, leftOpRightOpEquiv_unitIso_hom_app, IsLeftAdjoint.leftOp, leftOp_full, leftOp_additive, CategoryTheory.Limits.reflectsColimit_leftOp, CategoryTheory.Comma.opFunctorCompSnd_hom_app, CategoryTheory.Limits.coneLeftOpOfCocone_π_app, AlgebraicGeometry.PresheafedSpace.colimit_presheaf, CategoryTheory.Adjunction.leftOp_counit, CategoryTheory.Limits.preservesColimitsOfSize_leftOp, leftOpComp_inv_app, CategoryTheory.Comma.opFunctorCompFst_hom_app, CategoryTheory.Comma.opFunctorCompFst_inv_app, IsRightAdjoint.leftOp, CategoryTheory.simplicialCosimplicialEquiv_inverse_obj, CategoryTheory.Limits.coconeLeftOpOfConeEquiv_unitIso, leftOpRightOpEquiv_counitIso_hom_app_app, CategoryTheory.Limits.isColimitOfConeLeftOpOfCocone_desc, CategoryTheory.Adjunction.leftOp_eq, CategoryTheory.Limits.coneLeftOpOfCocone_pt, const.opObjUnop_inv_app, CategoryTheory.Comma.opEquiv_unitIso, CategoryTheory.Limits.hasColimit_leftOp_of_hasLimit, CategoryTheory.NatTrans.leftOpWhiskerRight, CategoryTheory.Limits.reflectsLimitsOfSize_leftOp, CategoryTheory.Limits.limitLeftOpIsoUnopColimit_inv_comp_π, CategoryTheory.CategoryOfElements.costructuredArrowYonedaEquivalenceFunctorProj_hom_app, CategoryTheory.Limits.isLimitConeLeftOpOfCocone_lift, CategoryTheory.simplicialCosimplicialEquiv_unitIso_inv_app, CategoryTheory.Limits.isLimitConeOfCoconeLeftOp_lift, CategoryTheory.Limits.coconeLeftOpOfCone_ι_app, CategoryTheory.NatTrans.leftOpWhiskerRight_assoc, leftOpComp_hom_app, CategoryTheory.Limits.coneLeftOpOfCoconeEquiv_functor_map_hom, CategoryTheory.Limits.coneLeftOpOfCoconeEquiv_functor_obj, CategoryTheory.Limits.coneLeftOpOfCoconeEquiv_unitIso, CategoryTheory.Limits.coneLeftOpOfCoconeEquiv_inverse_map, CategoryTheory.Limits.preservesColimitsOfShape_leftOp, CategoryTheory.ComposableArrows.opEquivalence_inverse_map, CategoryTheory.Comma.opFunctorCompSnd_inv_app, instIsEquivalenceOppositeLeftOp, CategoryTheory.Comma.opEquiv_inverse, instFinalOppositeLeftOpOfInitial, CategoryTheory.Limits.coconeOfConeLeftOp_ι_app, CategoryTheory.Limits.coneLeftOpOfCoconeEquiv_counitIso, CategoryTheory.CosimplicialObject.Augmented.leftOp_hom_app, leftOp_map, CategoryTheory.Presheaf.coconeOfRepresentable_ι_app, CategoryTheory.CategoryOfElements.costructuredArrowYonedaEquivalenceInverseπ_inv_app, CategoryTheory.Limits.isColimitCoconeOfConeLeftOp_desc, CategoryTheory.Limits.limitLeftOpIsoUnopColimit_hom_comp_ι_assoc, CategoryTheory.Limits.coconeLeftOpOfConeEquiv_counitIso, CategoryTheory.Limits.ι_comp_colimitLeftOpIsoUnopLimit_hom_assoc, CategoryTheory.Limits.limitLeftOpIsoUnopColimit_inv_comp_π_assoc, CategoryTheory.Limits.reflectsColimitsOfSize_leftOp, CategoryTheory.Limits.ι_comp_colimitLeftOpIsoUnopLimit_hom, CategoryTheory.Limits.limitLeftOpIsoUnopColimit_hom_comp_ι, CategoryTheory.Limits.reflectsFiniteProducts_leftOp, CategoryTheory.NatTrans.leftOp_comp, instEssSurjOppositeLeftOp, leftOpRightOpEquiv_inverse_obj, CategoryTheory.Limits.preservesFiniteCoproducts_leftOp, leftOpRightOpIso_inv_app, CategoryTheory.Limits.colimitCoyonedaHomIsoLimitLeftOp_π_apply, CategoryTheory.Limits.coconeLeftOpOfConeEquiv_inverse_map, CategoryTheory.Limits.preservesLimits_leftOp, AlgebraicGeometry.PresheafedSpace.colimitCocone_ι_app_c, CategoryTheory.Limits.coconeLeftOpOfConeEquiv_functor_obj, CategoryTheory.CategoryOfElements.costructuredArrowYonedaEquivalenceFunctorProj_inv_app, CategoryTheory.Limits.walkingParallelPairOpEquiv_inverse, CategoryTheory.Limits.coconeLeftOpOfCone_pt
leftOpComp 📖	CompOp	4 math math: leftOpComp_inv_app, CategoryTheory.NatTrans.leftOpWhiskerRight, CategoryTheory.NatTrans.leftOpWhiskerRight_assoc, leftOpComp_hom_app
leftOpId 📖	CompOp	2 math math: leftOpId_hom_app, leftOpId_inv_app
leftOpRightOpEquiv 📖	CompOp	14 math math: CategoryTheory.ComposableArrows.opEquivalence_counitIso_inv_app_app, leftOpRightOpEquiv_functor_obj_map, leftOpRightOpEquiv_counitIso_inv_app_app, leftOpRightOpEquiv_unitIso_inv_app, CategoryTheory.ComposableArrows.opEquivalence_unitIso_inv_app, leftOpRightOpEquiv_functor_obj_obj, CategoryTheory.ComposableArrows.opEquivalence_functor_map_app, CategoryTheory.ComposableArrows.opEquivalence_unitIso_hom_app, leftOpRightOpEquiv_inverse_map, leftOpRightOpEquiv_unitIso_hom_app, leftOpRightOpEquiv_counitIso_hom_app_app, leftOpRightOpEquiv_functor_map_app, CategoryTheory.ComposableArrows.opEquivalence_counitIso_hom_app_app, leftOpRightOpEquiv_inverse_obj
leftOpRightOpIso 📖	CompOp	2 math math: leftOpRightOpIso_hom_app, leftOpRightOpIso_inv_app
op 📖	CompOp	720 math math: CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence.functor_obj_hom_right, FullyFaithful.homNatIsoMaxRight_inv_app, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_skyscraperPresheafHomEquiv_symm, PresheafOfModules.instIsRightAdjointPushforwardCompFunctorOppositeRingCatWhiskerLeftOp, CategoryTheory.Adjunction.compUliftCoyonedaIso_hom_app_app_down, smoothSheafCommRing.ι_forgetStalk_inv, CategoryTheory.Limits.IndizationClosedUnderFilteredColimitsAux.exists_nonempty_limit_obj_of_isColimit, IsDenseSubsite.instIsIsoSheafAppCounitSheafAdjunctionCocontinuous, CategoryTheory.Equivalence.preregular_isSheaf_iff, CategoryTheory.preservesFiniteLimits_iff_lan_preservesFiniteLimits, CategoryTheory.Pretriangulated.shiftFunctorCompIsoId_op_hom_app, CategoryTheory.Localization.isLocalization_op, CategoryTheory.Pretriangulated.shift_opShiftFunctorEquivalence_counitIso_inv_app, LightProfinite.Extend.functorOp_obj, CategoryTheory.Limits.colimitHomIsoLimitYoneda_hom_comp_π, LightCondensed.lanPresheafIso_hom, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit_app, CategoryTheory.regularTopology.equalizerCondition_precomp_of_preservesPullback, AlgebraicGeometry.LocallyRingedSpace.Γ_def, CategoryTheory.GrothendieckTopology.yonedaOpCompCoyoneda_hom_app_app_down, CategoryTheory.Limits.coneOpEquiv_unitIso, CategoryTheory.Comma.opFunctor_obj, opUnopIso_hom_app, sheafAdjunctionCocontinuous_homEquiv_apply_hom, CategoryTheory.Localization.isoOfHom_unop, CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan.uliftYonedaEquiv_ι_presheafHom, CategoryTheory.Limits.Cone.unop_ι, CategoryTheory.Limits.reflectsFiniteColimits_op, Profinite.Extend.cocone_pt, mapTriangleOpCompTriangleOpEquivalenceFunctorApp_inv_hom₁, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverse_obj_π_app, IsOpenMap.pullbackIso_hom_app_app, CategoryTheory.MorphismProperty.LeftFraction.op_map, CategoryTheory.GrothendieckTopology.Point.skyscraperPresheafFunctor_map_app, AlgebraicGeometry.Scheme.restrictFunctorΓ_inv_app, CategoryTheory.ran_isSheaf_of_isCocontinuous, CategoryTheory.Limits.ι_comp_colimitOpIsoOpLimit_hom_assoc, CategoryTheory.lan_preservesFiniteLimits_of_flat, CategoryTheory.Limits.parallelPairOpIso_inv_app_zero, CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence.functor_obj_left_hom, Profinite.Extend.functorOp_map, CategoryTheory.NatIso.op_rightUnitor, CategoryTheory.Pretriangulated.Opposite.UnopUnopCommShift.iso_inv_app, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_inverse, unopOpIso_inv_app, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.inv_naturality, CategoryTheory.Limits.preservesFiniteLimits_op, CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence_inverse, Elements.shrinkYoneda_map_app_coconeπOpCompShrinkYonedaObj_ι_app_assoc, CategoryTheory.Pretriangulated.shiftFunctorZero_op_inv_app, sheafPushforwardContinuous_map_hom_app, CategoryTheory.Join.inclRightCompOpEquivInverse_inv_app_op, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.invApp_app_assoc, IsLeftAdjoint.op, FullyFaithful.homNatIsoMaxRight_hom_app_down, CategoryTheory.Adjunction.compCoyonedaIso_inv_app_app, CategoryTheory.sheafToPresheafCompYonedaCompWhiskeringLeftSheafToPresheaf_inv_app_app, LightCondensed.isoFinYoneda_inv_app, CategoryTheory.Presheaf.isSheaf_iff_isLimit_coverage, CondensedMod.isDiscrete_tfae, smoothSheafCommRing.ι_evalHom_apply, commShiftOp_iso_eq, CategoryTheory.Limits.PushoutCocone.unop_π_app, CategoryTheory.Equivalence.precoherent_isSheaf_iff_of_essentiallySmall, CategoryTheory.Pretriangulated.shiftFunctorCompIsoId_op_inv_app, OneHypercoverDenseData.isSheaf_iff, CategoryTheory.Limits.opCompYonedaSectionsEquiv_symm_apply_coe, CategoryTheory.Limits.yonedaCompLimIsoCocones_inv_app, IsCoverDense.restrictHomEquivHom_naturality_left_symm_assoc, CategoryTheory.Limits.Cocone.unop_π, LightCondensed.finYoneda_map, CategoryTheory.Limits.colimitCoyonedaHomIsoLimit'_π_apply, cones_map_app, op_comp_isSheaf_of_isSheaf, CategoryTheory.Presheaf.instIsLeftKanExtensionFunctorOppositeTypeLanOpHomCompULiftYonedaIsoULiftYonedaCompLan, CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan.natTrans_app_uliftYoneda_obj, CategoryTheory.Limits.limitOpIsoOpColimit_hom_comp_ι_assoc, CategoryTheory.sheafToPresheafCompCoyonedaCompWhiskeringLeftSheafToPresheaf_inv_app_app, CategoryTheory.IsFiltered.iff_nonempty_limit, CategoryTheory.Comma.opEquiv_counitIso, IsOpenMap.pullbackObjIso_hom_app, Elements.coconeπOpCompShrinkYonedaObj_pt, skyscraperPresheafCocone_pt, CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence.functor_obj_right_as, AlgebraicGeometry.AffineScheme.instIsEquivalenceOppositeCommRingCatOpRightOpΓ, IsCoverDense.isoOver_hom_app, FullyFaithful.compUliftCoyonedaCompWhiskeringLeft_hom_app_app_down, opComp_inv_app, smoothSheaf.ι_evalHom_apply, CategoryTheory.sheafToPresheafCompYonedaCompWhiskeringLeftSheafToPresheaf_app_app, IsOpenMap.pullbackObjIso_hom_naturality, TopCat.Presheaf.pushforwardToOfIso_app, CategoryTheory.SimplicialObject.instIsRightKanExtensionOppositeTruncatedSimplexCategoryObjCoskAppTruncatedCounitCoskAdjTruncation, CategoryTheory.Pretriangulated.shiftFunctorAdd'_op_hom_app, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivCounitIso_hom_app_hom, CategoryTheory.Adjunction.compYonedaIso_hom_app_app, AlgebraicGeometry.ProjectiveSpectrum.Proj.toOpen_toSpec_val_c_app_assoc, CategoryTheory.CostructuredArrow.overEquivPresheafCostructuredArrow_inverse_map_toOverCompCoyoneda, CategoryTheory.Pretriangulated.Opposite.OpOpCommShift.iso_inv_app, CategoryTheory.GrothendieckTopology.Point.toPresheafFiberMap_presheafFiberMapObjIso_hom_assoc, sheafPushforwardContinuousNatTrans_app_hom, CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence.functor_obj_left_right, CategoryTheory.Sheaf.isPullback_square_op_map_yoneda_presheafToSheaf_yoneda_iff, CategoryTheory.NatIso.op_trans, CategoryTheory.Limits.parallelPairOpIso_inv_app_one, LightProfinite.Extend.cocone_ι_app, sheafAdjunctionCocontinuous_counit_app_val, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.biprodAddEquiv_symm_biprodIsoProd_hom_toBiprod_apply, AlgebraicGeometry.PresheafedSpace.componentwiseDiagram_map, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.inv_naturality, Condensed.lanPresheafExt_inv, CategoryTheory.NatIso.op_isoWhiskerLeft, PresheafOfModules.pushforward₀_obj_obj_carrier, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.invApp_app, imageSieve_eq_imageSieve, CategoryTheory.HasLiftingProperty.transfiniteComposition.wellOrderInductionData.liftHom_fac_assoc, CategoryTheory.Pretriangulated.commShiftIso_opOp_inv_app, CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan.presheafHom_naturality, CategoryTheory.Presheaf.isLocallySurjective_whisker, rightOpComp_hom_app, rightAdjointObjIsDefined_iff, CategoryTheory.Comma.opFunctor_map, IsCoverDense.Types.appHom_valid_glue, SSet.OneTruncation₂.ofNerve₂.natIso_inv_app_obj_obj, CategoryTheory.Adjunction.Triple.op_adj₁, CategoryTheory.Adjunction.sheafPushforwardContinuous_unit_app_hom_app, mapTriangleOpCompTriangleOpEquivalenceFunctorApp_inv_hom₃, CategoryTheory.Pretriangulated.Opposite.UnopUnopCommShift.iso_inv_app_assoc, FullyFaithful.homNatIso_inv_app_down, opId_hom_app, CategoryTheory.GrothendieckTopology.Point.presheafFiberOfIsCofilteredCocone_ι_app, PresheafOfModules.pushforward_map_app_apply', shift_map_op_assoc, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverseObj_π_app, opUnopIso_inv_app, IsCoverDense.Types.pushforwardFamily_def, CategoryTheory.Adjunction.Quadruple.op_rightTriple, Condensed.isSheafStonean, CategoryTheory.Pretriangulated.commShiftIso_opOp_hom_app_assoc, instEssSurjOppositeOp, AlgebraicGeometry.PresheafedSpace.GlueData.snd_invApp_t_app_assoc, CategoryTheory.SimplicialObject.Truncated.rightExtensionInclusion_hom_app, smoothSheaf.ι_evalHom_assoc, CategoryTheory.Pretriangulated.shiftFunctorAdd'_op_inv_app, IsCoverDense.sheafHom_eq, CategoryTheory.Adjunction.Quadruple.op_leftTriple, CategoryTheory.NatTrans.op_id, CategoryTheory.Limits.reflectsLimits_op, instFaithfulOppositeOp, sheafAdjunctionCocontinuous_unit_app_hom, CategoryTheory.Limits.spanOp_hom_app, CategoryTheory.Presheaf.isLimit_iff_isSheafFor_presieve, CategoryTheory.Limits.coyonedaOpColimitIsoLimitCoyoneda_hom_comp_π_assoc, Profinite.Extend.functorOp_obj, IsCoverDense.restrictHomEquivHom_naturality_right_symm, CategoryTheory.Limits.reflectsFiniteProducts_op, CategoryTheory.Limits.preservesLimitsOfSize_op, Condensed.finYoneda_map, mapCoconeOp_inv_hom, CategoryTheory.Limits.Cocone.op_pt, CategoryTheory.GrothendieckTopology.Point.skyscraperPresheafHomEquiv_apply_app, CategoryTheory.NatTrans.op_whiskerRight_assoc, CategoryTheory.regularTopology.equalizerConditionMap_iff_nonempty_isLimit, FullyFaithful.compUliftYonedaCompWhiskeringLeft_inv_app_app_down, AlgebraicGeometry.StructureSheaf.toPushforwardStalkAlgHom_apply, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv_counitIso, CategoryTheory.Adjunction.compPreadditiveYonedaIso_inv_app_app_apply, CategoryTheory.HasLiftingProperty.transfiniteComposition.wellOrderInductionData.liftHom_fac, map_opShiftFunctorEquivalence_unitIso_inv_app_unop, TopCat.Presheaf.pushforward_map_app, AlgebraicGeometry.Scheme.Hom.coequifibered_normalizationDiagramMap, TopologicalSpace.OpenNhds.op_map_id_obj, IsCoverDense.restrictHomEquivHom_naturality_left, HomologicalComplex.cyclesOpNatIso_inv_app, CategoryTheory.preadditiveYonedaMap_app, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit_app', CategoryTheory.Enriched.FunctorCategory.diagram_map_app, CategoryTheory.Adjunction.compUliftCoyonedaIso_inv_app_app_down, CategoryTheory.presheafHom_obj, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctorObj_π_app, whiskerLeft_obj_map_bijective_of_isCoverDense, CategoryTheory.Limits.parallelPairOpIso_hom_app_one, IsCoverDense.sheafHom_restrict_eq, CategoryTheory.Limits.PullbackCone.op_ι_app, CategoryTheory.Pretriangulated.TriangleOpEquivalence.unitIso_hom_app, TopCat.Presheaf.pushforwardEq_hom_app, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.inv_naturality_assoc, CategoryTheory.Presieve.IsSheaf.comp_of_W_map_of_adjunction, CategoryTheory.Adjunction.Triple.op_rightToLeft, CategoryTheory.Limits.reflectsColimitsOfSize_op, CategoryTheory.SimplicialObject.Truncated.rightExtensionInclusion_left, CategoryTheory.Limits.coyonedaOpColimitIsoLimitCoyoneda_inv_comp_π_assoc, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.invApp_app_apply, IsCoverDense.sheafIso_inv_hom, CategoryTheory.instRepresentablyFlatOppositeOpOfRepresentablyCoflat, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivCounitIso_inv_app_hom, CategoryTheory.Limits.opHomCompWhiskeringLimYonedaIsoCocones_hom_app_app_app, TopCat.Sheaf.extend_hom_app, CategoryTheory.Limits.opParallelPairIso_hom_app_zero, CategoryTheory.Limits.coneOpEquiv_inverse_obj, CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence_counitIso, PresheafOfModules.pushforward₀_obj_obj_isAddCommGroup, CategoryTheory.Adjunction.Triple.op_adj₂, CategoryTheory.NatTrans.removeOp_id, CategoryTheory.Localization.isoOfHom_op_inv, CategoryTheory.CostructuredArrow.toStructuredArrow'_obj, sheafPushforwardContinuousCompSheafToPresheafIso_inv_app_app, CategoryTheory.NatTrans.rightOpWhiskerRight, CategoryTheory.Presheaf.isLocallyInjective_whisker, AlgebraicGeometry.Scheme.Γ_def, CategoryTheory.presheafHom_map_app, CategoryTheory.Limits.opParallelPairIso_hom_app_one, IsOpenMap.pullbackIso_inv_app_app, CategoryTheory.Limits.yonedaCompLimIsoCocones_hom_app_app, CategoryTheory.Limits.instHasColimitDiscreteOppositeCompInverseOppositeOpFunctor, CategoryTheory.Limits.preservesLimitsOfShape_op, mapTriangleOpCompTriangleOpEquivalenceFunctorApp_hom_hom₁, smoothSheafCommRing.ι_evalHom, CategoryTheory.representablyFlat_op_iff, AlgebraicGeometry.Scheme.Modules.restrictFunctorComp_hom_app_app, CategoryTheory.Limits.preservesColimitsOfSize_op, sheafAdjunctionCocontinuous_counit_app_hom, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_hom_app, CategoryTheory.NatIso.op_isoWhiskerRight, AlgebraicGeometry.Scheme.Modules.restrictFunctorComp_inv_app_app, CategoryTheory.Presieve.IsSheafFor.comp_iff_of_preservesPairwisePullbacks, CategoryTheory.Presheaf.restrictedULiftYoneda_map_app, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.inv_naturality_assoc, CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan_inv_app_app_apply_eq_id, CategoryTheory.Limits.spanOp_inv_app, CategoryTheory.Presheaf.isSeparated_iff_subsingleton, IsCoverDense.restrictHomEquivHom_naturality_left_assoc, map_shift_unop, IsCoverDense.presheafIso_hom_app, instIsRepresentableCompOppositeOpObjTypeYonedaObjRightAdjointObjIsDefined, op_commShiftIso_hom_app_assoc, FullyFaithful.compYonedaCompWhiskeringLeftMaxRight_inv_app_app, CategoryTheory.Pretriangulated.shift_opShiftFunctorEquivalence_counitIso_inv_app_assoc, CategoryTheory.Limits.coconeOpEquiv_inverse_map, CategoryTheory.GrothendieckTopology.Point.presheafFiberMapCocone_pt, CategoryTheory.GrothendieckTopology.Point.presheafFiberOfIsCofilteredCocone_pt, PresheafOfModules.pullback_assoc, CategoryTheory.Limits.opSpan_hom_app, CategoryTheory.NatTrans.op_whiskerRight, CategoryTheory.Limits.coconeOpEquiv_functor_obj, AlgebraicGeometry.ΓSpec.toOpen_comp_locallyRingedSpaceAdjunction_homEquiv_app, AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap_app, CategoryTheory.instRepresentablyCoflatOppositeOpOfRepresentablyFlat, CategoryTheory.NatTrans.op_comp_assoc, CategoryTheory.GrothendieckTopology.yonedaOpCompCoyoneda_inv_app_app, Condensed.isColimitLocallyConstantPresheaf_desc_apply, CategoryTheory.NatIso.op_symm, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctor_obj_pt, CategoryTheory.Pretriangulated.Opposite.UnopUnopCommShift.iso_hom_app_assoc, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.inv_invApp, CategoryTheory.Limits.opCompYonedaSectionsEquiv_apply_app, TopCat.Presheaf.whiskerIsoMapGenerateCocone_inv_hom, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.inv_naturality_assoc, TopologicalSpace.Opens.op_map_id_obj, op_iff, CategoryTheory.GrothendieckTopology.W_inverseImage_whiskeringLeft, initial_op_of_final, CategoryTheory.TwoSquare.guitartExact_op_iff, SSet.Truncated.rightExtensionInclusion_right_as, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_hom_app, CategoryTheory.NatTrans.op_whiskerLeft_assoc, AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap_app', AlgebraicGeometry.Scheme.Hom.appIso_inv_naturality, CategoryTheory.Limits.π_comp_colimitOpIsoOpLimit_inv_assoc, LightCondensed.isoLocallyConstantOfIsColimit_inv, AlgebraicGeometry.Scheme.AffineZariskiSite.coequifibered_iff_forall_isLocalizationAway, AlgebraicGeometry.PresheafedSpace.restrictStalkIso_inv_eq_germ_apply, AugmentedSimplexCategory.equivAugmentedSimplicialObjectFunctorCompDropIso_hom_app_app, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv_inverse, CategoryTheory.Pretriangulated.Opposite.OpOpCommShift.iso_hom_app_assoc, CategoryTheory.cosimplicialSimplicialEquiv_functor_map_app, CategoryTheory.Presheaf.isLocallyInjective_whisker_iff, CategoryTheory.Cat.opFunctor_map, CategoryTheory.RanIsSheafOfIsCocontinuous.fac', CategoryTheory.Limits.instHasLimitOppositeDiscreteOpFunctor, CategoryTheory.Pretriangulated.commShiftIso_unopUnop_inv_app, CategoryTheory.Limits.opCospan_hom_app, CategoryTheory.Limits.preservesFiniteColimits_op, IsOpenMap.pullbackObjIso_inv_app, CategoryTheory.lan_preservesFiniteLimits_of_preservesFiniteLimits, unopOpIso_hom_app, AlgebraicGeometry.ΓSpec.unop_locallyRingedSpaceAdjunction_counit_app', op_comp_isSheaf_of_types, TopCat.Presheaf.pullbackPushforwardAdjunction_unit_pullback_map_germToPullbackStalk, CategoryTheory.Enriched.FunctorCategory.enrichedComp_π_assoc, CategoryTheory.StructuredArrow.toCostructuredArrow_map, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.invApp_app, CategoryTheory.CostructuredArrow.overEquivPresheafCostructuredArrow_functor_map_toOverCompYoneda, CategoryTheory.NatTrans.equifibered_op_iff, IsCoverDense.Types.appHom_restrict, AlgebraicGeometry.PresheafedSpace.restrictStalkIso_hom_eq_germ_apply, CategoryTheory.Comma.unopFunctor_obj, CategoryTheory.Limits.colimitHomIsoLimitYoneda_hom_comp_π_assoc, CategoryTheory.Pretriangulated.Opposite.commShift_natTrans_op_int, Condensed.isoLocallyConstantOfIsColimit_inv, CategoryTheory.TwoSquare.natTrans_op, CategoryTheory.SimplicialObject.isCoskeletal_iff, AlgebraicGeometry.Scheme.restrictFunctorΓ_hom_app, smoothSheafCommRing.ι_forgetStalk_inv_apply, Condensed.lanPresheafNatIso_hom_app, CategoryTheory.Pretriangulated.shiftFunctorZero_op_hom_app, TopCat.Presheaf.IsSheaf.isSheafPreservesLimitPairwiseIntersections, CategoryTheory.Limits.coneOpEquiv_functor_obj, CategoryTheory.Limits.coneOpEquiv_counitIso, final_op_of_initial, IsCoverDense.Types.appIso_inv, AlgebraicGeometry.PresheafedSpace.GlueData.f_invApp_f_app_assoc, AlgebraicGeometry.Scheme.Hom.preservesLocalization_normalizationDiagramMap, op_commShiftIso_inv_app_assoc, CategoryTheory.CostructuredArrow.toStructuredArrow'_map, CategoryTheory.Limits.reflectsLimitsOfSize_op, AlgebraicGeometry.Spec.sheafedSpaceMap_hom_c_app, CategoryTheory.Equivalence.sheafCongrPreregular_inverse_map_hom_app, CategoryTheory.sheafToPresheafCompCoyonedaCompWhiskeringLeftSheafToPresheaf_app_app, AlgebraicGeometry.Scheme.Hom.appIso_inv_naturality_assoc, SSet.StrictSegal.isPointwiseRightKanExtensionAt.fac_aux₂, IsLocalization.op_iff, CategoryTheory.Presheaf.isSheaf_iff_isLimit, CategoryTheory.NatTrans.op_app, CategoryTheory.Limits.PushoutCocone.op_π_app, CategoryTheory.NatTrans.op_comp, CategoryTheory.Adjunction.rightOp_eq, CategoryTheory.Limits.π_comp_colimitOpIsoOpLimit_inv, IsCoverDense.restrictHomEquivHom_naturality_right_assoc, CategoryTheory.cones_obj_map_app, instIsEquivalenceOppositeOp, CategoryTheory.Limits.opCospan_inv_app, CategoryTheory.SimplicialObject.Truncated.rightExtensionInclusion_right_as, sheafPushforwardCocontinuousCompSheafToPresheafIso_inv, CategoryTheory.Limits.reflectsColimit_op, Condensed.instFinalOppositeDiscreteQuotientCarrierToTopTotallyDisconnectedSpaceCostructuredArrowFintypeCatProfiniteOpToProfiniteOpPtAsLimitConeFunctorOp, CategoryTheory.Adjunction.compPreadditiveYonedaIso_hom_app_app_apply, CategoryTheory.Limits.preservesFiniteProducts_op, CategoryTheory.Limits.PullbackCone.unop_ι_app, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIso_inv_app_hom, CategoryTheory.Limits.reflectsFiniteLimits_op, CategoryTheory.Adjunction.op_unit, IsCoverDense.Types.presheafIso_inv_app, CategoryTheory.cosimplicialSimplicialEquiv_unitIso_hom_app, CategoryTheory.Join.inclRightCompOpEquivInverse_hom_app_op, CategoryTheory.Limits.cospanOp_hom_app, CategoryTheory.NatIso.op_associator, smoothSheafCommRing.ι_forgetStalk_inv_assoc, skyscraperPresheafCocone_ι_app, op_commShiftIso_inv_app, CategoryTheory.Comma.unopFunctorCompSnd_inv_app, CategoryTheory.GrothendieckTopology.W_whiskerLeft_iff, CategoryTheory.GrothendieckTopology.diagramPullback_app, AlgebraicGeometry.PresheafedSpace.GlueData.f_invApp_f_app, CategoryTheory.MorphismProperty.op_inverseImage, CategoryTheory.sheafToPresheafCompYonedaCompWhiskeringLeftSheafToPresheaf_hom_app_app_hom, sheafPushforwardCocontinuousCompSheafToPresheafIso_hom, CompHausOpToFrame.faithful, CategoryTheory.Pretriangulated.Opposite.commShift_adjunction_op_int, SSet.StrictSegal.isPointwiseRightKanExtensionAt.fac_aux₃, CategoryTheory.Pretriangulated.TriangleOpEquivalence.unitIso_inv_app, TopCat.Presheaf.pullback_obj_obj_ext_iff, CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence.functor_map_left_right, CategoryTheory.Limits.hasLimit_op_of_hasColimit, CategoryTheory.Adjunction.compYonedaIso_inv_app_app, IsRightAdjoint.op, CategoryTheory.Equivalence.sheafCongrPreregular_functor_map_hom_app, CategoryTheory.Limits.hasColimit_op_iff_hasLimit, CategoryTheory.Presheaf.instIsLeftKanExtensionOppositeObjFunctorTypeYonedaYonedaMap, CategoryTheory.Limits.limitOpIsoOpColimit_inv_comp_π_assoc, LightProfinite.Extend.functorOp_map, CategoryTheory.Pretriangulated.Opposite.functor_isTriangulated_op, const.opObjOp_hom_app, CategoryTheory.MorphismProperty.RightFraction.op_map, CategoryTheory.Pretriangulated.Opposite.OpOpCommShift.iso_inv_app_assoc, CategoryTheory.MorphismProperty.IsInvertedBy.op, Alexandrov.lowerCone_π_app, CategoryTheory.Comma.opFunctorCompSnd_hom_app, CategoryTheory.Limits.instHasLimitDiscreteOppositeCompInverseOppositeOpFunctor, PresheafOfModules.pushforward_obj_obj, CategoryTheory.Join.inclLeftCompOpEquivInverse_inv_app_op, SSet.Truncated.rightExtensionInclusion_left, CategoryTheory.Adjunction.Triple.leftToRight_op, CategoryTheory.GrothendieckTopology.Point.toPresheafFiberMap_presheafFiberMapObjIso_hom, CategoryTheory.Presieve.isSheafFor_over_map_op_comp_ofArrows_iff, map_opShiftFunctorEquivalence_counitIso_hom_app_unop, CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence_unitIso, CategoryTheory.NatIso.op_leftUnitor, CategoryTheory.Limits.reflectsFiniteCoproducts_op, CategoryTheory.Presieve.isSheafFor_over_map_op_comp_iff, mapTriangleOpCompTriangleOpEquivalenceFunctorApp_hom_hom₃, CategoryTheory.NatTrans.coequifibered_op_iff, map_opShiftFunctorEquivalence_counitIso_inv_app_unop_assoc, const.opObjOp_inv_app, CategoryTheory.Limits.coyonedaOpColimitIsoLimitCoyoneda_inv_comp_π, CategoryTheory.Equivalence.op_functor, CategoryTheory.cosimplicialSimplicialEquiv_unitIso_inv_app, CategoryTheory.yonedaMap_app_apply, toSheafify_pullbackSheafificationCompatibility, IsCoverDense.Types.naturality_apply, CategoryTheory.NatTrans.rightOpWhiskerRight_assoc, CategoryTheory.Limits.reflectsColimitsOfShape_op, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctor_obj_π_app, CategoryTheory.SimplicialObject.Truncated.trunc_map_app, TopCat.Presheaf.whiskerIsoMapGenerateCocone_hom_hom, const.unop_functor_op_obj_map, leftOpComp_inv_app, CategoryTheory.Comma.opFunctorCompFst_hom_app, CategoryTheory.Limits.limitCompYonedaIsoCocone_inv, CategoryTheory.Limits.Fork.op_ι_app, CategoryTheory.Limits.reflectsColimits_op, CategoryTheory.CategoryOfElements.costructuredArrow_yoneda_equivalence_naturality, mapTriangleOpCompTriangleOpEquivalenceFunctorApp_inv_hom₂, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctor_map_hom, CategoryTheory.Comma.opFunctorCompFst_inv_app, CategoryTheory.CostructuredArrow.toStructuredArrow_obj, CategoryTheory.Comma.unopFunctor_map, CondensedSet.isDiscrete_tfae, CategoryTheory.Equivalence.precoherent_isSheaf_iff, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverse_obj_pt, CategoryTheory.Adjunction.leftOp_eq, CategoryTheory.ShortComplex.rightHomologyFunctorOpNatIso_hom_app, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIso_hom_app_hom, opHom_obj, AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap_app_assoc, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_hom_app_assoc, CategoryTheory.Limits.hasLimit_op_iff_hasColimit, CategoryTheory.yonedaPairing_map, CategoryTheory.Limits.preservesLimits_op, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIsoApp_hom_hom, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_inv_app, CategoryTheory.NatTrans.Coequifibered.op, rightOpComp_inv_app, CategoryTheory.Limits.coconeOpEquiv_inverse_obj, CategoryTheory.Comma.opEquiv_functor, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctorObj_pt, opId_inv_app, AlgebraicGeometry.PresheafedSpace.GlueData.snd_invApp_t_app, CategoryTheory.Limits.FormalCoproduct.evalOpCompInlIsoId_hom_app_app, IsCoverDense.presheafIso_inv, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.inv_naturality, CategoryTheory.flat_iff_lan_flat, CategoryTheory.yonedaPairingExt_iff, CategoryTheory.Comma.opEquiv_unitIso, opUnopEquiv_unitIso, CategoryTheory.LocalizerMorphism.op_functor, FullyFaithful.compYonedaCompWhiskeringLeftMaxRight_hom_app_app_down, CategoryTheory.lan_flat_of_flat, CategoryTheory.Join.inclLeftCompOpEquivInverse_hom_app_op, CategoryTheory.Presheaf.coconeOfRepresentable_naturality, PresheafOfModules.Derivation.d_app, CategoryTheory.GrothendieckTopology.Point.presheafFiberMapCocone_ι_app, CategoryTheory.NatIso.op_refl, CategoryTheory.Adjunction.compCoyonedaIso_hom_app_app, closedIhom_map_app, CategoryTheory.NatTrans.leftOpWhiskerRight, CategoryTheory.Limits.coneOpEquiv_functor_map_hom, SSet.horn.spineId_vertex_coe, CategoryTheory.Presieve.FamilyOfElements.Compatible.functorPullback, map_opShiftFunctorEquivalence_unitIso_hom_app_unop_assoc, AlgebraicGeometry.Scheme.AffineZariskiSite.opensRange_relativeGluingData_map, CategoryTheory.GrothendieckTopology.uliftYonedaOpCompCoyoneda_hom_app_app_down, CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan.uliftYonedaEquiv_presheafHom_uliftYoneda_obj, shift_map_op, Elements.shrinkYoneda_map_app_coconeπOpCompShrinkYonedaObj_ι_app, CategoryTheory.Limits.ι_comp_colimitOpIsoOpLimit_hom, Elements.shrinkYonedaCompWhiskeringLeftObjπCompColimIso_inv_app_apply, CategoryTheory.Pretriangulated.shiftFunctor_op_map, SSet.horn.spineId_arrow_coe, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.invApp_app, CategoryTheory.Limits.reflectsLimitsOfShape_op, CategoryTheory.SimplicialObject.IsCoskeletal.isRightKanExtension, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverse_map_hom, Profinite.Extend.functorOp_final, FullyFaithful.compUliftYonedaCompWhiskeringLeft_hom_app_app_down, CategoryTheory.Limits.coconeOpEquiv_unitIso, CategoryTheory.Pretriangulated.commShiftIso_opOp_hom_app, CategoryTheory.Comma.unopFunctorCompFst_inv_app, CategoryTheory.ShortComplex.leftHomologyFunctorOpNatIso_hom_app, FullyFaithful.compUliftCoyonedaCompWhiskeringLeft_inv_app_app_down, LightCondensed.lanPresheafNatIso_hom_app, smoothSheafCommRing.ι_forgetStalk_hom_apply, IsCoverDense.homOver_app, CategoryTheory.Adjunction.Quadruple.op_adj₁, CategoryTheory.NatIso.removeOp_inv, smoothSheafCommRing.ι_forgetStalk_hom_assoc, AlgebraicGeometry.Proj.sheafedSpaceMap_hom_c_app_hom_apply_coe, CategoryTheory.GrothendieckTopology.uliftYonedaOpCompCoyoneda_app_app, CategoryTheory.Presheaf.map_comp_uliftYonedaEquiv_down_assoc, SSet.Subcomplex.liftPath_arrow_coe, IsCoverDense.Types.presheafIso_hom_app, CategoryTheory.NatTrans.leftOpWhiskerRight_assoc, leftOpComp_hom_app, CategoryTheory.Limits.preservesLimit_op, sheafPushforwardContinuousCompSheafToPresheafIso_hom_app_app, LightProfinite.Extend.cocone_pt, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.invApp_app_assoc, CategoryTheory.Limits.preservesFiniteCoproducts_op, TopologicalSpace.Opens.op_map_comp_obj, CategoryTheory.GrothendieckTopology.overMapPullbackCongr_inv_app_hom_app, Condensed.isColimitLocallyConstantPresheafDiagram_desc_apply, CategoryTheory.TwoSquare.instGuitartExactOppositeOp, opComp_hom_app, CategoryTheory.NatIso.op_hom, CategoryTheory.Limits.coyonedaOpColimitIsoLimitCoyoneda_hom_comp_π, instFullOppositeOp, Profinite.Extend.cocone_ι_app, CategoryTheory.Limits.preservesColimit_op, CategoryTheory.Presheaf.isLocallySurjective_whisker_iff, CategoryTheory.MonoidalCategory.dayConvolutionInternalHomDiagramFunctor_obj_map_app_app, CategoryTheory.Limits.whiskeringLimYonedaIsoCones_hom_app_app_app, CategoryTheory.Pretriangulated.commShiftIso_opOp_inv_app_assoc, CategoryTheory.Pretriangulated.commShiftIso_unopUnop_inv_app_assoc, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIsoApp_inv_hom, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_presheafFiberMapObjIso_inv_assoc, mapConeOp_inv_hom, AlgebraicGeometry.ΓSpec.toSpecΓ_of, map_shift_unop_assoc, TopCat.Presheaf.IsSheaf.isSheafPairwiseIntersections, CategoryTheory.Comma.opFunctorCompSnd_inv_app, CategoryTheory.Pretriangulated.Opposite.UnopUnopCommShift.iso_hom_app, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_hom_app_assoc, TopCat.Presheaf.isGluing_iff_pairwise, Alexandrov.lowerCone_pt, op_isTriangulated_iff, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_inv_app_assoc, CategoryTheory.Equivalence.sheafCongr.functor_map_hom_app, CategoryTheory.NatIso.op_inv, PresheafOfModules.pushforward_assoc, Condensed.isoFinYoneda_hom_app, CategoryTheory.Equivalence.sheafCongrPrecoherent_functor_map_hom_app, CategoryTheory.Limits.coneOpEquiv_inverse_map, CategoryTheory.Equivalence.sheafCongrPrecoherent_inverse_map_hom_app, IsCoverDense.isoOver_inv_app, CategoryTheory.Limits.limitOpIsoOpColimit_inv_comp_π, smoothSheaf.ι_evalHom, AlgebraicGeometry.SheafedSpace.Γ_def, CategoryTheory.Limits.preservesColimits_op, smoothSheafCommRing.ι_evalHom_assoc, CategoryTheory.Limits.Cone.unop_pt, CategoryTheory.Comma.opEquiv_inverse, CategoryTheory.PresheafHom.IsSheafFor.app_cond, mapCoconeOp_hom_hom, CategoryTheory.NatTrans.op_whiskerLeft, CategoryTheory.GrothendieckTopology.uliftYonedaOpCompCoyoneda_inv_app_app, AugmentedSimplexCategory.equivAugmentedSimplicialObjectFunctorCompDropIso_inv_app_app, CategoryTheory.ShortComplex.rightHomologyFunctorOpNatIso_inv_app, LightCondensed.lanPresheafExt_inv, Condensed.lanPresheafExt_hom, SSet.Subcomplex.liftPath_vertex_coe, op_comp_isSheaf, CategoryTheory.Presheaf.map_comp_uliftYonedaEquiv_down, CategoryTheory.NatIso.removeOp_hom, WellOrderInductionData.map_lift, smoothSheafCommRing.ι_forgetStalk_hom, CategoryTheory.Limits.limitCompYonedaIsoCocone_hom_app, SSet.OneTruncation₂.ofNerve₂.natIso_inv_app_obj_map, CategoryTheory.GrothendieckTopology.Point.presheafFiberMapIso_inv_app, AlgebraicGeometry.PresheafedSpace.congr_app, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_skyscraperPresheafHomEquiv_symm_assoc, CategoryTheory.Limits.Cocone.op_π, CategoryTheory.HasLiftingProperty.transfiniteComposition.wellOrderInductionData.map_lift, CategoryTheory.Limits.colimitYonedaHomIsoLimitOp_π_apply, LightCondensed.isColimitLocallyConstantPresheaf_desc_apply, CategoryTheory.SimplicialObject.instIsLeftKanExtensionOppositeTruncatedSimplexCategoryObjSkAppTruncatedUnitSkAdjTruncation, CategoryTheory.Limits.cospanOp_inv_app, PresheafOfModules.pushforward_obj_map_apply', CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan.presheafHom_naturality_assoc, HomologicalComplex.cyclesOpNatIso_hom_app, SSet.StrictSegal.isPointwiseRightKanExtensionAt.fac_aux₁, skyscraperPresheafCoconeOfSpecializes_ι_app, CategoryTheory.PresheafHom.IsSheafFor.exists_app, CategoryTheory.CostructuredArrow.overEquivPresheafCostructuredArrow_functor_map_toOverCompCoyoneda, CategoryTheory.GrothendieckTopology.overMapPullbackCongr_hom_app_hom_app, Elements.coconeπOpCompShrinkYonedaObj_ι_app, mapConeOp_hom_hom, LightCondensed.instFinalNatCostructuredArrowOppositeFintypeCatLightProfiniteOpToLightProfiniteOpPtAsLimitConeFunctorOp, AlgebraicGeometry.Scheme.AffineZariskiSite.PreservesLocalization.opensRange_map, SSet.OneTruncation₂.nerveHomEquiv_apply, CategoryTheory.Limits.coconeOpEquiv_counitIso, CategoryTheory.Equivalence.preregular_isSheaf_iff_of_essentiallySmall, op_comp_isSheaf_of_preservesOneHypercovers, CategoryTheory.PreGaloisCategory.PointedGaloisObject.instHasColimitOppositeFunctorTypeCompOpInclCoyoneda, CategoryTheory.Limits.instHasColimitOppositeDiscreteOpFunctor, opHom_map_app, map_opShiftFunctorEquivalence_counitIso_inv_app_unop, CategoryTheory.sheafToPresheafCompCoyonedaCompWhiskeringLeftSheafToPresheaf_hom_app_app_hom, CategoryTheory.cones_map_app_app, LightCondensed.isoFinYoneda_hom_app, CategoryTheory.Pretriangulated.shiftFunctorCompIsoId_op_hom_app_assoc, AlgebraicGeometry.SheafedSpace.congr_hom_app, map_opShiftFunctorEquivalence_unitIso_inv_app_unop_assoc, CategoryTheory.NatTrans.Equifibered.op, CategoryTheory.Limits.Cone.op_pt, CategoryTheory.GrothendieckTopology.Point.presheafFiberCocone_ι_app, CategoryTheory.Adjunction.representableBy_homEquiv, AlgebraicGeometry.PresheafedSpace.restrict_presheaf, ContinuousMap.comp_yonedaPresheaf', CategoryTheory.Enriched.FunctorCategory.enrichedComp_π, TopCat.Presheaf.pullbackPushforwardAdjunction_unit_pullback_map_germToPullbackStalk_assoc, CategoryTheory.Presheaf.isLimit_iff_isSheafFor, CategoryTheory.Pretriangulated.commShiftIso_unopUnop_hom_app, CategoryTheory.ShortComplex.leftHomologyFunctorOpNatIso_inv_app, op_commShiftIso_hom_app, CategoryTheory.Equivalence.op_unitIso, skyscraperPresheafCoconeOfSpecializes_pt, CategoryTheory.PresheafHom.isAmalgamation_iff, LightCondensed.instHasColimitsOfShapeCostructuredArrowOppositeFintypeCatLightProfiniteOpToLightProfiniteType, CategoryTheory.Pretriangulated.Opposite.OpOpCommShift.iso_hom_app, IsContinuous.op_comp_isSheaf_of_types, CategoryTheory.Equivalence.op_inverse, CategoryTheory.Limits.Cocone.unop_pt, AlgebraicGeometry.LocallyRingedSpace.toΓSpecMapBasicOpen_eq, CategoryTheory.uliftYonedaMap_app_apply, WellOrderInductionData.Extension.map_limit, CategoryTheory.MonoidalCategory.dayConvolutionInternalHomDiagramFunctor_obj_obj_map_app, IsCoverDense.Types.appIso_hom, IsCoverDense.restrictHomEquivHom_naturality_right_symm_assoc, SSet.Truncated.rightExtensionInclusion_hom_app, TopCat.Presheaf.toPushforwardOfIso_app, FullyFaithful.homNatIso_hom_app_down, CategoryTheory.Limits.IndizationClosedUnderFilteredColimitsAux.exists_nonempty_limit_obj_of_colimit, CategoryTheory.CostructuredArrow.overEquivPresheafCostructuredArrow_inverse_map_toOverCompYoneda, SmallCategories.instPreservesFiniteLimitsSheafSheafPullbackOfRepresentablyFlat, CategoryTheory.Pretriangulated.commShiftIso_unopUnop_hom_app_assoc, CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan.coconeApp_naturality, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_presheafFiberMapObjIso_inv, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.inv_invApp, TopCat.Presheaf.IsSheaf.isSheafOpensLeCover, CategoryTheory.Limits.preservesColimitsOfShape_op, CategoryTheory.Adjunction.sheafPushforwardContinuous_counit_app_hom_app, AlgebraicGeometry.LocallyRingedSpace.toΓSpecCBasicOpens_app, IsDenseSubsite.isIso_ranCounit_app_of_isDenseSubsite, CategoryTheory.Limits.opParallelPairIso_inv_app_zero, SheafOfModules.pushforwardSections_coe, CategoryTheory.GrothendieckTopology.Point.presheafFiberMapIso_hom_app, CategoryTheory.Limits.opParallelPairIso_inv_app_one, CategoryTheory.Limits.colimitHomIsoLimitYoneda_inv_comp_π, AlgebraicGeometry.SheafedSpace.congr_app, CategoryTheory.yonedaYonedaColimit_app_inv, CategoryTheory.Limits.Cone.extensions_app, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv_functor, map_opShiftFunctorEquivalence_counitIso_hom_app_unop_assoc, CategoryTheory.Comma.unopFunctorCompSnd_hom_app, LightCondensed.lanPresheafExt_hom, CategoryTheory.PreGaloisCategory.PointedGaloisObject.cocone_app, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv_unitIso, CategoryTheory.StructuredArrow.toCostructuredArrow'_map, CategoryTheory.Adjunction.op_counit, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverseObj_pt, CategoryTheory.Adjunction.Quadruple.op_adj₂, sheafAdjunctionCocontinuous_homEquiv_apply_val, PresheafOfModules.pushforward₀_obj_obj_isModule, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.invApp_app_assoc, CategoryTheory.StructuredArrow.toCostructuredArrow'_obj, CategoryTheory.Limits.opSpan_inv_app, AlgebraicGeometry.ProjectiveSpectrum.Proj.toOpen_toSpec_val_c_app, CategoryTheory.Limits.coyonedaCompLimIsoCones_hom_app_app, Condensed.isoFinYoneda_inv_app, CategoryTheory.Equivalence.op_counitIso, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.inv_invApp, op_additive, CategoryTheory.Comma.unopFunctorCompFst_hom_app, CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence_functor, CategoryTheory.RanIsSheafOfIsCocontinuous.liftAux_map, CategoryTheory.Presheaf.subsingleton_iff_isSeparatedFor, CategoryTheory.Limits.reflectsLimit_op, Condensed.isSheafProfinite, CategoryTheory.representablyCoflat_op_iff, AlgebraicGeometry.PresheafedSpace.GlueData.snd_invApp_t_app', pushforwardContinuousSheafificationCompatibility_hom_app_val, IsLocalization.op, CategoryTheory.NatTrans.removeOp_app, IsCoverDense.restrictHomEquivHom_naturality_right, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_inv_app, CategoryTheory.Limits.colimitHomIsoLimitYoneda_inv_comp_π_assoc, Condensed.locallyConstantIsoFinYoneda_hom_app, AlgebraicGeometry.nonempty_isColimit_Γ_mapCocone, SSet.StrictSegal.isRightKanExtension, CategoryTheory.GrothendieckTopology.uliftYonedaOpCompCoyoneda_inv_app_app_hom_app, CategoryTheory.Limits.opHomCompWhiskeringLimYonedaIsoCocones_inv_app_app, AlgebraicGeometry.LocallyRingedSpace.toΓSpec_preimage_basicOpen_eq, sheafAdjunctionCocontinuous_unit_app_val, map_opShiftFunctorEquivalence_unitIso_hom_app_unop, op_obj, CategoryTheory.Limits.coconeOpEquiv_functor_map_hom, CategoryTheory.Limits.FormalCoproduct.evalOpCompInlIsoId_inv_app_app, CategoryTheory.Adjunction.Quadruple.op_adj₃, mapTriangleOpCompTriangleOpEquivalenceFunctorApp_hom_hom₂, CategoryTheory.CostructuredArrow.toStructuredArrow_map, IsCoverDense.Types.pushforwardFamily_apply, CategoryTheory.GrothendieckTopology.pseudofunctorOver_toPrelaxFunctor_toPrelaxFunctorStruct_toPrefunctor_map_toFunctor_map_hom_app, LightProfinite.Extend.functorOp_final, CategoryTheory.presheafHom_map_app_op_mk_id, Condensed.lanPresheafIso_hom, pushforwardContinuousSheafificationCompatibility_hom_app_hom, CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence.functor_obj_left_left_as, CategoryTheory.Limits.limitOpIsoOpColimit_hom_comp_ι, TopCat.Presheaf.isSheaf_of_isOpenEmbedding, AlgebraicGeometry.ΓSpec.toSpecΓ_unop, CategoryTheory.Equivalence.sheafCongr.inverse_map_hom_app, CategoryTheory.StructuredArrow.toCostructuredArrow_obj, IsCoverDense.restrictHomEquivHom_naturality_left_symm, CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan.coconeApp_naturality_assoc, CategoryTheory.Pretriangulated.shiftFunctorCompIsoId_op_inv_app_assoc, CategoryTheory.colimitYonedaHomEquiv_π_apply, CategoryTheory.Presheaf.instIsLeftKanExtensionOppositeObjFunctorTypeUliftYonedaUliftYonedaMap, AlgebraicGeometry.PresheafedSpace.ofRestrict_c_app, IsCoverDense.Types.naturality_assoc, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit, CategoryTheory.Limits.Cone.op_ι, CategoryTheory.Limits.hasColimit_op_of_hasLimit, op_map, PresheafOfModules.pushforward₀_obj_map, CategoryTheory.Presheaf.isSheaf_iff_isLimit_pretopology, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_inv_app_assoc, CategoryTheory.GrothendieckTopology.Point.presheafFiberCocone_pt, CategoryTheory.MonoidalCategory.dayConvolutionInternalHomDiagramFunctor_map_app_app_app, CategoryTheory.Limits.parallelPairOpIso_hom_app_zero, IsCoverDense.Types.naturality
opComp 📖	CompOp	11 math math: CategoryTheory.NatIso.op_rightUnitor, opComp_inv_app, CategoryTheory.NatIso.op_isoWhiskerLeft, CategoryTheory.NatTrans.op_whiskerRight_assoc, CategoryTheory.NatIso.op_isoWhiskerRight, CategoryTheory.NatTrans.op_whiskerRight, CategoryTheory.NatTrans.op_whiskerLeft_assoc, CategoryTheory.NatIso.op_associator, CategoryTheory.NatIso.op_leftUnitor, opComp_hom_app, CategoryTheory.NatTrans.op_whiskerLeft
opHom 📖	CompOp	11 math math: CategoryTheory.cosimplicialSimplicialEquiv_counitIso_hom_app_app, opUnopEquiv_counitIso, CategoryTheory.Limits.opHomCompWhiskeringLimYonedaIsoCocones_hom_app_app_app, opUnopEquiv_functor, CategoryTheory.cosimplicialSimplicialEquiv_counitIso_inv_app_app, CategoryTheory.cosimplicialSimplicialEquiv_unitIso_hom_app, CategoryTheory.cosimplicialSimplicialEquiv_unitIso_inv_app, opHom_obj, opUnopEquiv_unitIso, opHom_map_app, CategoryTheory.Limits.opHomCompWhiskeringLimYonedaIsoCocones_inv_app_app
opId 📖	CompOp	4 math math: CategoryTheory.NatIso.op_rightUnitor, opId_hom_app, CategoryTheory.NatIso.op_leftUnitor, opId_inv_app
opInv 📖	CompOp	9 math math: CategoryTheory.cosimplicialSimplicialEquiv_counitIso_hom_app_app, opInv_obj, opUnopEquiv_counitIso, CategoryTheory.cosimplicialSimplicialEquiv_counitIso_inv_app_app, opUnopEquiv_inverse, CategoryTheory.cosimplicialSimplicialEquiv_unitIso_hom_app, opInv_map, CategoryTheory.cosimplicialSimplicialEquiv_unitIso_inv_app, opUnopEquiv_unitIso
opUnopEquiv 📖	CompOp	4 math math: opUnopEquiv_counitIso, opUnopEquiv_functor, opUnopEquiv_inverse, opUnopEquiv_unitIso
opUnopIso 📖	CompOp	5 math math: opUnopIso_hom_app, opUnopIso_inv_app, CategoryTheory.cosimplicialSimplicialEquiv_unitIso_hom_app, CategoryTheory.cosimplicialSimplicialEquiv_unitIso_inv_app, opUnopEquiv_unitIso
rightOp 📖	CompOp	184 math math: CategoryTheory.Limits.colimitHomIsoLimitYoneda'_hom_comp_π, functorHomEquiv_apply_app, natTransEquiv_apply_app, CategoryTheory.Limits.preservesColimitsOfShape_rightOp, CategoryTheory.Limits.isLimitConeRightOpOfCocone_lift, CategoryTheory.Limits.preservesFiniteCoproducts_rightOp, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit_app, rightOp_map, CategoryTheory.Limits.coneRightOpOfCoconeEquiv_functor_obj, AlgebraicGeometry.ΓSpec.isIso_adjunction_counit, CategoryTheory.FunctorToTypes.functorHomEquiv_symm_apply_app_app, AlgebraicGeometry.instPreservesLimitSchemeOppositeCommRingCatRightOpΓOfIsAffineHomMapOfCompactSpaceOfQuasiSeparatedSpaceCarrierCarrierObj, CategoryTheory.Limits.coneRightOpOfCoconeEquiv_functor_map_hom, CategoryTheory.Limits.reflectsLimits_rightOp, CategoryTheory.Limits.preservesColimits_rightOp, leftOpRightOpEquiv_counitIso_inv_app_app, CategoryTheory.simplicialCosimplicialEquiv_counitIso_inv_app_app, CategoryTheory.NatTrans.Equifibered.rightOp, AlgebraicGeometry.AffineScheme.instIsEquivalenceOppositeCommRingCatOpRightOpΓ, AlgebraicGeometry.Scheme.SpecΓIdentity_hom_app, AlgebraicGeometry.AffineScheme.instIsEquivalenceOppositeCommRingCatRightOpΓ, CategoryTheory.Limits.isColimitOfConeRightOpOfCocone_desc, CategoryTheory.NatTrans.rightOp_comp, CategoryTheory.Limits.isLimitOfCoconeOfConeRightOp_lift, CategoryTheory.Limits.colimitHomIsoLimitYoneda'_hom_comp_π_assoc, rightOpComp_hom_app, rightOpLeftOpIso_hom_app, instFinalOppositeRightOpOfInitial, CategoryTheory.Limits.coneOfCoconeRightOp_π, CategoryTheory.Limits.preservesColimit_rightOp, CategoryTheory.Limits.coconeRightOpOfConeEquiv_functor_map_hom, CategoryTheory.Limits.ι_comp_colimitRightOpIsoUnopLimit_hom_assoc, AlgebraicGeometry.Scheme.kerAdjunction_unit_app, CategoryTheory.Limits.coneRightOpOfCoconeEquiv_inverse_obj, leftOpRightOpEquiv_unitIso_inv_app, CategoryTheory.ComposableArrows.opEquivalence_unitIso_inv_app, AlgebraicGeometry.ΓSpec.adjunction_counit_app', AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit_app', CategoryTheory.Limits.coyonedaOpColimitIsoLimitCoyoneda'_inv_comp_π, CategoryTheory.LocalizerMorphism.LeftResolution.opEquivalence_inverse, CategoryTheory.Limits.preservesFiniteColimits_rightOp, CategoryTheory.Limits.reflectsLimit_rightOp, AlgebraicGeometry.AffineScheme.Γ_preservesLimits, CategoryTheory.Limits.reflectsColimitsOfShape_rightOp, natTransEquiv_symm_apply_app, CategoryTheory.Limits.reflectsColimitsOfSize_rightOp, CategoryTheory.Limits.hasColimit_rightOp_iff_hasLimit, CategoryTheory.simplicialCosimplicialEquiv_unitIso_hom_app, CategoryTheory.NatTrans.rightOpWhiskerRight, AlgebraicGeometry.ΓSpec.adjunction_counit_app, leftOpRightOpIso_hom_app, rightOpId_inv_app, rightOp_obj, CategoryTheory.simplicialCosimplicialEquiv_counitIso_hom_app_app, AlgebraicGeometry.Scheme.SpecΓIdentity_app, LightCondensed.isColimitLocallyConstantPresheafDiagram_desc_apply, CategoryTheory.MorphismProperty.IsInvertedBy.rightOp, CategoryTheory.Limits.hasLimit_rightOp_iff_hasColimit, AlgebraicGeometry.Scheme.kerAdjunction_counit_app, CategoryTheory.FunctorToTypes.rightAdj_map_app, AlgebraicGeometry.ΓSpec.toOpen_comp_locallyRingedSpaceAdjunction_homEquiv_app, AlgebraicGeometry.ΓSpec.adjunction_unit_app, CategoryTheory.Limits.preservesLimitsOfSize_rightOp, IsLeftAdjoint.rightOp, AlgebraicGeometry.Scheme.AffineZariskiSite.restrictIsoSpec_hom_app, CategoryTheory.FunctorToTypes.rightAdj_map_app_app, CategoryTheory.Limits.limitRightOpIsoOpColimit_hom_comp_ι, CategoryTheory.CategoryOfElements.costructuredArrowYonedaEquivalence_counitIso, CategoryTheory.CategoryOfElements.costructuredArrowYonedaEquivalence_inverse, AlgebraicGeometry.ΓSpec.left_triangle, CategoryTheory.Enriched.Functor.functorHom_whiskerLeft_natTransEquiv_symm_app, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_homEquiv_apply', CategoryTheory.Limits.coyonedaOpColimitIsoLimitCoyoneda'_hom_comp_π, AlgebraicGeometry.ΓSpec.unop_locallyRingedSpaceAdjunction_counit_app', rightOpLeftOpIso_inv_app, CategoryTheory.Limits.coneRightOpOfCocone_π, functorHom_ext_iff, rightOp_faithful, rightOp_full, rightOp_leftOp_eq, CategoryTheory.ComposableArrows.opEquivalence_unitIso_hom_app, CategoryTheory.Limits.coconeOfConeRightOp_ι, AlgebraicGeometry.instIsIsoSchemeAppUnitOppositeCommRingCatAdjunctionOfIsAffine, CategoryTheory.Limits.coconeRightOpOfConeEquiv_inverse_obj, CategoryTheory.Limits.coconeRightOpOfConeEquiv_unitIso, CategoryTheory.Adjunction.rightOp_counit, CategoryTheory.Limits.reflectsLimitsOfSize_rightOp, AlgebraicGeometry.ΓSpec.isIso_locallyRingedSpaceAdjunction_counit, CategoryTheory.Enriched.Functor.natTransEquiv_symm_app_app_apply, CategoryTheory.Adjunction.rightOp_eq, CategoryTheory.Join.InclLeftCompRightOpOpEquivFunctor_hom_app, leftOpRightOpEquiv_unitIso_hom_app, CategoryTheory.Limits.colimitHomIsoLimitYoneda'_inv_comp_π_assoc, CategoryTheory.NatTrans.removeRightOp_app, CategoryTheory.Limits.limitRightOpIsoOpColimit_hom_comp_ι_assoc, CategoryTheory.Limits.coconeRightOpOfConeEquiv_inverse_map, CategoryTheory.Limits.isLimitOfCoconeRightOpOfCone_lift, CategoryTheory.Limits.coconeOfConeRightOp_pt, CategoryTheory.Limits.coneRightOpOfCoconeEquiv_inverse_map, CategoryTheory.Limits.isLimitConeOfCoconeRightOp_lift, CategoryTheory.Limits.coneOfCoconeRightOp_pt, CategoryTheory.Limits.ι_comp_colimitRightOpIsoUnopLimit_hom, CategoryTheory.Limits.reflectsColimit_rightOp, CategoryTheory.Limits.hasLimit_rightOp_of_hasColimit, CategoryTheory.NatTrans.rightOpWhiskerRight_assoc, CategoryTheory.Limits.preservesFiniteProducts_rightOp, AlgebraicGeometry.ΓSpec.adjunction_homEquiv_apply, CategoryTheory.simplicialCosimplicialEquiv_functor_map_app, leftOpRightOpEquiv_counitIso_hom_app_app, CategoryTheory.CategoryOfElements.costructuredArrowYonedaEquivalence_unitIso, CategoryTheory.Limits.coneRightOpOfCocone_pt, rightOpComp_inv_app, CategoryTheory.Adjunction.rightOp_unit, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_homEquiv_apply, AlgebraicGeometry.ΓSpecIso_inv_ΓSpec_adjunction_homEquiv, CategoryTheory.NatTrans.rightOp_app, CategoryTheory.Enriched.Functor.natTransEquiv_symm_whiskerRight_functorHom_app, CategoryTheory.Limits.reflectsFiniteProducts_rightOp, CategoryTheory.Limits.reflectsLimitsOfShape_rightOp, AlgebraicGeometry.Scheme.AffineZariskiSite.PreservesLocalization.isLocallyDirected, CategoryTheory.Limits.coyonedaOpColimitIsoLimitCoyoneda'_inv_comp_π_assoc, AlgebraicGeometry.Scheme.SpecΓIdentity_inv_app, CategoryTheory.simplicialCosimplicialEquiv_unitIso_inv_app, AlgebraicGeometry.preservesLimit_rightOp_Γ, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_unit, CategoryTheory.Limits.preservesFiniteLimits_rightOp, CategoryTheory.FunctorToTypes.rightAdj_obj_map_app, CategoryTheory.Limits.isColimitCoconeRightOpOfCone_desc, CategoryTheory.Limits.isColimitCoconeOfConeRightOp_desc, CategoryTheory.Limits.π_comp_colimitRightOpIsoUnopLimit_inv, CategoryTheory.NatTrans.Coequifibered.rightOp, CategoryTheory.Limits.reflectsFiniteColimits_rightOp, CategoryTheory.Limits.coconeRightOpOfCone_ι, AlgebraicGeometry.ΓSpec.right_triangle, CategoryTheory.Limits.coconeRightOpOfConeEquiv_counitIso, CategoryTheory.Join.InclLeftCompRightOpOpEquivFunctor_inv_app, CategoryTheory.SimplicialObject.Augmented.rightOp_hom_app, CategoryTheory.Limits.coconeRightOpOfConeEquiv_functor_obj, AlgebraicGeometry.ΓSpec.adjunction_homEquiv_symm_apply, CategoryTheory.Limits.coyonedaOpColimitIsoLimitCoyoneda'_hom_comp_π_assoc, CategoryTheory.Limits.hasColimit_rightOp_of_hasLimit, CategoryTheory.Join.InclRightCompRightOpOpEquivFunctor_inv_app, CategoryTheory.Limits.preservesColimitsOfSize_rightOp, CategoryTheory.Limits.isColimitOfConeOfCoconeRightOp_desc, instEssSurjOppositeRightOp, CategoryTheory.Limits.colimitCoyonedaHomIsoLimit_π_apply, rightOpId_hom_app, AlgebraicGeometry.Scheme.AffineZariskiSite.cocone_ι_app, rightOp_additive, CategoryTheory.coyonedaPairing_map, CategoryTheory.Limits.π_comp_colimitRightOpIsoUnopLimit_inv_assoc, instIsEquivalenceOppositeRightOp, CategoryTheory.LocalizerMorphism.LeftResolution.opEquivalence_counitIso, AlgebraicGeometry.Scheme.AffineZariskiSite.cocone_pt, AlgebraicGeometry.ΓSpec_adjunction_homEquiv_eq, CategoryTheory.Limits.limitRightOpIsoOpColimit_inv_comp_π_assoc, CategoryTheory.coyonedaPairingExt_iff, CategoryTheory.Limits.coconeRightOpOfCone_pt, AlgebraicGeometry.LocallyRingedSpace.SpecΓIdentity_hom_app, CategoryTheory.Limits.reflectsColimits_rightOp, leftOpRightOpEquiv_functor_map_app, CategoryTheory.Limits.coneRightOpOfCoconeEquiv_unitIso, CategoryTheory.Limits.preservesLimitsOfShape_rightOp, CategoryTheory.NatTrans.rightOp_id, CategoryTheory.NatTrans.removeRightOp_id, CategoryTheory.Limits.FormalCoproduct.cosimplicialObjectFunctor_map_app, CategoryTheory.Limits.colimitYonedaHomIsoLimitRightOp_π_apply, rightOp_map_unop, AlgebraicGeometry.LocallyRingedSpace.SpecΓIdentity_inv_app, leftOpRightOpIso_inv_app, instInitialOppositeRightOpOfFinal, CategoryTheory.Limits.reflectsFiniteLimits_rightOp, CategoryTheory.Join.InclRightCompRightOpOpEquivFunctor_hom_app, CategoryTheory.Limits.preservesLimits_rightOp, AlgebraicGeometry.Scheme.AffineZariskiSite.PreservesLocalization.isOpenImmersion, IsRightAdjoint.rightOp, AlgebraicGeometry.Scheme.AffineZariskiSite.restrictIsoSpec_inv_app, CategoryTheory.Limits.coneRightOpOfCoconeEquiv_counitIso, CategoryTheory.Limits.reflectsFiniteCoproducts_rightOp, CategoryTheory.Limits.preservesLimit_rightOp, CategoryTheory.Limits.limitRightOpIsoOpColimit_inv_comp_π, functorHomEquiv_symm_apply_app_app, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit, CategoryTheory.Limits.colimitHomIsoLimitYoneda'_inv_comp_π
rightOpComp 📖	CompOp	4 math math: rightOpComp_hom_app, CategoryTheory.NatTrans.rightOpWhiskerRight, CategoryTheory.NatTrans.rightOpWhiskerRight_assoc, rightOpComp_inv_app
rightOpId 📖	CompOp	2 math math: rightOpId_inv_app, rightOpId_hom_app
rightOpLeftOpIso 📖	CompOp	8 math math: rightOpLeftOpIso_hom_app, leftOpRightOpEquiv_unitIso_inv_app, CategoryTheory.ComposableArrows.opEquivalence_unitIso_inv_app, CategoryTheory.simplicialCosimplicialEquiv_unitIso_hom_app, rightOpLeftOpIso_inv_app, CategoryTheory.ComposableArrows.opEquivalence_unitIso_hom_app, leftOpRightOpEquiv_unitIso_hom_app, CategoryTheory.simplicialCosimplicialEquiv_unitIso_inv_app
unop 📖	CompOp	114 math math: CategoryTheory.Limits.limitUnopIsoUnopColimit_hom_comp_ι, CategoryTheory.NatTrans.unop_whiskerLeft, opUnopIso_hom_app, CategoryTheory.Limits.reflectsColimitsOfShape_unop, CategoryTheory.Limits.coneUnopOfCoconeEquiv_counitIso, unopOpIso_inv_app, CategoryTheory.Limits.reflectsFiniteProducts_unop, CategoryTheory.NatIso.unop_rightUnitor, CategoryTheory.NatTrans.unop_comp_assoc, CategoryTheory.NatTrans.unop_whiskerLeft_assoc, CategoryTheory.Limits.reflectsColimitsOfSize_unop, opInv_obj, CategoryTheory.Limits.reflectsColimits_unop, CategoryTheory.Limits.coconeUnopOfConeEquiv_unitIso, CategoryTheory.Limits.preservesLimits_unop, CategoryTheory.Limits.coneUnopOfCocone_pt, opUnopIso_inv_app, unopId_inv_app, CategoryTheory.Limits.π_comp_colimitUnopIsoOpLimit_inv, CategoryTheory.Limits.coconeOfConeUnop_pt, CategoryTheory.Limits.preservesColimit_unop, CategoryTheory.Limits.reflectsLimitsOfShape_unop, CategoryTheory.NatIso.unop_trans, unop_additive, CategoryTheory.NatTrans.removeUnop_app, CategoryTheory.Limits.preservesColimitsOfSize_unop, CategoryTheory.Limits.colimitYonedaHomIsoLimit_π_apply, CategoryTheory.Limits.coconeUnopOfConeEquiv_counitIso, CategoryTheory.Limits.coneOfCoconeUnop_π, CategoryTheory.Limits.coconeUnopOfConeEquiv_functor_obj, CategoryTheory.Limits.isLimitConeOfCoconeUnop_lift, CategoryTheory.Limits.isColimitOfConeUnopOfCocone_desc, CategoryTheory.cosimplicialSimplicialEquiv_inverse_obj, CategoryTheory.Limits.reflectsColimit_unop, CategoryTheory.Limits.coconeUnopOfConeEquiv_functor_map_hom, CategoryTheory.MorphismProperty.IsInvertedBy.unop, CategoryTheory.Limits.isColimitCoconeUnopOfCone_desc, CategoryTheory.Limits.limitUnopIsoUnopColimit_inv_comp_π_assoc, CategoryTheory.NatTrans.unop_whiskerRight, CategoryTheory.Limits.colimitCoyonedaHomIsoLimitUnop_π_apply, CategoryTheory.Limits.coconeUnopOfConeEquiv_inverse_map, CategoryTheory.Limits.preservesLimitsOfShape_unop, CategoryTheory.Limits.reflectsFiniteCoproducts_unop, CategoryTheory.Equivalence.unop_counitIso, CategoryTheory.Limits.preservesFiniteCoproducts_unop, CategoryTheory.Limits.isLimitConeUnopOfCocone_lift, CategoryTheory.NatIso.unop_symm, unopOpIso_hom_app, CategoryTheory.Adjunction.unop_counit, CategoryTheory.Limits.coneUnopOfCoconeEquiv_functor_map_hom, CategoryTheory.Limits.preservesColimits_unop, unop_obj, CategoryTheory.NatIso.unop_whiskerRight, CategoryTheory.Limits.hasLimit_unop_of_hasColimit, CategoryTheory.cosimplicialSimplicialEquiv_unitIso_hom_app, unop_map, opInv_map, CategoryTheory.Limits.coneUnopOfCoconeEquiv_functor_obj, CategoryTheory.NatIso.unop_leftUnitor, CategoryTheory.Limits.reflectsLimits_unop, CategoryTheory.NatIso.unop_associator, CategoryTheory.NatTrans.unop_id, CategoryTheory.Limits.coneUnopOfCoconeEquiv_inverse_obj, CategoryTheory.Limits.preservesFiniteProducts_unop, CategoryTheory.Limits.isLimitOfCoconeOfConeUnop_lift, CategoryTheory.cosimplicialSimplicialEquiv_unitIso_inv_app, CategoryTheory.Limits.preservesLimitsOfSize_unop, CategoryTheory.Limits.coconeUnopOfCone_pt, IsLocalization.unop, CategoryTheory.Limits.reflectsFiniteColimits_unop, opUnopEquiv_unitIso, CategoryTheory.NatTrans.equifibered_unop_iff, CategoryTheory.Limits.limitUnopIsoUnopColimit_hom_comp_ι_assoc, CategoryTheory.Limits.preservesFiniteLimits_unop, CategoryTheory.Limits.ι_comp_colimitUnopIsoOpLimit_hom_assoc, CategoryTheory.NatTrans.coequifibered_unop_iff, unopComp_inv_app, CategoryTheory.NatTrans.Coequifibered.unop, CategoryTheory.Limits.preservesLimit_unop, CategoryTheory.Limits.coconeUnopOfConeEquiv_inverse_obj, CategoryTheory.Limits.coconeOfConeUnop_ι, CategoryTheory.NatTrans.removeUnop_id, CategoryTheory.Equivalence.unop_inverse, CategoryTheory.Limits.coneUnopOfCoconeEquiv_inverse_map, CategoryTheory.Limits.isLimitOfCoconeUnopOfCone_lift, CategoryTheory.Limits.preservesFiniteColimits_unop, CategoryTheory.NatIso.unop_refl, CategoryTheory.NatTrans.Equifibered.unop, CategoryTheory.Equivalence.unop_unitIso, CategoryTheory.cosimplicialSimplicialEquiv_inverse_map, CategoryTheory.Limits.π_comp_colimitUnopIsoOpLimit_inv_assoc, CategoryTheory.Equivalence.unop_functor, CategoryTheory.Limits.isColimitOfConeOfCoconeUnop_desc, CategoryTheory.NatIso.unop_inv, CategoryTheory.Limits.coconeUnopOfCone_ι, CategoryTheory.Limits.reflectsFiniteLimits_unop, CategoryTheory.NatTrans.unop_whiskerRight_assoc, CategoryTheory.Limits.preservesColimitsOfShape_unop, CategoryTheory.Limits.coneUnopOfCocone_π, CategoryTheory.Limits.hasColimit_unop_of_hasLimit, CategoryTheory.NatIso.unop_whiskerLeft, CategoryTheory.Limits.isColimitCoconeOfConeUnop_desc, CategoryTheory.NatIso.unop_hom, unopId_hom_app, CategoryTheory.Adjunction.unop_unit, CategoryTheory.Limits.reflectsLimitsOfSize_unop, CategoryTheory.Limits.limitUnopIsoUnopColimit_inv_comp_π, CategoryTheory.NatTrans.unop_app, CategoryTheory.Limits.coneOfCoconeUnop_pt, unopComp_hom_app, CategoryTheory.Limits.coneUnopOfCoconeEquiv_unitIso, CategoryTheory.Limits.reflectsLimit_unop, CategoryTheory.Limits.ι_comp_colimitUnopIsoOpLimit_hom, CategoryTheory.NatTrans.unop_comp
unopComp 📖	CompOp	11 math math: CategoryTheory.NatTrans.unop_whiskerLeft, CategoryTheory.NatIso.unop_rightUnitor, CategoryTheory.NatTrans.unop_whiskerLeft_assoc, CategoryTheory.NatTrans.unop_whiskerRight, CategoryTheory.NatIso.unop_whiskerRight, CategoryTheory.NatIso.unop_leftUnitor, CategoryTheory.NatIso.unop_associator, unopComp_inv_app, CategoryTheory.NatTrans.unop_whiskerRight_assoc, CategoryTheory.NatIso.unop_whiskerLeft, unopComp_hom_app
unopId 📖	CompOp	4 math math: CategoryTheory.NatIso.unop_rightUnitor, unopId_inv_app, CategoryTheory.NatIso.unop_leftUnitor, unopId_hom_app
unopOpIso 📖	CompOp	3 math math: unopOpIso_inv_app, opUnopEquiv_counitIso, unopOpIso_hom_app

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
instEssSurjOppositeLeftOp 📖	mathematical	—	EssSurj Opposite CategoryTheory.Category.opposite leftOp	—	EssSurj.mem_essImage
instEssSurjOppositeOp 📖	mathematical	—	EssSurj Opposite CategoryTheory.Category.opposite op	—	EssSurj.mem_essImage
instEssSurjOppositeRightOp 📖	mathematical	—	EssSurj Opposite CategoryTheory.Category.opposite rightOp	—	EssSurj.mem_essImage
instFaithfulOppositeOp 📖	mathematical	—	Faithful Opposite CategoryTheory.Category.opposite op	—	map_injective
instFullOppositeOp 📖	mathematical	—	Full Opposite CategoryTheory.Category.opposite op	—	map_preimage
instIsEquivalenceOppositeLeftOp 📖	mathematical	—	IsEquivalence Opposite CategoryTheory.Category.opposite leftOp	—	leftOp_faithful IsEquivalence.faithful leftOp_full IsEquivalence.full instEssSurjOppositeLeftOp IsEquivalence.essSurj
instIsEquivalenceOppositeOp 📖	mathematical	—	IsEquivalence Opposite CategoryTheory.Category.opposite op	—	instFaithfulOppositeOp IsEquivalence.faithful instFullOppositeOp IsEquivalence.full instEssSurjOppositeOp IsEquivalence.essSurj
instIsEquivalenceOppositeRightOp 📖	mathematical	—	IsEquivalence Opposite CategoryTheory.Category.opposite rightOp	—	rightOp_faithful IsEquivalence.faithful rightOp_full IsEquivalence.full instEssSurjOppositeRightOp IsEquivalence.essSurj
leftOpComp_hom_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Category.opposite leftOp comp CategoryTheory.Iso.hom CategoryTheory.Functor category op leftOpComp CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct Opposite.unop obj	—	—
leftOpComp_inv_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Category.opposite leftOp comp CategoryTheory.Iso.inv CategoryTheory.Functor category op leftOpComp CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct Opposite.unop obj	—	—
leftOpId_hom_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Category.opposite leftOp id CategoryTheory.Iso.hom CategoryTheory.Functor category CategoryTheory.unopUnop leftOpId CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct Opposite.unop	—	—
leftOpId_inv_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Category.opposite leftOp id CategoryTheory.Iso.inv CategoryTheory.Functor category CategoryTheory.unopUnop leftOpId CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct Opposite.unop	—	—
leftOpRightOpEquiv_counitIso_hom_app_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Category.opposite obj CategoryTheory.Functor category comp Opposite.op leftOp Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct CategoryTheory.NatTrans.leftOp rightOp Opposite.unop CategoryTheory.NatTrans.rightOp Quiver.Hom.unop id CategoryTheory.Iso.hom CategoryTheory.Equivalence.counitIso leftOpRightOpEquiv CategoryTheory.CategoryStruct.id	—	—
leftOpRightOpEquiv_counitIso_inv_app_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Category.opposite obj CategoryTheory.Functor category id comp Opposite.op leftOp Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct CategoryTheory.NatTrans.leftOp rightOp Opposite.unop CategoryTheory.NatTrans.rightOp Quiver.Hom.unop CategoryTheory.Iso.inv CategoryTheory.Equivalence.counitIso leftOpRightOpEquiv CategoryTheory.CategoryStruct.id	—	—
leftOpRightOpEquiv_functor_map_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Category.opposite rightOp Opposite.unop CategoryTheory.Functor map category CategoryTheory.Equivalence.functor leftOpRightOpEquiv Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct obj Opposite.op Quiver.Hom.unop	—	—
leftOpRightOpEquiv_functor_obj_map 📖	mathematical	—	map Opposite CategoryTheory.Category.opposite obj CategoryTheory.Functor category CategoryTheory.Equivalence.functor leftOpRightOpEquiv Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct Opposite.unop Opposite.op	—	—
leftOpRightOpEquiv_functor_obj_obj 📖	mathematical	—	obj Opposite CategoryTheory.Category.opposite CategoryTheory.Functor category CategoryTheory.Equivalence.functor leftOpRightOpEquiv Opposite.op Opposite.unop	—	—
leftOpRightOpEquiv_inverse_map 📖	mathematical	—	map CategoryTheory.Functor Opposite CategoryTheory.Category.opposite category CategoryTheory.Equivalence.inverse leftOpRightOpEquiv Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct leftOp CategoryTheory.NatTrans.leftOp	—	—
leftOpRightOpEquiv_inverse_obj 📖	mathematical	—	obj CategoryTheory.Functor Opposite CategoryTheory.Category.opposite category CategoryTheory.Equivalence.inverse leftOpRightOpEquiv Opposite.op leftOp	—	—
leftOpRightOpEquiv_unitIso_hom_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Functor CategoryTheory.Category.opposite category id comp rightOp Opposite.unop CategoryTheory.NatTrans.rightOp Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct Opposite.op leftOp Quiver.Hom.op CategoryTheory.NatTrans.leftOp CategoryTheory.Iso.hom CategoryTheory.Equivalence.unitIso leftOpRightOpEquiv rightOpLeftOpIso	—	—
leftOpRightOpEquiv_unitIso_inv_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Functor CategoryTheory.Category.opposite category comp rightOp Opposite.unop CategoryTheory.NatTrans.rightOp Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct Opposite.op leftOp Quiver.Hom.op CategoryTheory.NatTrans.leftOp id CategoryTheory.Iso.inv CategoryTheory.Equivalence.unitIso leftOpRightOpEquiv rightOpLeftOpIso	—	—
leftOpRightOpIso_hom_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Category.opposite rightOp leftOp CategoryTheory.Iso.hom CategoryTheory.Functor category leftOpRightOpIso CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct obj	—	—
leftOpRightOpIso_inv_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Category.opposite rightOp leftOp CategoryTheory.Iso.inv CategoryTheory.Functor category leftOpRightOpIso CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct obj	—	—
leftOp_faithful 📖	mathematical	—	Faithful Opposite CategoryTheory.Category.opposite leftOp	—	map_injective
leftOp_full 📖	mathematical	—	Full Opposite CategoryTheory.Category.opposite leftOp	—	map_preimage
leftOp_map 📖	mathematical	—	map Opposite CategoryTheory.Category.opposite leftOp Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct obj Opposite.unop	—	—
leftOp_obj 📖	mathematical	—	obj Opposite CategoryTheory.Category.opposite leftOp Opposite.unop	—	—
opComp_hom_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Category.opposite op comp CategoryTheory.Iso.hom CategoryTheory.Functor category opComp CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct Opposite.op obj Opposite.unop	—	—
opComp_inv_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Category.opposite op comp CategoryTheory.Iso.inv CategoryTheory.Functor category opComp CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct Opposite.op obj Opposite.unop	—	—
opHom_map_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Category.opposite op Opposite.unop CategoryTheory.Functor map category opHom Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct obj Quiver.Hom.unop	—	—
opHom_obj 📖	mathematical	—	obj Opposite CategoryTheory.Functor CategoryTheory.Category.opposite category opHom op Opposite.unop	—	—
opId_hom_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Category.opposite op id CategoryTheory.Iso.hom CategoryTheory.Functor category opId CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct	—	—
opId_inv_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Category.opposite op id CategoryTheory.Iso.inv CategoryTheory.Functor category opId CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct	—	—
opInv_map 📖	mathematical	—	map CategoryTheory.Functor Opposite CategoryTheory.Category.opposite category opInv Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct unop Quiver.Hom.unop obj Opposite.op CategoryTheory.NatTrans.app	—	—
opInv_obj 📖	mathematical	—	obj CategoryTheory.Functor Opposite CategoryTheory.Category.opposite category opInv Opposite.op unop	—	—
opUnopEquiv_counitIso 📖	mathematical	—	CategoryTheory.Equivalence.counitIso Opposite CategoryTheory.Functor CategoryTheory.Category.opposite category opUnopEquiv CategoryTheory.NatIso.ofComponents comp opInv opHom id unopOpIso	—	—
opUnopEquiv_functor 📖	mathematical	—	CategoryTheory.Equivalence.functor Opposite CategoryTheory.Functor CategoryTheory.Category.opposite category opUnopEquiv opHom	—	—
opUnopEquiv_inverse 📖	mathematical	—	CategoryTheory.Equivalence.inverse Opposite CategoryTheory.Functor CategoryTheory.Category.opposite category opUnopEquiv opInv	—	—
opUnopEquiv_unitIso 📖	mathematical	—	CategoryTheory.Equivalence.unitIso Opposite CategoryTheory.Functor CategoryTheory.Category.opposite category opUnopEquiv CategoryTheory.NatIso.ofComponents id comp opHom opInv CategoryTheory.Iso.op unop op Opposite.unop opUnopIso	—	—
opUnopIso_hom_app 📖	mathematical	—	CategoryTheory.NatTrans.app unop op CategoryTheory.Iso.hom CategoryTheory.Functor category opUnopIso CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct obj	—	—
opUnopIso_inv_app 📖	mathematical	—	CategoryTheory.NatTrans.app unop op CategoryTheory.Iso.inv CategoryTheory.Functor category opUnopIso CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct obj	—	—
op_map 📖	mathematical	—	map Opposite CategoryTheory.Category.opposite op Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct obj Opposite.unop Quiver.Hom.unop	—	—
op_obj 📖	mathematical	—	obj Opposite CategoryTheory.Category.opposite op Opposite.op Opposite.unop	—	—
rightOpComp_hom_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Category.opposite rightOp comp CategoryTheory.Iso.hom CategoryTheory.Functor category op rightOpComp CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct Opposite.op obj	—	—
rightOpComp_inv_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Category.opposite rightOp comp CategoryTheory.Iso.inv CategoryTheory.Functor category op rightOpComp CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct Opposite.op obj	—	—
rightOpId_hom_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Category.opposite rightOp id CategoryTheory.Iso.hom CategoryTheory.Functor category CategoryTheory.opOp rightOpId CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct Opposite.op	—	—
rightOpId_inv_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Category.opposite rightOp id CategoryTheory.Iso.inv CategoryTheory.Functor category CategoryTheory.opOp rightOpId CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct Opposite.op	—	—
rightOpLeftOpIso_hom_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Category.opposite leftOp rightOp CategoryTheory.Iso.hom CategoryTheory.Functor category rightOpLeftOpIso CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct obj	—	—
rightOpLeftOpIso_inv_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Category.opposite leftOp rightOp CategoryTheory.Iso.inv CategoryTheory.Functor category rightOpLeftOpIso CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct obj	—	—
rightOp_faithful 📖	mathematical	—	Faithful Opposite CategoryTheory.Category.opposite rightOp	—	map_injective
rightOp_full 📖	mathematical	—	Full Opposite CategoryTheory.Category.opposite rightOp	—	map_preimage
rightOp_leftOp_eq 📖	mathematical	—	leftOp rightOp	—	—
rightOp_map 📖	mathematical	—	map Opposite CategoryTheory.Category.opposite rightOp Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct obj Opposite.op	—	—
rightOp_map_unop 📖	mathematical	—	Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct obj Opposite CategoryTheory.Category.opposite rightOp map Opposite.op Quiver.Hom.op	—	—
rightOp_obj 📖	mathematical	—	obj Opposite CategoryTheory.Category.opposite rightOp Opposite.op	—	—
unopComp_hom_app 📖	mathematical	—	CategoryTheory.NatTrans.app unop comp Opposite CategoryTheory.Category.opposite CategoryTheory.Iso.hom CategoryTheory.Functor category unopComp CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct Opposite.unop obj Opposite.op	—	—
unopComp_inv_app 📖	mathematical	—	CategoryTheory.NatTrans.app unop comp Opposite CategoryTheory.Category.opposite CategoryTheory.Iso.inv CategoryTheory.Functor category unopComp CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct Opposite.unop obj Opposite.op	—	—
unopId_hom_app 📖	mathematical	—	CategoryTheory.NatTrans.app unop id Opposite CategoryTheory.Category.opposite CategoryTheory.Iso.hom CategoryTheory.Functor category unopId CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct	—	—
unopId_inv_app 📖	mathematical	—	CategoryTheory.NatTrans.app unop id Opposite CategoryTheory.Category.opposite CategoryTheory.Iso.inv CategoryTheory.Functor category unopId CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct	—	—
unopOpIso_hom_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Category.opposite op unop CategoryTheory.Iso.hom CategoryTheory.Functor category unopOpIso CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct obj	—	—
unopOpIso_inv_app 📖	mathematical	—	CategoryTheory.NatTrans.app Opposite CategoryTheory.Category.opposite op unop CategoryTheory.Iso.inv CategoryTheory.Functor category unopOpIso CategoryTheory.CategoryStruct.id CategoryTheory.Category.toCategoryStruct obj	—	—
unop_map 📖	mathematical	—	map unop Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct obj Opposite CategoryTheory.Category.opposite Opposite.op Quiver.Hom.op	—	—
unop_obj 📖	mathematical	—	obj unop Opposite.unop Opposite CategoryTheory.Category.opposite Opposite.op	—	—

CategoryTheory.Functor.FullyFaithful

Definitions

Name	Category	Theorems
leftOp 📖	CompOp	—
op 📖	CompOp	—
rightOp 📖	CompOp	—

CategoryTheory.Iso

Definitions

Name	Category	Theorems
op 📖	CompOp	31 math math: AlgebraicGeometry.Scheme.germ_stalkClosedPointTo, CategoryTheory.CategoryOfElements.costructuredArrowULiftYonedaEquivalence_unitIso, CategoryTheory.MonoidalCategory.MonoidalLeftAction.oppositeLeftAction_actionAssocIso, op_inv, op_symm, CategoryTheory.Limits.opCoproductIsoProduct'_comp_self, op_hom, CategoryTheory.op_leftUnitor, op_trans, CategoryTheory.Limits.opProductIsoCoproduct'_comp_self, CategoryTheory.op_braiding, CategoryTheory.MonoidalCategory.MonoidalRightAction.oppositeRightAction_actionAssocIso_op, CategoryTheory.op_rightUnitor, AlgebraicGeometry.Scheme.Hom.comp_appIso, AlgebraicGeometry.Scheme.Hom.id_appIso, AlgebraicGeometry.Scheme.isoSpec_Spec, CategoryTheory.Equivalence.symmEquiv_counitIso, CategoryTheory.CategoryOfElements.costructuredArrowYonedaEquivalence_unitIso, CategoryTheory.isoOpEquiv_symm_apply, CategoryTheory.Functor.opUnopEquiv_unitIso, op_refl, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo_assoc, op_unop, CategoryTheory.ShortComplex.HomologyData.op_iso, CategoryTheory.MonoidalCategory.MonoidalLeftAction.oppositeLeftAction_actionAssocIso_op, CompHausLike.LocallyConstant.sigmaComparison_comp_sigmaIso, unop_op, CategoryTheory.MonoidalCategory.MonoidalRightAction.oppositeRightAction_actionAssocIso, CategoryTheory.op_associator, CategoryTheory.MonoidalCategory.MonoidalLeftAction.oppositeLeftAction_actionUnitIso, CategoryTheory.MonoidalCategory.MonoidalRightAction.oppositeRightAction_actionUnitIso
unop 📖	CompOp	21 math math: CategoryTheory.Localization.isoOfHom_unop, CategoryTheory.MonoidalCategory.MonoidalLeftAction.leftActionOfOppositeLeftAction_actionAssocIso, unop_refl, unop_trans, CategoryTheory.MonoidalCategory.MonoidalRightAction.rightActionOfOppositeRightAction_actionUnitIso, CategoryTheory.MonoidalCategory.MonoidalRightAction.rightActionOfOppositeRightAction_actionAssocIso_unop, unop_symm, CategoryTheory.MonoidalCategory.MonoidalLeftAction.leftActionOfOppositeLeftAction_actionUnitIso, CategoryTheory.Equivalence.inverseFunctorMapIso_symm_eq_isoInverseOfIsoFunctor, CategoryTheory.unop_leftUnitor, CategoryTheory.unop_braiding, unop_inv, CategoryTheory.MonoidalCategory.MonoidalLeftAction.leftActionOfOppositeLeftAction_actionAssocIso_unop, CategoryTheory.unop_rightUnitor, CategoryTheory.MonoidalCategory.MonoidalRightAction.rightActionOfOppositeRightAction_actionAssocIso, unop_hom, op_unop, CategoryTheory.ShortComplex.HomologyData.unop_iso, CategoryTheory.isoOpEquiv_apply, unop_op, CategoryTheory.unop_associator

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
op_hom 📖	mathematical	—	hom Opposite CategoryTheory.Category.opposite Opposite.op op Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct	—	—
op_inv 📖	mathematical	—	inv Opposite CategoryTheory.Category.opposite Opposite.op op Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct	—	—
op_refl 📖	mathematical	—	op refl Opposite CategoryTheory.Category.opposite Opposite.op	—	—
op_symm 📖	mathematical	—	op symm Opposite CategoryTheory.Category.opposite Opposite.op	—	—
op_trans 📖	mathematical	—	op trans Opposite CategoryTheory.Category.opposite Opposite.op	—	—
op_unop 📖	mathematical	—	unop Opposite.op op	—	ext
unop_hom 📖	mathematical	—	hom Opposite.unop unop Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct Opposite CategoryTheory.Category.opposite	—	—
unop_hom_inv_id_app 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CategoryTheory.Category.toCategoryStruct Opposite.unop CategoryTheory.Functor.obj Opposite CategoryTheory.Category.opposite Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver CategoryTheory.NatTrans.app hom CategoryTheory.Functor CategoryTheory.Functor.category inv CategoryTheory.CategoryStruct.id	—	CategoryTheory.unop_comp inv_hom_id_app CategoryTheory.unop_id
unop_hom_inv_id_app_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CategoryTheory.Category.toCategoryStruct Opposite.unop CategoryTheory.Functor.obj Opposite CategoryTheory.Category.opposite Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver CategoryTheory.NatTrans.app hom CategoryTheory.Functor CategoryTheory.Functor.category inv	—	CategoryTheory.Category.assoc CategoryTheory.Category.id_comp Mathlib.Tactic.Reassoc.eq_whisker' unop_hom_inv_id_app
unop_inv 📖	mathematical	—	inv Opposite.unop unop Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct Opposite CategoryTheory.Category.opposite	—	—
unop_inv_hom_id_app 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CategoryTheory.Category.toCategoryStruct Opposite.unop CategoryTheory.Functor.obj Opposite CategoryTheory.Category.opposite Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver CategoryTheory.NatTrans.app inv CategoryTheory.Functor CategoryTheory.Functor.category hom CategoryTheory.CategoryStruct.id	—	CategoryTheory.unop_comp hom_inv_id_app CategoryTheory.unop_id
unop_inv_hom_id_app_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CategoryTheory.Category.toCategoryStruct Opposite.unop CategoryTheory.Functor.obj Opposite CategoryTheory.Category.opposite Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver CategoryTheory.NatTrans.app inv CategoryTheory.Functor CategoryTheory.Functor.category hom	—	CategoryTheory.Category.assoc CategoryTheory.Category.id_comp Mathlib.Tactic.Reassoc.eq_whisker' unop_inv_hom_id_app
unop_op 📖	mathematical	—	op Opposite.unop unop	—	ext
unop_refl 📖	mathematical	—	unop refl Opposite CategoryTheory.Category.opposite Opposite.unop	—	—
unop_symm 📖	mathematical	—	unop symm Opposite CategoryTheory.Category.opposite Opposite.unop	—	—
unop_trans 📖	mathematical	—	unop trans Opposite CategoryTheory.Category.opposite Opposite.unop	—	—

CategoryTheory.NatIso

Definitions

Name	Category	Theorems
op 📖	CompOp	14 math math: op_rightUnitor, CategoryTheory.Functor.commShiftOp_iso_eq, op_trans, op_isoWhiskerLeft, op_isoWhiskerRight, op_symm, op_associator, op_leftUnitor, op_refl, op_hom, op_inv, CategoryTheory.Equivalence.op_unitIso, CategoryTheory.Equivalence.op_counitIso, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit
removeOp 📖	CompOp	3 math math: removeOp_inv, removeOp_hom, CategoryTheory.Functor.commShiftUnop_commShiftIso
unop 📖	CompOp	12 math math: unop_rightUnitor, unop_trans, CategoryTheory.Equivalence.unop_counitIso, unop_symm, unop_whiskerRight, unop_leftUnitor, unop_associator, unop_refl, CategoryTheory.Equivalence.unop_unitIso, unop_inv, unop_whiskerLeft, unop_hom

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
op_associator 📖	mathematical	—	op CategoryTheory.Functor.comp CategoryTheory.Functor.associator CategoryTheory.Iso.trans CategoryTheory.Functor Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.category CategoryTheory.Functor.op CategoryTheory.Functor.opComp CategoryTheory.Functor.isoWhiskerLeft CategoryTheory.Iso.symm CategoryTheory.Functor.isoWhiskerRight	—	CategoryTheory.Iso.ext CategoryTheory.NatTrans.ext' CategoryTheory.Functor.map_id CategoryTheory.Category.comp_id
op_hom 📖	mathematical	—	CategoryTheory.Iso.hom CategoryTheory.Functor Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.category CategoryTheory.Functor.op op CategoryTheory.NatTrans.op	—	—
op_inv 📖	mathematical	—	CategoryTheory.Iso.inv CategoryTheory.Functor Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.category CategoryTheory.Functor.op op CategoryTheory.NatTrans.op	—	—
op_isoWhiskerLeft 📖	mathematical	—	op CategoryTheory.Functor.comp CategoryTheory.Functor.isoWhiskerLeft CategoryTheory.Iso.trans CategoryTheory.Functor Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.category CategoryTheory.Functor.op CategoryTheory.Functor.opComp CategoryTheory.Iso.symm	—	CategoryTheory.Iso.ext CategoryTheory.NatTrans.ext' CategoryTheory.Category.comp_id CategoryTheory.Category.id_comp
op_isoWhiskerRight 📖	mathematical	—	op CategoryTheory.Functor.comp CategoryTheory.Functor.isoWhiskerRight CategoryTheory.Iso.trans CategoryTheory.Functor Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.category CategoryTheory.Functor.op CategoryTheory.Functor.opComp CategoryTheory.Iso.symm	—	CategoryTheory.Iso.ext CategoryTheory.NatTrans.ext' CategoryTheory.Category.comp_id CategoryTheory.Category.id_comp
op_leftUnitor 📖	mathematical	—	op CategoryTheory.Functor.comp CategoryTheory.Functor.id CategoryTheory.Functor.leftUnitor CategoryTheory.Iso.trans CategoryTheory.Functor Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.category CategoryTheory.Functor.op CategoryTheory.Iso.symm CategoryTheory.Functor.isoWhiskerRight CategoryTheory.Functor.opId CategoryTheory.Functor.opComp	—	CategoryTheory.Iso.ext CategoryTheory.NatTrans.ext' CategoryTheory.Functor.map_id CategoryTheory.Category.comp_id
op_refl 📖	mathematical	—	op CategoryTheory.Iso.refl CategoryTheory.Functor CategoryTheory.Functor.category Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.op	—	—
op_rightUnitor 📖	mathematical	—	op CategoryTheory.Functor.comp CategoryTheory.Functor.id CategoryTheory.Functor.rightUnitor CategoryTheory.Iso.trans CategoryTheory.Functor Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.category CategoryTheory.Functor.op CategoryTheory.Iso.symm CategoryTheory.Functor.isoWhiskerLeft CategoryTheory.Functor.opId CategoryTheory.Functor.opComp	—	CategoryTheory.Iso.ext CategoryTheory.NatTrans.ext' CategoryTheory.Category.comp_id
op_symm 📖	mathematical	—	op CategoryTheory.Iso.symm CategoryTheory.Functor CategoryTheory.Functor.category Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.op	—	—
op_trans 📖	mathematical	—	op CategoryTheory.Iso.trans CategoryTheory.Functor CategoryTheory.Functor.category Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.op	—	—
removeOp_hom 📖	mathematical	—	CategoryTheory.Iso.hom CategoryTheory.Functor CategoryTheory.Functor.category removeOp CategoryTheory.NatTrans.removeOp Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.op	—	—
removeOp_inv 📖	mathematical	—	CategoryTheory.Iso.inv CategoryTheory.Functor CategoryTheory.Functor.category removeOp CategoryTheory.NatTrans.removeOp Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.op	—	—
unop_associator 📖	mathematical	—	unop CategoryTheory.Functor.comp Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.associator CategoryTheory.Iso.trans CategoryTheory.Functor CategoryTheory.Functor.category CategoryTheory.Functor.unop CategoryTheory.Functor.unopComp CategoryTheory.Functor.isoWhiskerLeft CategoryTheory.Iso.symm CategoryTheory.Functor.isoWhiskerRight	—	CategoryTheory.Iso.ext CategoryTheory.NatTrans.ext' CategoryTheory.Functor.map_id CategoryTheory.Category.comp_id
unop_hom 📖	mathematical	—	CategoryTheory.Iso.hom CategoryTheory.Functor CategoryTheory.Functor.category CategoryTheory.Functor.unop unop CategoryTheory.NatTrans.unop Opposite CategoryTheory.Category.opposite	—	—
unop_inv 📖	mathematical	—	CategoryTheory.Iso.inv CategoryTheory.Functor CategoryTheory.Functor.category CategoryTheory.Functor.unop unop CategoryTheory.NatTrans.unop Opposite CategoryTheory.Category.opposite	—	—
unop_leftUnitor 📖	mathematical	—	unop CategoryTheory.Functor.comp Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.id CategoryTheory.Functor.leftUnitor CategoryTheory.Iso.trans CategoryTheory.Functor CategoryTheory.Functor.category CategoryTheory.Functor.unop CategoryTheory.Iso.symm CategoryTheory.Functor.isoWhiskerRight CategoryTheory.Functor.unopId CategoryTheory.Functor.unopComp	—	CategoryTheory.Iso.ext CategoryTheory.NatTrans.ext' CategoryTheory.Functor.map_id CategoryTheory.Category.comp_id
unop_refl 📖	mathematical	—	unop CategoryTheory.Iso.refl CategoryTheory.Functor Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.category CategoryTheory.Functor.unop	—	—
unop_rightUnitor 📖	mathematical	—	unop CategoryTheory.Functor.comp Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.id CategoryTheory.Functor.rightUnitor CategoryTheory.Iso.trans CategoryTheory.Functor CategoryTheory.Functor.category CategoryTheory.Functor.unop CategoryTheory.Iso.symm CategoryTheory.Functor.isoWhiskerLeft CategoryTheory.Functor.unopId CategoryTheory.Functor.unopComp	—	CategoryTheory.Iso.ext CategoryTheory.NatTrans.ext' CategoryTheory.Category.comp_id
unop_symm 📖	mathematical	—	unop CategoryTheory.Iso.symm CategoryTheory.Functor Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.category CategoryTheory.Functor.unop	—	—
unop_trans 📖	mathematical	—	unop CategoryTheory.Iso.trans CategoryTheory.Functor Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.category CategoryTheory.Functor.unop	—	—
unop_whiskerLeft 📖	mathematical	—	unop CategoryTheory.Functor.comp Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.isoWhiskerLeft CategoryTheory.Iso.trans CategoryTheory.Functor CategoryTheory.Functor.category CategoryTheory.Functor.unop CategoryTheory.Functor.unopComp CategoryTheory.Iso.symm	—	CategoryTheory.Iso.ext CategoryTheory.NatTrans.ext' CategoryTheory.Category.comp_id CategoryTheory.Category.id_comp
unop_whiskerRight 📖	mathematical	—	unop CategoryTheory.Functor.comp Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.isoWhiskerRight CategoryTheory.Iso.trans CategoryTheory.Functor CategoryTheory.Functor.category CategoryTheory.Functor.unop CategoryTheory.Functor.unopComp CategoryTheory.Iso.symm	—	CategoryTheory.Iso.ext CategoryTheory.NatTrans.ext' CategoryTheory.Category.comp_id CategoryTheory.Category.id_comp

CategoryTheory.NatTrans

Definitions

Name	Category	Theorems
leftOp 📖	CompOp	17 math math: CategoryTheory.Functor.leftOpRightOpEquiv_counitIso_inv_app_app, CategoryTheory.simplicialCosimplicialEquiv_counitIso_inv_app_app, CategoryTheory.simplicialCosimplicialEquiv_inverse_map, leftOp_app, CategoryTheory.Functor.leftOpRightOpEquiv_unitIso_inv_app, leftOp_id, CategoryTheory.simplicialCosimplicialEquiv_unitIso_hom_app, CategoryTheory.simplicialCosimplicialEquiv_counitIso_hom_app_app, CategoryTheory.Functor.leftOpRightOpEquiv_inverse_map, CategoryTheory.Functor.leftOpRightOpEquiv_unitIso_hom_app, CategoryTheory.Functor.leftOpRightOpEquiv_counitIso_hom_app_app, leftOpWhiskerRight, CategoryTheory.simplicialCosimplicialEquiv_unitIso_inv_app, leftOpWhiskerRight_assoc, CategoryTheory.ComposableArrows.opEquivalence_inverse_map, CategoryTheory.cosimplicialToSimplicialAugmented_map, leftOp_comp
op 📖	CompOp	29 math math: CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence.functor_obj_hom_right, CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence.functor_obj_left_hom, CategoryTheory.Functor.sheafPushforwardContinuousNatTrans_app_hom, CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence.functor_obj_left_right, op_id, op_whiskerRight_assoc, CategoryTheory.Adjunction.Triple.op_rightToLeft, op_whiskerRight, op_comp_assoc, op_whiskerLeft_assoc, equifibered_op_iff, CategoryTheory.Pretriangulated.Opposite.commShift_natTrans_op_int, CategoryTheory.TwoSquare.natTrans_op, CategoryTheory.Adjunction.rightOp_counit, op_app, op_comp, CategoryTheory.Adjunction.op_unit, CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence.functor_map_left_right, CategoryTheory.Adjunction.Triple.leftToRight_op, CategoryTheory.Adjunction.leftOp_counit, coequifibered_op_iff, Coequifibered.op, CategoryTheory.NatIso.op_hom, CategoryTheory.NatIso.op_inv, op_whiskerLeft, CategoryTheory.Limits.Cocone.op_π, Equifibered.op, CategoryTheory.Adjunction.op_counit, CategoryTheory.Limits.Cone.op_ι
removeLeftOp 📖	CompOp	2 math math: removeLeftOp_id, removeLeftOp_app
removeOp 📖	CompOp	6 math math: CategoryTheory.Limits.Cone.unop_ι, CategoryTheory.Limits.Cocone.unop_π, removeOp_id, CategoryTheory.NatIso.removeOp_inv, CategoryTheory.NatIso.removeOp_hom, removeOp_app
removeRightOp 📖	CompOp	4 math math: CategoryTheory.Limits.coneOfCoconeRightOp_π, CategoryTheory.Limits.coconeOfConeRightOp_ι, removeRightOp_app, removeRightOp_id
removeUnop 📖	CompOp	4 math math: removeUnop_app, CategoryTheory.Limits.coneOfCoconeUnop_π, CategoryTheory.Limits.coconeOfConeUnop_ι, removeUnop_id
rightOp 📖	CompOp	18 math math: CategoryTheory.Functor.leftOpRightOpEquiv_counitIso_inv_app_app, CategoryTheory.simplicialCosimplicialEquiv_counitIso_inv_app_app, Equifibered.rightOp, rightOp_comp, CategoryTheory.Functor.leftOpRightOpEquiv_unitIso_inv_app, CategoryTheory.simplicialCosimplicialEquiv_unitIso_hom_app, rightOpWhiskerRight, CategoryTheory.simplicialCosimplicialEquiv_counitIso_hom_app_app, CategoryTheory.Limits.coneRightOpOfCocone_π, CategoryTheory.Functor.leftOpRightOpEquiv_unitIso_hom_app, rightOpWhiskerRight_assoc, CategoryTheory.Functor.leftOpRightOpEquiv_counitIso_hom_app_app, rightOp_app, CategoryTheory.simplicialCosimplicialEquiv_unitIso_inv_app, Coequifibered.rightOp, CategoryTheory.Limits.coconeRightOpOfCone_ι, CategoryTheory.simplicialToCosimplicialAugmented_map_right, rightOp_id
unop 📖	CompOp	20 math math: CategoryTheory.Adjunction.leftOp_unit, unop_whiskerLeft, unop_comp_assoc, unop_whiskerLeft_assoc, unop_whiskerRight, CategoryTheory.Adjunction.unop_counit, unop_id, CategoryTheory.Adjunction.rightOp_unit, equifibered_unop_iff, coequifibered_unop_iff, Coequifibered.unop, Equifibered.unop, CategoryTheory.NatIso.unop_inv, CategoryTheory.Limits.coconeUnopOfCone_ι, unop_whiskerRight_assoc, CategoryTheory.Limits.coneUnopOfCocone_π, CategoryTheory.NatIso.unop_hom, CategoryTheory.Adjunction.unop_unit, unop_app, unop_comp

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
leftOpWhiskerRight 📖	mathematical	—	leftOp CategoryTheory.Functor.comp Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.whiskerLeft CategoryTheory.CategoryStruct.comp CategoryTheory.Functor CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.category CategoryTheory.Functor.leftOp CategoryTheory.Functor.op CategoryTheory.Iso.hom CategoryTheory.Functor.leftOpComp CategoryTheory.Iso.inv	—	ext' CategoryTheory.Category.comp_id CategoryTheory.Category.id_comp
leftOpWhiskerRight_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CategoryTheory.Functor Opposite CategoryTheory.Category.opposite CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.category CategoryTheory.Functor.leftOp CategoryTheory.Functor.comp leftOp CategoryTheory.Functor.whiskerLeft CategoryTheory.Functor.op CategoryTheory.Iso.hom CategoryTheory.Functor.leftOpComp CategoryTheory.Iso.inv	—	CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' leftOpWhiskerRight
leftOp_app 📖	mathematical	—	app Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.leftOp leftOp Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.obj Opposite.unop	—	—
leftOp_comp 📖	mathematical	—	leftOp CategoryTheory.CategoryStruct.comp CategoryTheory.Functor Opposite CategoryTheory.Category.opposite CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.category CategoryTheory.Functor.leftOp	—	—
leftOp_id 📖	mathematical	—	leftOp CategoryTheory.CategoryStruct.id CategoryTheory.Functor Opposite CategoryTheory.Category.opposite CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.category CategoryTheory.Functor.leftOp	—	—
op_app 📖	mathematical	—	app Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.op op Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.obj Opposite.unop	—	—
op_comp 📖	mathematical	—	op CategoryTheory.CategoryStruct.comp CategoryTheory.Functor CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.category Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.op	—	—
op_comp_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CategoryTheory.Functor Opposite CategoryTheory.Category.opposite CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.category CategoryTheory.Functor.op op	—	CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' op_comp
op_id 📖	mathematical	—	op CategoryTheory.CategoryStruct.id CategoryTheory.Functor CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.category Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.op	—	—
op_whiskerLeft 📖	mathematical	—	op CategoryTheory.Functor.comp CategoryTheory.Functor.whiskerLeft CategoryTheory.CategoryStruct.comp CategoryTheory.Functor Opposite CategoryTheory.Category.opposite CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.category CategoryTheory.Functor.op CategoryTheory.Iso.hom CategoryTheory.Functor.opComp CategoryTheory.Iso.inv	—	ext' CategoryTheory.Category.comp_id CategoryTheory.Category.id_comp
op_whiskerLeft_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CategoryTheory.Functor Opposite CategoryTheory.Category.opposite CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.category CategoryTheory.Functor.op CategoryTheory.Functor.comp op CategoryTheory.Functor.whiskerLeft CategoryTheory.Iso.hom CategoryTheory.Functor.opComp CategoryTheory.Iso.inv	—	CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' op_whiskerLeft
op_whiskerRight 📖	mathematical	—	op CategoryTheory.Functor.comp CategoryTheory.Functor.whiskerRight CategoryTheory.CategoryStruct.comp CategoryTheory.Functor Opposite CategoryTheory.Category.opposite CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.category CategoryTheory.Functor.op CategoryTheory.Iso.hom CategoryTheory.Functor.opComp CategoryTheory.Iso.inv	—	ext' CategoryTheory.Category.comp_id CategoryTheory.Category.id_comp
op_whiskerRight_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CategoryTheory.Functor Opposite CategoryTheory.Category.opposite CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.category CategoryTheory.Functor.op CategoryTheory.Functor.comp op CategoryTheory.Functor.whiskerRight CategoryTheory.Iso.hom CategoryTheory.Functor.opComp CategoryTheory.Iso.inv	—	CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' op_whiskerRight
removeLeftOp_app 📖	mathematical	—	app Opposite CategoryTheory.Category.opposite removeLeftOp Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.obj CategoryTheory.Functor.leftOp Opposite.op	—	—
removeLeftOp_id 📖	mathematical	—	removeLeftOp CategoryTheory.CategoryStruct.id CategoryTheory.Functor Opposite CategoryTheory.Category.opposite CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.category CategoryTheory.Functor.leftOp	—	—
removeOp_app 📖	mathematical	—	app removeOp Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.obj Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.op Opposite.op	—	—
removeOp_id 📖	mathematical	—	removeOp CategoryTheory.CategoryStruct.id CategoryTheory.Functor Opposite CategoryTheory.Category.opposite CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.category CategoryTheory.Functor.op	—	—
removeRightOp_app 📖	mathematical	—	app Opposite CategoryTheory.Category.opposite removeRightOp Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.obj CategoryTheory.Functor.rightOp Opposite.unop	—	—
removeRightOp_id 📖	mathematical	—	removeRightOp CategoryTheory.CategoryStruct.id CategoryTheory.Functor Opposite CategoryTheory.Category.opposite CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.category CategoryTheory.Functor.rightOp	—	—
removeUnop_app 📖	mathematical	—	app Opposite CategoryTheory.Category.opposite removeUnop Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.obj CategoryTheory.Functor.unop Opposite.unop	—	—
removeUnop_id 📖	mathematical	—	removeUnop CategoryTheory.CategoryStruct.id CategoryTheory.Functor CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.category CategoryTheory.Functor.unop Opposite CategoryTheory.Category.opposite	—	—
rightOpWhiskerRight 📖	mathematical	—	rightOp CategoryTheory.Functor.comp Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.whiskerRight CategoryTheory.CategoryStruct.comp CategoryTheory.Functor CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.category CategoryTheory.Functor.rightOp CategoryTheory.Functor.op CategoryTheory.Iso.hom CategoryTheory.Functor.rightOpComp CategoryTheory.Iso.inv	—	ext' CategoryTheory.Category.comp_id CategoryTheory.Category.id_comp
rightOpWhiskerRight_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CategoryTheory.Functor Opposite CategoryTheory.Category.opposite CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.category CategoryTheory.Functor.rightOp CategoryTheory.Functor.comp rightOp CategoryTheory.Functor.whiskerRight CategoryTheory.Functor.op CategoryTheory.Iso.hom CategoryTheory.Functor.rightOpComp CategoryTheory.Iso.inv	—	CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' rightOpWhiskerRight
rightOp_app 📖	mathematical	—	app Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.rightOp rightOp Quiver.Hom.op CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.obj Opposite.op	—	—
rightOp_comp 📖	mathematical	—	rightOp CategoryTheory.CategoryStruct.comp CategoryTheory.Functor Opposite CategoryTheory.Category.opposite CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.category CategoryTheory.Functor.rightOp	—	—
rightOp_id 📖	mathematical	—	rightOp CategoryTheory.CategoryStruct.id CategoryTheory.Functor Opposite CategoryTheory.Category.opposite CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.category CategoryTheory.Functor.rightOp	—	—
unop_app 📖	mathematical	—	app CategoryTheory.Functor.unop unop Quiver.Hom.unop CategoryTheory.CategoryStruct.toQuiver CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.obj Opposite CategoryTheory.Category.opposite Opposite.op	—	—
unop_comp 📖	mathematical	—	unop CategoryTheory.CategoryStruct.comp CategoryTheory.Functor Opposite CategoryTheory.Category.opposite CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.category CategoryTheory.Functor.unop	—	—
unop_comp_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CategoryTheory.Functor CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.category CategoryTheory.Functor.unop unop Opposite CategoryTheory.Category.opposite	—	CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' unop_comp
unop_id 📖	mathematical	—	unop CategoryTheory.CategoryStruct.id CategoryTheory.Functor Opposite CategoryTheory.Category.opposite CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.category CategoryTheory.Functor.unop	—	—
unop_whiskerLeft 📖	mathematical	—	unop CategoryTheory.Functor.comp Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.whiskerLeft CategoryTheory.CategoryStruct.comp CategoryTheory.Functor CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.category CategoryTheory.Functor.unop CategoryTheory.Iso.hom CategoryTheory.Functor.unopComp CategoryTheory.Iso.inv	—	ext' CategoryTheory.Category.comp_id CategoryTheory.Category.id_comp
unop_whiskerLeft_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CategoryTheory.Functor CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.category CategoryTheory.Functor.unop CategoryTheory.Functor.comp Opposite CategoryTheory.Category.opposite unop CategoryTheory.Functor.whiskerLeft CategoryTheory.Iso.hom CategoryTheory.Functor.unopComp CategoryTheory.Iso.inv	—	CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' unop_whiskerLeft
unop_whiskerRight 📖	mathematical	—	unop CategoryTheory.Functor.comp Opposite CategoryTheory.Category.opposite CategoryTheory.Functor.whiskerRight CategoryTheory.CategoryStruct.comp CategoryTheory.Functor CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.category CategoryTheory.Functor.unop CategoryTheory.Iso.hom CategoryTheory.Functor.unopComp CategoryTheory.Iso.inv	—	ext' CategoryTheory.Category.comp_id CategoryTheory.Category.id_comp
unop_whiskerRight_assoc 📖	mathematical	—	CategoryTheory.CategoryStruct.comp CategoryTheory.Functor CategoryTheory.Category.toCategoryStruct CategoryTheory.Functor.category CategoryTheory.Functor.unop CategoryTheory.Functor.comp Opposite CategoryTheory.Category.opposite unop CategoryTheory.Functor.whiskerRight CategoryTheory.Iso.hom CategoryTheory.Functor.unopComp CategoryTheory.Iso.inv	—	CategoryTheory.Category.assoc Mathlib.Tactic.Reassoc.eq_whisker' unop_whiskerRight

Quiver.Hom

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
op_inj 📖	mathematical	—	Quiver.Hom Opposite Quiver.opposite Opposite.op op	—	—
op_unop 📖	mathematical	—	op Opposite.unop unop	—	—
unop_inj 📖	mathematical	—	Quiver.Hom Opposite Quiver.opposite Opposite.unop unop	—	—
unop_mk 📖	mathematical	—	unop Opposite Quiver.opposite Opposite.op Quiver.Hom Opposite.unop	—	—
unop_op 📖	mathematical	—	unop Opposite.op op	—	—
unop_op' 📖	mathematical	—	unop	—	—

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