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	5026 math math: CategoryTheory.Functor.IsDenseSubsite.mapPreimage_map_of_fac, CategoryTheory.Sheaf.cartesianMonoidalCategoryLift_val, CategoryTheory.GrothendieckTopology.coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation_pt, HomotopicalAlgebra.fibration_op_iff, CategoryTheory.Presheaf.instIsCardinalPresentableFunctorOppositeFreeYonedaOfHasColimitsOfSize, CategoryTheory.Comon.tensorObj_comul', Condensed.finYoneda_obj, AlgebraicGeometry.Scheme.toSpecΓ_apply, CategoryTheory.LocalizerMorphism.LeftResolution.opFunctor_map_f, PresheafOfModules.Monoidal.tensorObj_obj, SSet.op_δ, AlgebraicGeometry.Γ_map_morphismRestrict, CategoryTheory.Limits.pushoutIsoOpPullback_inr_hom_assoc, CategoryTheory.CommSq.LiftStruct.op_l, CategoryTheory.GrothendieckTopology.overMapPullbackId_hom_app_val_app, 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, PresheafOfModules.instIsRightAdjointPushforwardCompFunctorOppositeRingCatWhiskerLeftOp, CategoryTheory.Adjunction.compUliftCoyonedaIso_hom_app_app_down, CategoryTheory.sheafBotEquivalence_functor, smoothSheafCommRing.ι_forgetStalk_inv, 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, 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', AddCommGrpCat.coyoneda_obj_obj_coe, AlgebraicGeometry.Proj.awayMap_awayToSection_assoc, CategoryTheory.Equivalence.leftOp_unitIso_hom_app, CategoryTheory.linearCoyoneda_obj_additive, CategoryTheory.Equivalence.sheafCongrPrecoherent_counitIso_hom_app_val_app, CategoryTheory.SimplicialObject.whiskering_obj_map_app, 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, 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, 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.ParametrizedAdjunction.homEquiv_symm_naturality_three_assoc, 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.Pretriangulated.shift_opShiftFunctorEquivalence_counitIso_inv_app, CategoryTheory.Equivalence.sheafCongr.counitIso_hom_app_val_app, CategoryTheory.Limits.preservesColimitsOfShape_rightOp, AlgebraicGeometry.Spec.toPresheafedSpace_obj, CategoryTheory.GrothendieckTopology.liftToDiagramLimitObjAux_fac, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_appTop, LightProfinite.proj_comp_transitionMap, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo, CategoryTheory.ULiftYoneda.instFullFunctorOppositeTypeUliftYoneda, CategoryTheory.yoneda_preservesLimit, 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, 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, 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.ShortComplex.hasRightHomology_iff_op, CategoryTheory.regularTopology.EqualizerCondition.bijective_mapToEqualizer_pullback', CategoryTheory.Limits.limitUnopIsoUnopColimit_hom_comp_ι, CategoryTheory.ObjectProperty.colimitsOfShape_op, AlgebraicGeometry.Scheme.Hom.germ_stalkMap_assoc, AlgebraicGeometry.LocallyQuasiFinite.quasiFinite_appLE, 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, CategoryTheory.ShortComplex.SnakeInput.op_δ, CategoryTheory.Limits.isLimitConeRightOpOfCocone_lift, CategoryTheory.ObjectProperty.isClosedUnderColimitsOfShape_iff_op, AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToSection_apply, CategoryTheory.Limits.preservesFiniteCoproducts_rightOp, SSet.Truncated.tensor_map_apply_snd, CategoryTheory.ShortComplex.cyclesOpIso_inv_op_iCycles_assoc, CategoryTheory.MorphismProperty.instHasOfPrecompPropertyOppositeOpOfHasOfPostcompProperty, CategoryTheory.Limits.pullbackIsoUnopPushout_inv_snd, AlgebraicGeometry.PresheafedSpace.stalkMap_germ, CategoryTheory.CategoryOfElements.costructuredArrowYonedaEquivalenceInverseπ_hom_app, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_add_unitIso_hom_app_eq, CategoryTheory.op_hom_leftUnitor, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit_app, CategoryTheory.shrinkYoneda_map, CategoryTheory.isSeparator_iff_faithful_preadditiveCoyoneda, CategoryTheory.regularTopology.equalizerCondition_precomp_of_preservesPullback, CategoryTheory.ParametrizedAdjunction.homEquiv_naturality_two, CategoryTheory.Limits.coconeEquivalenceOpConeOp_unitIso, 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, PresheafOfModules.pullback_id_comp, AlgebraicGeometry.Scheme.ofRestrict_appIso, CategoryTheory.isSeparating_unop_iff, CategoryTheory.Equivalence.sheafCongrPreregular_counitIso_inv_app_val_app, CategoryTheory.NatTrans.unop_whiskerLeft, ContinuousMap.yonedaPresheaf'_obj, CategoryTheory.sheafOver_val, AlgebraicGeometry.RingedSpace.basicOpen_res, CategoryTheory.Comma.opFunctor_obj, AlgebraicGeometry.Scheme.Hom.app_invApp'_assoc, CategoryTheory.Classifier.SubobjectRepresentableBy.pullback_homEquiv_symm_obj_Ω₀, 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, AlgebraicGeometry.Scheme.app_eq, CategoryTheory.shrinkYonedaEquiv_comp, CategoryTheory.ShortComplex.LeftHomologyData.unop_p, CategoryTheory.eval_app, 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, CategoryTheory.Groupoid.invEquivalence_inverse_map, AlgebraicTopology.NormalizedMooreComplex.obj_d, CategoryTheory.SimplicialObject.δ_comp_σ_of_gt, CategoryTheory.ObjectProperty.essentiallySmall_unop_iff, CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan.uliftYonedaEquiv_ι_presheafHom, CategoryTheory.Pseudofunctor.DescentData.pullFunctorCompIso_inv_app_hom, CategoryTheory.Limits.Cone.unop_ι, CategoryTheory.Pseudofunctor.isEquivalence_toDescentData, CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinsetLimitCone_isLimit_lift, CategoryTheory.GrothendieckTopology.Cover.index_fst, SSet.horn₃₁.desc.multicofork_π_two, AlgebraicGeometry.LocallyOfFiniteType.finiteType_of_affine_subset, AlgebraicGeometry.Scheme.zeroLocus_empty_eq_univ, CategoryTheory.CategoryOfElements.costructuredArrowYonedaEquivalence_functor, CategoryTheory.Limits.walkingSpanOpEquiv_inverse_map, CategoryTheory.Limits.widePushoutShapeUnop_obj, CategoryTheory.Subfunctor.isSeparated, 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, PresheafOfModules.add_app, CategoryTheory.Functor.mapTriangleOpCompTriangleOpEquivalenceFunctorApp_inv_hom₁, CategoryTheory.Subfunctor.Subpresheaf.range_eq_ofSection, SSet.ι₀_snd_assoc, CategoryTheory.Limits.widePushoutShapeOpEquiv_functor, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverse_obj_π_app, CategoryTheory.Functor.sheafPushforwardContinuousComp'_inv_app_val_app, condensedSetToTopCat_obj_carrier, CategoryTheory.Functor.rightOp_map, SSet.Subcomplex.mem_ofSimplex_obj_iff, AlgebraicGeometry.instIsIsoSchemeCoprodComparisonOppositeCommRingCatSpec, CategoryTheory.MorphismProperty.IsStableUnderCobaseChange.op, CategoryTheory.Limits.preservesLimits_of_leftOp, CondensedMod.IsSolid.isIso_solidification_map, CategoryTheory.instFullMonFunctorOppositeMonCatYonedaMon, AlgebraicGeometry.IsAffineOpen.map_fromSpec, AlgebraicGeometry.StructureSheaf.const_mul_rev, CategoryTheory.NonemptyParallelPairPresentationAux.hf, AlgebraicGeometry.ΓSpec.isIso_adjunction_counit, CategoryTheory.sheafToPresheafCompCoyonedaCompWhiskeringLeftSheafToPresheaf_hom_app_app_val, 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, CategoryTheory.MorphismProperty.LeftFraction.op_map, CategoryTheory.WithInitial.opEquiv_unitIso_inv_app, AlgebraicGeometry.AffineScheme.Spec_faithful, CategoryTheory.Limits.pushoutIsoOpPullback_inl_hom, AlgebraicGeometry.IsLocallyArtinian.isArtinianRing_presheaf_obj, SSet.Truncated.mapHomotopyCategory_homMk, AlgebraicGeometry.Scheme.Opens.toSpecΓ_naturality, 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.opOp_obj, AlgebraicGeometry.Scheme.restrictFunctorΓ_inv_app, CategoryTheory.op_inv_associator, Condensed.hom_ext_iff, CategoryTheory.SimplicialObject.σ_naturality_assoc, CategoryTheory.Limits.walkingParallelPairOp_left, CategoryTheory.Arrow.augmentedCechNerve_hom_app, CategoryTheory.shrinkYonedaEquiv_symm_map_assoc, AlgebraicGeometry.Scheme.Modules.restrictAdjunction_counit_app_app, CategoryTheory.ShortComplex.op_g, CategoryTheory.Limits.ι_comp_colimitOpIsoOpLimit_hom_assoc, CategoryTheory.GrothendieckTopology.diagram_obj, AlgebraicGeometry.instIsDomainCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTopOfIsIntegral, AlgebraicGeometry.instPreservesLimitSchemeOppositeCommRingCatRightOpΓOfIsAffineHomMapOfCompactSpaceOfQuasiSeparatedSpaceCarrierCarrierObj, LightCondensed.forget_obj_val_map, CategoryTheory.GrothendieckTopology.yonedaULiftEquiv_apply, 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.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, CategoryTheory.GrothendieckTopology.liftToDiagramLimitObjAux_fac_assoc, 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.eval_map, CategoryTheory.isCoseparator_iff_faithful_preadditiveYoneda, 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, AlgebraicGeometry.Scheme.Modules.pushforwardCongr_hom_app_app, AlgebraicGeometry.AffineSpace.SpecIso_hom_appTop, PresheafOfModules.restrictScalars_map_app, CategoryTheory.MonoidalCategory.MonoidalLeftAction.oppositeLeftAction_actionHomRight, CategoryTheory.MorphismProperty.LeftFraction.op_X', AlgebraicGeometry.LocallyRingedSpace.basicOpen_eq_bot_iff_forall_evaluation_eq_zero, CategoryTheory.NatIso.op_rightUnitor, CategoryTheory.op_tensorHom, SSet.σ_mem_degenerate, AlgebraicGeometry.Scheme.Modules.pushforward_obj_obj, SSet.ofSimplex_le_skeleton, CategoryTheory.Comon.MonOpOpToComon_obj, SSet.Subcomplex.toSSetFunctor_map, CategoryTheory.Pretriangulated.Opposite.UnopUnopCommShift.iso_inv_app, AlgebraicGeometry.Spec.locallyRingedSpaceObj_presheaf_map, CategoryTheory.MonoidalCategory.MonoidalLeftAction.oppositeLeftAction_actionAssocIso, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_inverse, SSet.iSup_subcomplexOfSimplex_prod_eq_top, CategoryTheory.Equivalence.rightOp_counitIso_inv_app, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_inv_eq_germ_assoc, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.app_invApp, CategoryTheory.Functor.PullbackObjObj.mapArrowRight_right, CategoryTheory.Arrow.AugmentedCechNerve.ExtraDegeneracy.s_comp_base, HomologicalComplex.op_d, CategoryTheory.Functor.unopOpIso_inv_app, 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.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, CategoryTheory.typeEquiv_functor_obj_val_obj, 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.GrothendieckTopology.overMapPullbackId_inv_app_val_app, 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.instPreservesFiniteProductsOppositeVal, CategoryTheory.yonedaGrp_obj, CategoryTheory.MonoidalCategory.MonoidalRightAction.oppositeRightAction_actionObj, CategoryTheory.Functor.Monoidal.RepresentableBy.tensorObj_homEquiv, 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, CategoryTheory.MonoidalCategory.MonoidalLeftAction.leftActionOfOppositeLeftAction_actionHomRight, PresheafOfModules.comp_app, CategoryTheory.Limits.FormalCoproduct.cosimplicialObjectFunctor_obj_obj, CategoryTheory.Square.unop_X₁, CategoryTheory.ShortComplex.RightHomologyMapData.op_φH, CategoryTheory.nerve.functorOfNerveMap_map, SSet.opFunctorCompOpFunctorIso_inv_app_app, CategoryTheory.Pretriangulated.shiftFunctorZero_op_inv_app, HomologicalComplex.unopFunctor_obj, CategoryTheory.HasLiftingProperty.op, LightCondSet.continuous_coinducingCoprod, CategoryTheory.Join.inclRightCompOpEquivInverse_inv_app_op, CategoryTheory.Injective.instProjectiveOppositeOp, CategoryTheory.Equivalence.leftOp_functor_obj, CategoryTheory.Presieve.isSheaf_iff_preservesFiniteProducts, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.invApp_app_assoc, SSet.Truncated.StrictSegal.spine_spineToSimplex, 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.WithTerminal.opEquiv_inverse_obj, CategoryTheory.Limits.opProdIsoCoprod_hom_snd_assoc, 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.Limits.preservesColimits_rightOp, 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, CategoryTheory.ParametrizedAdjunction.whiskerLeft_map_counit, CategoryTheory.OverPresheafAux.restrictedYoneda_obj, LightCondensed.finYoneda_obj, AlgebraicGeometry.Scheme.inv_app, 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.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, AlgebraicGeometry.Proj.basicOpenIsoAway_hom, CategoryTheory.linearYoneda_obj_map, CategoryTheory.Limits.coend.map_id, smoothSheafCommRing.ι_evalHom_apply, CategoryTheory.PreOneHypercover.multicospanIndex_fst, CategoryTheory.NatIso.unop_rightUnitor, CategoryTheory.Pretriangulated.instIsTriangulatedOppositeOpOpEquivalence, CategoryTheory.NatTrans.unop_comp_assoc, TopCat.Presheaf.germ_exist_of_isBasis, PresheafOfModules.homEquivOfIsLocallyBijective_symm_apply, CategoryTheory.Limits.widePullbackShapeUnop_map, CategoryTheory.prodOpEquiv_inverse_map, CategoryTheory.ShortComplex.opMap_τ₁, 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.ShiftedHom.opEquiv_symm_add, CategoryTheory.Equivalence.precoherent_isSheaf_iff_of_essentiallySmall, 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, 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.SequentialProduct.cone_π_app_comp_Pi_π_pos_assoc, CategoryTheory.IsPullback.op, CategoryTheory.Limits.opCompYonedaSectionsEquiv_symm_apply_coe, CategoryTheory.Yoneda.naturality, CategoryTheory.GrothendieckTopology.yoneda_map_val, CondensedSet.hom_naturality_apply, CategoryTheory.Limits.yonedaCompLimIsoCocones_inv_app, CategoryTheory.nerve.homEquiv_edgeMk_map_nerveMap, 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, CompHausLike.LocallyConstant.adjunction_counit, CategoryTheory.Limits.hasLimit_leftOp_of_hasColimit, CategoryTheory.MorphismProperty.IsMultiplicative.of_unop, AlgebraicGeometry.Scheme.ideal_ker_le_ker_ΓSpecIso_inv_comp, SSet.Truncated.Edge.map_fst, CategoryTheory.instFaithfulGrpFunctorOppositeGrpCatYonedaGrp, CategoryTheory.Functor.IsCoverDense.restrictHomEquivHom_naturality_left_symm_assoc, HomologicalComplex.opcyclesOpIso_inv_naturality_assoc, smoothSheafCommRing.eval_germ, CategoryTheory.Limits.Cocone.unop_π, PresheafOfModules.pushforward_map_app_apply, CategoryTheory.preadditiveYonedaObj_obj_carrier, LightCondensed.finYoneda_map, CategoryTheory.Presheaf.isSeparating, SimplicialObject.Splitting.cofan_inj_epi_naturality_assoc, CategoryTheory.sheafificationNatIso_inv_app_val, CategoryTheory.Limits.colimitCoyonedaHomIsoLimit'_π_apply, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_map, AlgebraicGeometry.Scheme.isoSpec_inv_toSpecΓ, 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.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.Subfunctor.ofSection_eq_range, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.shortComplex_X₁, HomologicalComplex.instHasHomologyOppositeOp, CategoryTheory.Limits.walkingParallelPairOpEquiv_counitIso_inv_app_op_zero, AlgebraicGeometry.isIso_ΓSpec_adjunction_unit_app_basicOpen, AlgebraicGeometry.isNoetherian_iff_of_finite_affine_openCover, CategoryTheory.Limits.multispanIndexCoend_right, CategoryTheory.OverPresheafAux.YonedaCollection.map₂_yonedaEquivFst, CondensedMod.LocallyConstant.instFullModuleCatSheafCompHausCoherentTopologyConstantSheaf, CategoryTheory.ParametrizedAdjunction.homEquiv_symm_naturality_one, CategoryTheory.SimplicialObject.Truncated.whiskering_map_app_app, Condensed.discrete_map, PresheafOfModules.toSheafify_app_apply', CategoryTheory.isConnected_op, CategoryTheory.opOp_map, CategoryTheory.Limits.limitOpIsoOpColimit_hom_comp_ι_assoc, PresheafOfModules.instPreservesLimitsOfShapeModuleCatCarrierObjOppositeRingCatEvaluation, CategoryTheory.Limits.pushoutIsoUnopPullback_inr_hom, PresheafOfModules.toFreeYonedaCoproduct_fromFreeYonedaCoproduct, HomotopicalAlgebra.instFibrationOppositeOpOfCofibration, CategoryTheory.Equalizer.firstObjEqFamily_inv, AlgebraicGeometry.SmoothOfRelativeDimension.exists_isStandardSmoothOfRelativeDimension, CategoryTheory.unop_mono_iff, CategoryTheory.yonedaEquiv_apply, 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.instIsIsoFunctorOppositeValAppSheafCounitSheafificationAdjunction, CategoryTheory.sheafToPresheafCompCoyonedaCompWhiskeringLeftSheafToPresheaf_inv_app_app, CategoryTheory.IsFiltered.iff_nonempty_limit, CategoryTheory.Comon.monoidal_rightUnitor_inv_hom, CategoryTheory.Comma.opEquiv_counitIso, CategoryTheory.sheafificationNatIso_hom_app_val, SSet.horn₃₂.desc.multicofork_π_one, PresheafOfModules.Derivation.d_map, CategoryTheory.Pseudofunctor.CoGrothendieck.instIsEquivalenceαCategoryObjLocallyDiscreteOppositeCatMkOpFiberForgetInducedFunctor, CategoryTheory.ShortComplex.hasLeftHomology_iff_op, CategoryTheory.ShortComplex.shortExact_iff_op, AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine_apply, AlgebraicGeometry.structureSheafInType.mul_apply, TopCat.Presheaf.germ_stalkSpecializes_apply, SSet.Subcomplex.topIso_inv_app_coe, SSet.prodStdSimplex.orderHomOfSimplex_coe, HomologicalComplex.extend_op_d_assoc, CategoryTheory.constantCommuteCompose_hom_app_val, CategoryTheory.Square.op_f₁₂, CategoryTheory.cokernelUnopUnop_inv, AlgebraicGeometry.StructureSheaf.exists_const, CategoryTheory.Limits.desc_op_comp_opCoproductIsoProduct'_hom, skyscraperPresheafCocone_pt, CategoryTheory.forgetEnrichmentOppositeEquivalence_functor, AlgebraicGeometry.Scheme.Modules.Hom.zero_app, CategoryTheory.WithInitial.opEquiv_functor_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, HomotopicalAlgebra.weakEquivalences_op, CategoryTheory.GrothendieckTopology.preservesLimitsOfShape_diagramFunctor, AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine_symm_apply, CategoryTheory.Iso.op_inv, AlgebraicGeometry.Scheme.congr_app, CategoryTheory.unop_tensor_unop, SimplicialObject.opFunctor_obj_map, AlgebraicGeometry.Scheme.IdealSheafData.ideal_mul, CategoryTheory.kernelOpUnop_hom, CategoryTheory.Functor.IsCoverDense.isoOver_hom_app, AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.structurePresheafInCommRing_obj_carrier, AlgebraicGeometry.Scheme.SpecΓIdentity_hom_app, CategoryTheory.Pretriangulated.triangleOpEquivalence_unitIso, CategoryTheory.Functor.FullyFaithful.compUliftCoyonedaCompWhiskeringLeft_hom_app_app_down, CategoryTheory.MonoidalCategory.MonoidalRightAction.oppositeRightAction_actionHomRight, AlgebraicGeometry.locallyQuasiFinite_iff, CategoryTheory.ShortComplex.Homotopy.op_h₀, CategoryTheory.Coyoneda.objOpOp_inv_app, CategoryTheory.Functor.opComp_inv_app, smoothSheaf.ι_evalHom_apply, HomologicalComplex.isStrictlySupportedOutside_op_iff, TopCat.Presheaf.SheafConditionEqualizerProducts.fork_π_app_walkingParallelPair_one, SSet.Truncated.Edge.CompStruct.d₂, 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, CategoryTheory.Equivalence.instPreservesFiniteLimitsFunctorOppositeSheafTransportAndSheafify, CategoryTheory.SimplicialObject.Augmented.w₀, 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, 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.WithInitial.opEquiv_counitIso_hom_app, CategoryTheory.Pretriangulated.shiftFunctorAdd'_op_hom_app, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.ofRestrict_invApp, CategoryTheory.uliftCoyonedaIsoCoyoneda_inv_app_app_down, CategoryTheory.Presheaf.isSheaf_iff_preservesFiniteProducts, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivCounitIso_hom_app_hom, CategoryTheory.CostructuredArrow.unop_left_comp_ofMkLEMk_unop, 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, CategoryTheory.Equivalence.sheafCongrPreregular_unitIso_hom_app_val_app, CategoryTheory.PresheafOfGroups.OneCocycle.ev_trans, CategoryTheory.Adjunction.compYonedaIso_hom_app_app, CategoryTheory.Functor.WellOrderInductionData.sectionsMk_val_op_bot, PresheafOfModules.freeYonedaEquiv_symm_app, CommRingCat.coyoneda_map_app, HomologicalComplex.unopEquivalence_functor, CategoryTheory.ShortComplex.HomologyMapData.unop_right, TopCat.toSheafCompHausLike_val_obj, CategoryTheory.Limits.hasCoequalizers_opposite, 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, PresheafOfModules.restrictScalarsObj_map, CategoryTheory.Over.opEquivOpUnder_unitIso, CategoryTheory.Limits.coconeOfConeLeftOp_pt, PresheafOfModules.forgetToPresheafModuleCatObj_map, 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.SimplicialObject.Augmented.toArrow_map_left, imageToKernel_op, AlgebraicGeometry.Scheme.Modules.toOpen_fromTildeΓ_app, TopCat.Presheaf.stalkPushforward_germ_apply, InfiniteGalois.toAlgEquivAux_eq_proj_of_mem, CategoryTheory.op_epi_iff, CategoryTheory.sheafCompose_map_val, CategoryTheory.Sheaf.comp_val, CategoryTheory.Pseudofunctor.DescentData.comp_hom, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.s'_comp_ε, StalkSkyscraperPresheafAdjunctionAuxs.germ_fromStalk, SSet.prodStdSimplex.objEquiv_apply_fst, CategoryTheory.Presieve.FamilyOfElements.compatible_singleton_iff, SheafOfModules.evaluationPreservesLimitsOfShape, LightCondensed.discrete_map, CategoryTheory.monoidalOpOp_δ, SimplicialObject.Splitting.cofan_inj_eq, CategoryTheory.Limits.instHasProductOppositeOp, CategoryTheory.MonObj.ofRepresentableBy_one, AlgebraicGeometry.Scheme.exists_germ_injective, SSet.Truncated.Path.map_arrow, CategoryTheory.Limits.hasColimit_of_hasLimit_leftOp, 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, AlgebraicGeometry.StructureSheaf.instIsLocalizedModuleObjOppositeOpensCarrierTopValStructureSheafInTypeOpBasicOpenPowersToOpenₗ, CategoryTheory.Limits.coend.condition, CategoryTheory.sheafToPresheaf_μ, AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_app_eq, CategoryTheory.LocalizerMorphism.hasLeftResolutions_iff_op, LightProfinite.proj_comp_transitionMapLE', CategoryTheory.Presheaf.isSheaf_iff_preservesFiniteProducts_and_equalizerCondition, TopCat.Presheaf.germ_res_apply, 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, 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, CategoryTheory.Limits.closedUnderLimitsOfShape_walkingParallelPair_isIndObject, 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, HomologicalComplex.opcyclesOpIso_inv_naturality, AlgebraicGeometry.Scheme.Hom.toImage_app_injective, CategoryTheory.sheafToPresheafCompYonedaCompWhiskeringLeftSheafToPresheaf_hom_app_app_val, AlgebraicGeometry.Scheme.isoSpec_hom, CategoryTheory.Presheaf.isLocallyInjective_presheafToSheaf_map_iff, CategoryTheory.Groupoid.invFunctor_map, CategoryTheory.GrothendieckTopology.uliftYonedaEquiv_naturality, AlgebraicGeometry.StructureSheaf.to_basicOpen_epi, CategoryTheory.Functor.IsRepresentedBy.of_natIso, AlgebraicGeometry.IsAffineOpen.opensRange_fromSpec, HomologicalComplex.extend.XOpIso_hom_d_op, CategoryTheory.Limits.parallelPairOpIso_inv_app_one, CategoryTheory.Functor.instFullOppositeTypeRestrictedULiftYonedaOfIsDense, 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.Limits.PreservesFiniteLimitsOfIsFilteredCostructuredArrowYonedaAux.functorToInterchangeIso_inv_app_app, AlgebraicGeometry.Scheme.eqToHom_c_app, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.biprodAddEquiv_symm_biprodIsoProd_hom_toBiprod_apply, 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.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, 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.HasLiftingProperty.transfiniteComposition.wellOrderInductionData.liftHom_fac_assoc, CategoryTheory.Equivalence.sheafCongrPreregular_inverse_obj_val_map, PresheafOfModules.id_app, CategoryTheory.Pretriangulated.commShiftIso_opOp_inv_app, CategoryTheory.ShortComplex.RightHomologyMapData.op_φK, CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan.presheafHom_naturality, SSet.Truncated.StrictSegal.spineToSimplex_spine, CategoryTheory.ShortComplex.opMap_id, CategoryTheory.ShortComplex.LeftHomologyMapData.op_φH, CategoryTheory.Functor.IsDenseSubsite.mapPreimage_map, SimplexCategory.II_σ, 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, AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex_f, CategoryTheory.Presheaf.isLocallySurjective_whisker, 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, CategoryTheory.MorphismProperty.instHasLeftCalculusOfFractionsOppositeOpOfHasRightCalculusOfFractions, 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.Presheaf.comp_isLocallySurjective_iff, CategoryTheory.OverPresheafAux.unitAuxAuxAux_hom, AlgebraicGeometry.IsAffineOpen.isLocalization_basicOpen, CategoryTheory.Functor.leftOp_obj, CategoryTheory.Limits.coneUnopOfCocone_pt, CategoryTheory.Equivalence.inverseFunctor_obj, AlgebraicTopology.DoldKan.PInfty_f_naturality_assoc, 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, 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, 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, LightCondSet.topCatAdjunctionUnit_val_app_apply, CommRingCat.coyoneda_obj_map, CategoryTheory.yonedaFunctor_reflectsLimits, CategoryTheory.Limits.piConst_map_app, AlgebraicGeometry.IsIntegralHom.isIntegral_app, AlgebraicGeometry.Scheme.ker_ideal_of_isPullback_of_isOpenImmersion, CategoryTheory.isIso_toSheafify, CategoryTheory.ParametrizedAdjunction.arrowHomEquiv_symm_apply_right, CategoryTheory.Functor.instFinalOppositeRightOpOfInitial, CategoryTheory.Under.mapFunctor_obj, 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.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.Pretriangulated.Opposite.UnopUnopCommShift.iso_inv_app_assoc, CategoryTheory.Presieve.FamilyOfElements.singletonEquiv_symm_apply_self, HomologicalComplex.opFunctor_obj, CategoryTheory.Limits.walkingParallelPairOpEquiv_counitIso_zero, 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', CategoryTheory.sheafificationIso_inv_val, 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', AlgebraicGeometry.PresheafedSpace.comp_c_app, CategoryTheory.MorphismProperty.op_isomorphisms, CategoryTheory.GrothendieckTopology.uliftYonedaEquiv_naturality', CategoryTheory.Limits.opCoproductIsoProduct'_comp_self, PresheafOfModules.map_comp_apply, AlgebraicGeometry.Scheme.toSpecΓ_isoSpec_inv_assoc, AlgebraicGeometry.LocallyRingedSpace.component_nontrivial, CategoryTheory.uliftCoyonedaEquiv_symm_apply_app, CategoryTheory.ShortComplex.leftHomologyMap_op, AlgebraicGeometry.Scheme.IdealSheafData.ideal_map, CategoryTheory.Functor.representableByUliftFunctorEquiv_symm_apply_homEquiv, CategoryTheory.Functor.opId_hom_app, CategoryTheory.LocalizerMorphism.RightResolution.unop_X₁, PresheafOfModules.pushforward_map_app_apply', AlgebraicGeometry.Proj.sub_apply, AlgebraicGeometry.isLocallyArtinian_iff_of_isOpenCover, 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.Functor.IsDenseSubsite.mapPreimage_comp_assoc, CategoryTheory.SimplicialObject.Truncated.trunc_obj_obj, SSet.mem_degenerate_iff_notMem_nonDegenerate, CategoryTheory.GrothendieckTopology.diagramCompIso_hom_ι_assoc, 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.countableCategoryOpposite, PresheafOfModules.Derivation.d_mul, AddCommMonCat.coyonedaType_obj_map, AlgebraicGeometry.instIsScalarTowerObjOppositeOpensCarrierTopValStructureSheafInType, CategoryTheory.MorphismProperty.RespectsIso.op, CategoryTheory.OverPresheafAux.counitForward_naturality₁, HomologicalComplex.cyclesOpIso_inv_naturality_assoc, CategoryTheory.linearYoneda_obj_obj_carrier, CategoryTheory.Limits.walkingCospanOpEquiv_counitIso_hom_app, CategoryTheory.op_tensorUnit, CategoryTheory.Pretriangulated.commShiftIso_opOp_hom_app_assoc, CategoryTheory.ObjectProperty.limitsOfShape_op, CategoryTheory.Functor.instEssSurjOppositeOp, 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.GrothendieckTopology.uliftYonedaEquiv_symm_map, 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.Cone.extend_π, CategoryTheory.Limits.fst_opProdIsoCoprod_hom_assoc, CategoryTheory.CategoryOfElements.instHasInitialElementsOppositeOfIsRepresentable, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_inv_naturality, Profinite.isIso_indexCone_lift, CategoryTheory.Limits.coneOfCoconeRightOp_π, CategoryTheory.extensiveTopology.isSheaf_yoneda_obj, CategoryTheory.GrothendieckTopology.pseudofunctorOver_toPrelaxFunctor_toPrelaxFunctorStruct_toPrefunctor_obj_str_comp_val_app, CategoryTheory.Pretriangulated.shiftFunctorAdd'_op_inv_app, AlgebraicGeometry.Scheme.Hom.ι_toNormalization, CompHausLike.LocallyConstant.counitApp_app, 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, SSet.PtSimplex.RelStruct.δ_castSucc_map, CategoryTheory.eHomFunctor_obj_obj, Condensed.isoFinYonedaComponents_hom_apply, AlgebraicGeometry.Scheme.Hom.smooth_appLE, SSet.stdSimplex.obj₀Equiv_apply, SSet.horn₃₁.desc.multicofork_pt, AlgebraicTopology.AlternatingFaceMapComplex.map_f, CategoryTheory.Pretriangulated.instIsTriangulatedOppositeOpOp, CategoryTheory.GrothendieckTopology.pseudofunctorOver_mapComp_inv_toNatTrans_app_val_app, 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.Equivalence.sheafCongrPrecoherent_functor_obj_val_obj, CategoryTheory.Limits.preservesColimit_rightOp, CategoryTheory.Functor.sheafPushforwardContinuousNatTrans_app_val, CompHausLike.LocallyConstantModule.functor_map_val, CategoryTheory.Functor.PullbackObjObj.π_iso_of_iso_right_hom, CompHausLike.LocallyConstant.functor_map_val, SSet.skeleton_le_skeletonOfMono, CategoryTheory.Limits.PullbackCone.unop_inr, CategoryTheory.Sheaf.instIsIsoAppCounitConstantSheafAdjOfFaithfulOfFullConstantSheafOfIsConstant, SSet.prodStdSimplex.objEquiv_δ_apply, CategoryTheory.Enriched.FunctorCategory.enrichedId_π_assoc, Subobject.presheaf_obj, CategoryTheory.Functor.opUnopEquiv_counitIso, CategoryTheory.Adjunction.Quadruple.op_leftTriple, AddCommMonCat.coyonedaType_map_app, CategoryTheory.NatTrans.op_id, CategoryTheory.Limits.pullbackIsoUnopPushout_hom_inl, CategoryTheory.Limits.preservesColimitsOfShape_of_leftOp, SheafOfModules.evaluationPreservesLimitsOfSize, CategoryTheory.GrothendieckTopology.Point.instPreservesColimitsOfSizeFunctorOppositePresheafFiber, CategoryTheory.Comon.MonOpOpToComon_map_hom, CategoryTheory.Functor.instFaithfulOppositeOp, Condensed.instPreservesFiniteProductsOppositeCompHausVal, 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, CategoryTheory.Join.opEquiv_inverse_map_inclRight_op, Profinite.Extend.functorOp_obj, AlgebraicGeometry.Scheme.Hom.congr_app, CategoryTheory.HasClassifier.reflectsIsomorphismsOp, AlgebraicGeometry.instFullOppositeCommRingCatLocallyRingedSpaceToLocallyRingedSpace, CategoryTheory.Functor.PullbackObjObj.mapArrowRight_comp_assoc, CategoryTheory.Limits.preservesColimitsOfSize_of_rightOp, CategoryTheory.Limits.hasLimitsOfShape_op_of_hasColimitsOfShape, CategoryTheory.Presheaf.isLocallySurjective_toSheafify, 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.uliftYonedaEquiv_apply, CategoryTheory.Limits.ι_comp_colimitRightOpIsoUnopLimit_hom_assoc, CategoryTheory.Functor.sheafPushforwardContinuous_map_val_app, AlgebraicGeometry.Scheme.Opens.toSpecΓ_naturality_assoc, AlgebraicGeometry.Scheme.ofRestrict_appLE, AlgebraicGeometry.Scheme.Hom.QuasiFiniteAt.quasiFiniteAt, CategoryTheory.SimplicialObject.hom_ext_iff, 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, PresheafOfModules.unit_map_one, CategoryTheory.ShortComplex.exact_unop_iff, CategoryTheory.Functor.instInitialOppositeLeftOpOfFinal, SSet.Subcomplex.toSSetFunctor_obj, PresheafOfModules.zsmul_app, AlgebraicGeometry.isLocallyNoetherian_iff_of_affine_openCover, AlgebraicGeometry.Scheme.kerAdjunction_unit_app, Condensed.finYoneda_map, AlgebraicTopology.alternatingFaceMapComplex_map_f, CategoryTheory.Classifier.SubobjectRepresentableBy.homEquiv_eq, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.s_comp_δ, SSet.hasDimensionLT_subcomplex_top_iff, PresheafOfModules.toPresheaf_obj_coe, SSet.Truncated.Edge.id_tensor_id, 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, topCatToSheafCompHausLike_map_val_app, TopCat.Presheaf.germ_stalkSpecializes_assoc, CategoryTheory.Limits.Cocone.op_pt, CategoryTheory.Presieve.FamilyOfElements.singletonEquiv_apply, CategoryTheory.prodOpEquiv_unitIso_hom_app, 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, 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, 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.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', TopologicalSpace.OpenNhds.op_map_id_obj, CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheafMap_restriction_assoc, CategoryTheory.PreOneHypercover.multifork_ι, AugmentedSimplexCategory.equivAugmentedSimplicialObjectFunctorCompPointIso_inv_app, AlgebraicGeometry.Scheme.isoSpec_hom_naturality_assoc, AugmentedSimplexCategory.equivAugmentedSimplicialObjectFunctorCompPointIso_hom_app, CategoryTheory.ParametrizedAdjunction.hasLiftingProperty_iff, AlgebraicGeometry.Scheme.isoSpec_inv_image_zeroLocus, CategoryTheory.Pretriangulated.TriangleOpEquivalence.functor_map_hom₃, AlgebraicGeometry.Scheme.IdealSheafData.range_glueDataObjι_ι, CommGrpCat.coyonedaType_map_app, CategoryTheory.instIsIsoFunctorOppositeSheafSheafComposeNatTrans, AlgebraicGeometry.Scheme.germToFunctionField_injective, CategoryTheory.NatTrans.removeLeftOp_app, CommRingCat.moduleCatRestrictScalarsPseudofunctor_mapId, CategoryTheory.Subfunctor.IsGeneratedBy.ofSection_le, CategoryTheory.op_tensor_op, CategoryTheory.CommSq.HasLift.iff_op, CategoryTheory.Functor.IsCoverDense.restrictHomEquivHom_naturality_left, 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.homOfLE_op_comp_eqToHom_assoc, SSet.stdSimplex.spineId_arrow_apply_zero, CategoryTheory.Square.unop_f₁₂, CategoryTheory.shrinkYoneda_obj, 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, PresheafOfModules.map_smul, CategoryTheory.Pseudofunctor.DescentData.iso_inv, CategoryTheory.Abelian.subobjectIsoSubobjectOp_apply, AlgebraicGeometry.isIso_morphismRestrict_iff_isIso_app, CategoryTheory.Sheaf.imageι_val, CategoryTheory.Limits.preservesLimit_leftOp, CategoryTheory.kernelUnopUnop_inv, CategoryTheory.Limits.coyonedaOpColimitIsoLimitCoyoneda'_inv_comp_π, CategoryTheory.NatTrans.leftOp_id, CategoryTheory.Enriched.FunctorCategory.diagram_map_app, 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.RelCat.instIsEquivalenceOppositeUnopFunctor, CategoryTheory.Limits.parallelPairOpIso_hom_app_one, CategoryTheory.Limits.walkingCospanOpEquiv_counitIso_inv_app, CategoryTheory.Functor.IsCoverDense.sheafHom_restrict_eq, TopCat.Presheaf.stalk_open_algebraMap, CategoryTheory.ShortComplex.opMap_τ₃, CategoryTheory.PreOneHypercover.multicospanIndex_snd, CategoryTheory.IsCommMon.ofRepresentableBy, AlgebraicGeometry.Scheme.Hom.app_surjective, CategoryTheory.Pseudofunctor.CoGrothendieck.instFullαCategoryObjLocallyDiscreteOppositeCatMkOpFiberForgetInducedFunctor, CategoryTheory.Limits.PullbackCone.op_ι_app, CategoryTheory.ShortComplex.opcyclesOpIso_hom_naturality_assoc, 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.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.Pseudofunctor.DescentData.nonempty_fullyFaithful_toDescentData_iff_of_sieve_eq, 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.ShortComplex.unop_f, CategoryTheory.Adjunction.Triple.op_rightToLeft, AlgebraicGeometry.Scheme.Opens.toScheme_presheaf_obj, PresheafOfModules.instPreservesLimitsOfSizeModuleCatCarrierObjOppositeRingCatEvaluation, CategoryTheory.GrothendieckTopology.W_iff_isIso_map_of_adjunction, AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal_ideal, AlgebraicGeometry.PresheafedSpace.Γ_map_op, CategoryTheory.simplicialToCosimplicialAugmented_map_left, CategoryTheory.Equivalence.sheafCongrPreregular_functor_obj_val_obj, 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, CategoryTheory.Functor.isDense_iff_fullyFaithful_restrictedULiftYoneda, CategoryTheory.Sheaf.ΓHomEquiv_naturality_left, CategoryTheory.SimplicialObject.Truncated.rightExtensionInclusion_left, 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.Arrow.cechNerve_obj, CategoryTheory.yonedaMon_naturality_assoc, SSet.instFiniteElemObjOppositeSimplexCategoryOpMkNonDegenerateOfFinite, CategoryTheory.coyoneda_preservesLimits, CategoryTheory.Comon.ComonToMonOpOp_map, SheafOfModules.pushforwardComp_inv_app_val_app, AlgebraicGeometry.AffineScheme.Γ_preservesLimits, 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.ShortComplex.homologyMap'_op, SSet.spine_arrow, CategoryTheory.instRepresentablyFlatOppositeOpOfRepresentablyCoflat, CategoryTheory.Cat.opFunctor_obj, CategoryTheory.Equivalence.sheafCongrPreregular_functor_map_val_app, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_hom_naturality_assoc, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivCounitIso_inv_app_hom, CategoryTheory.Subfunctor.ofSection_obj, CategoryTheory.uliftYonedaEquiv_naturality, CategoryTheory.SimplicialObject.δ_comp_σ_self'_assoc, CategoryTheory.Limits.opHomCompWhiskeringLimYonedaIsoCocones_hom_app_app_app, CategoryTheory.Subpresheaf.isSeparated, CategoryTheory.CosimplicialObject.Augmented.leftOpRightOpIso_inv_left, CategoryTheory.Limits.Fork.unop_op_ι, CategoryTheory.Limits.FormalCoproduct.powerBifunctor_obj, TopCat.Sheaf.extend_hom_app, CategoryTheory.ObjectProperty.isSeparating_unop_iff, CategoryTheory.OverPresheafAux.YonedaCollection.yonedaEquivFst_eq, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_inv_app_shift, CategoryTheory.GrothendieckTopology.pseudofunctorOver_toPrelaxFunctor_toPrelaxFunctorStruct_toPrefunctor_map_toFunctor_obj_val_obj, CategoryTheory.op_inv_braiding, CategoryTheory.Pretriangulated.TriangleOpEquivalence.counitIso_hom_app_hom₁, CategoryTheory.id_app, 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, AlgebraicGeometry.Scheme.IdealSheafData.ideal_bot, CategoryTheory.Functor.coreprW_hom_app, AlgebraicGeometry.Spec.toLocallyRingedSpace_obj, CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence_counitIso, CategoryTheory.ShortComplex.unopFunctor_obj, PresheafOfModules.pushforward₀_obj_obj_isAddCommGroup, PartialOrder.mem_nerve_degenerate_of_eq, SSet.Augmented.stdSimplex_map_right, SSet.Subcomplex.preimage_range, SSet.Subcomplex.toImage_app_coe, CategoryTheory.Limits.walkingSpanOpEquiv_inverse_obj, 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.Pretriangulated.instIsHomologicalOppositeAddCommGrpCatObjFunctorPreadditiveYoneda, CategoryTheory.Limits.piConst_obj_map, CommMonCat.coyonedaForget_inv_app_app, Condensed.instPreservesLimitsOfShapeOppositeProfiniteDiscreteCarrierToTopTotallyDisconnectedSpaceOfFinite, PartialOrder.mem_nerve_nonDegenerate_iff_injective, CategoryTheory.ShortComplex.unopMap_τ₁, CategoryTheory.Equalizer.Presieve.Arrows.compatible_iff, SSet.prodStdSimplex.objEquiv_apply_snd, CategoryTheory.Pretriangulated.TriangleOpEquivalence.counitIso_inv_app_hom₃, AlgebraicGeometry.Scheme.Γevaluation_naturality_apply, CategoryTheory.Adjunction.Triple.op_adj₂, CategoryTheory.NatTrans.removeUnop_app, AlgebraicGeometry.locallyOfFiniteType_iff, 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, 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.cosimplicialSimplicialEquiv_functor_obj_obj, CategoryTheory.CostructuredArrow.toStructuredArrow'_obj, PresheafOfModules.freeObj_map, 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.simplicialCosimplicialEquiv_unitIso_hom_app, CategoryTheory.Limits.pullbackIsoOpPushout_inv_fst_assoc, AlgebraicGeometry.Scheme.Opens.toSpecΓ_preimage_basicOpen, TopCat.Presheaf.isSheaf_iff_isSheaf_comp, 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, CategoryTheory.equivYoneda'_inv_val, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.ofRestrict_invApp_apply, CategoryTheory.Presheaf.isLocallyInjective_whisker, AlgebraicGeometry.SheafedSpace.id_hom_c_app, CategoryTheory.cokernelUnopOp_hom, CategoryTheory.Equivalence.sheafCongrPreregular_inverse_map_val_app, CategoryTheory.Equivalence.sheafCongrPrecoherent_functor_obj_val_map, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_w, AlgebraicGeometry.Scheme.zeroLocus_def, CategoryTheory.Subfunctor.Subpresheaf.mem_ofSection_obj, 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, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv_assoc, CategoryTheory.Pseudofunctor.DescentData.instIsIsoαCategoryObjLocallyDiscreteOppositeCatMkOpHom, CategoryTheory.GrothendieckTopology.W_adj_unit_app, CategoryTheory.Limits.preservesLimitsOfSize_leftOp, CategoryTheory.Limits.coneOfCoconeLeftOp_π_app, CategoryTheory.ObjectProperty.instSmallOppositeOp, CommAlgCat.lift_unop_hom, AlgebraicGeometry.ΓSpec.adjunction_counit_app, SSet.S.le_iff, CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheafObj_condition_assoc, CategoryTheory.Limits.coneOfCoconeUnop_π, CategoryTheory.Equivalence.rightOp_inverse_map, AlgebraicGeometry.StructureSheaf.algebraMap_self_map, CategoryTheory.sheafToPresheaf_obj, AlgebraicGeometry.tilde.isoTop_hom, CategoryTheory.OverPresheafAux.unitAuxAuxAux_inv, CategoryTheory.ShortComplex.RightHomologyMapData.unop_φK, AlgebraicGeometry.Scheme.restrict_presheaf_obj, CategoryTheory.additive_yonedaObj', SSet.hasDimensionLT_iff, CategoryTheory.Limits.isColimitOfConeOfCoconeLeftOp_desc, CategoryTheory.nerveMap_app_mk₁, CategoryTheory.OverPresheafAux.yonedaCollectionPresheafToA_app, CategoryTheory.Limits.SequentialProduct.functorMap_commSq_aux, CategoryTheory.sheafToPresheaf_map, 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.ParametrizedAdjunction.homEquiv_symm_naturality_one_assoc, CategoryTheory.op_leftUnitor, AlgebraicGeometry.germ_comp_stalkToFiberRingHom, CategoryTheory.Over.opEquivOpUnder_inverse_map, AlgebraicGeometry.Scheme.IdealSheafData.mem_supportSet_iff, CategoryTheory.Equalizer.Presieve.isSheafFor_singleton_iff, CategoryTheory.Projective.projective_iff_preservesEpimorphisms_coyoneda_obj, 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.Pretriangulated.Opposite.contractibleTriangleIso_inv_hom₃, CategoryTheory.presheafHom_map_app, 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, CompHausLike.LocallyConstant.functor_obj_val, CategoryTheory.Functor.rightOpId_inv_app, CategoryTheory.Limits.isLimitConeOfCoconeUnop_lift, CategoryTheory.faithful_preadditiveCoyoneda, CategoryTheory.Equivalence.sheafCongrPreregular_functor_obj_val_map, InfiniteGalois.mulEquivToLimit_symm_continuous, SSet.horn₃₁.desc.multicofork_π_two_assoc, CategoryTheory.Ind.isIndObject_inclusion_obj, CategoryTheory.Limits.isColimitOfConeUnopOfCocone_desc, CategoryTheory.Limits.yonedaCompLimIsoCocones_hom_app_app, PresheafOfModules.instEpiModuleCatCarrierObjOppositeRingCatApp, AlgebraicGeometry.LocallyOfFiniteType.finiteType_appLE, CategoryTheory.Limits.coend.condition_assoc, AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_stalkMap_toSpec_assoc, AlgebraicGeometry.Spec.map_appLE, CategoryTheory.Limits.instHasColimitDiscreteOppositeCompInverseOppositeOpFunctor, CategoryTheory.Functor.rightOp_obj, CategoryTheory.Limits.preservesLimitsOfShape_op, AlgebraicGeometry.Scheme.Hom.finitePresentation_appLE, CategoryTheory.Functor.mapTriangleOpCompTriangleOpEquivalenceFunctorApp_hom_hom₁, smoothSheafCommRing.ι_evalHom, AlgebraicGeometry.Γ_restrict_isLocalization, CategoryTheory.Subfunctor.ofSection_eq_range', CategoryTheory.simplicialCosimplicialEquiv_counitIso_hom_app_app, 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, AlgebraicGeometry.Scheme.Hom.app_eq, CategoryTheory.linearYoneda_map_app, AlgebraicGeometry.RingedSpace.res_zero, 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, 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, CommRingCat.moduleCatRestrictScalarsPseudofunctor_map, HomologicalComplex.opcyclesOpIso_hom_naturality_assoc, AlgebraicGeometry.Spec.map_preimage_unop, CategoryTheory.Limits.opCoproductIsoProduct'_inv_comp_inj, 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, TopCat.Presheaf.germToPullbackStalk_stalkPullbackHom, CondensedSet.epi_iff_surjective_on_stonean, 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, SSet.leftUnitor_inv_app_apply, CategoryTheory.Comon.monoidal_leftUnitor_hom_hom, CategoryTheory.ShortComplex.HomologyData.unop_left, AlgebraicGeometry.Scheme.SpecΓIdentity_app, CategoryTheory.Functor.representable_preservesLimit, CategoryTheory.Equivalence.sheafCongrPrecoherent_inverse_map_val_app, 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.RelCat.unopFunctor_comp_opFunctor_eq, AlgebraicGeometry.Spec.germ_stalkMapIso_hom_assoc, 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.Functor.ofOpSequence_obj, LightCondensed.isColimitLocallyConstantPresheafDiagram_desc_apply, AlgebraicGeometry.StructureSheaf.IsLocalization.to_basicOpen, CategoryTheory.Equivalence.rightOp_unitIso_hom_app, CategoryTheory.fullyFaithfulSheafToPresheaf_preimage_val, CategoryTheory.Abelian.IsGrothendieckAbelian.OppositeModuleEmbedding.full_embedding, AlgebraicGeometry.Scheme.Opens.germ_stalkIso_inv_assoc, CategoryTheory.Presieve.isSheafFor_arrows_iff, 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, 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, CategoryTheory.Limits.preservesFiniteLimits_of_rightOp, AlgebraicGeometry.instPreservesColimitsOfShapeOppositeCommRingCatSchemeDiscreteSpecOfFinite, CommAlgCat.fst_unop_hom, CategoryTheory.isIso_iff_isIso_yoneda_map, smoothSheafCommRing.evalHom_germ, CategoryTheory.essImage_yonedaGrp, CategoryTheory.ParametrizedAdjunction.homEquiv_symm_naturality_three, CategoryTheory.Sheaf.natTransΓRes_app, CategoryTheory.Limits.Fork.op_pt, SSet.instFiniteObjOppositeSimplexCategoryOfFinite, 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.GrothendieckTopology.uliftYonedaEquiv_uliftYoneda_map, AlgebraicGeometry.StructureSheaf.toOpen_comp_comap_assoc, CategoryTheory.MonoidalCategory.MonoidalRightAction.rightActionOfOppositeRightAction_actionHom, SSet.RelativeMorphism.le_preimage, CategoryTheory.Functor.map_shift_unop, CategoryTheory.LocalizerMorphism.LeftResolution.opFunctor_obj, AlgebraicGeometry.IsAffineOpen.self_le_basicOpen_union_iff, CategoryTheory.Limits.hasFiniteColimits_opposite, CategoryTheory.Functor.IsCoverDense.presheafIso_hom_app, AlgebraicGeometry.exists_appTop_π_eq_of_isAffine_of_isLimit, CategoryTheory.sheafifyMap_sheafifyLift, AlgebraicGeometry.LocallyRingedSpace.GlueData.ι_isoSheafedSpace_inv, SSet.Truncated.id_app, Condensed.instPreservesFiniteProductsOppositeProfiniteVal, 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.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.MonoidalCategory.MonoidalLeftAction.leftActionOfOppositeLeftAction_actionObj, CategoryTheory.unop_inv_leftUnitor, CompHausLike.LocallyConstantModule.functor_obj_val, ContinuousMap.yonedaPresheaf_map, CategoryTheory.Functor.op_commShiftIso_hom_app_assoc, CategoryTheory.Functor.FullyFaithful.compYonedaCompWhiskeringLeftMaxRight_inv_app_app, 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.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, CategoryTheory.sheafifyLift_id_toSheafify, CategoryTheory.evalEquiv_apply, SheafOfModules.toSheaf_map_val, 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.Limits.IndObjectPresentation.toCostructuredArrow_obj_right_as, CategoryTheory.Presheaf.freeYoneda_map, 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, 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.Sheaf.isSheaf_of_isLimit, CategoryTheory.Limits.coconeEquivalenceOpConeOp_functor_obj, AlgebraicTopology.AlternatingFaceMapComplex.ε_app_f_zero, HomologicalComplex.unop_X, 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.LocalizerMorphism.LeftResolution.unop_w, CategoryTheory.ShortComplex.opEquiv_functor, 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, AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap_app, AlgebraicGeometry.ΓSpec.adjunction_unit_app, CategoryTheory.ShortComplex.HomologyMapData.op_right, CategoryTheory.Functor.homObjFunctor_map_app, SSet.ι₀_snd, CategoryTheory.MonoidalCategory.DayConvolution.corepresentableBy₂'_homEquiv, AlgebraicGeometry.Scheme.basicOpen_pow, CategoryTheory.Limits.preservesLimitsOfSize_rightOp, 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.instIsSplitEpiOppositeOpOfIsSplitMono, CategoryTheory.ShortComplex.fromOpcycles_op_cyclesOpIso_inv_assoc, 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, CategoryTheory.sheafBotEquivalence_inverse_map_val, 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.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, AlgebraicGeometry.AffineSpace.functor_obj_map, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_inv_app'_assoc, CategoryTheory.MorphismProperty.instHasTwoOutOfThreePropertyOppositeOp, CategoryTheory.NatTrans.op_comp_assoc, CategoryTheory.Limits.endFunctor_obj, CategoryTheory.ParametrizedAdjunction.inr_arrowHomEquiv_symm_apply_left_assoc, SSet.StrictSegal.spineInjective, 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, CompHausLike.LocallyConstant.functorToPresheaves_map_app, AlgebraicGeometry.Scheme.AffineZariskiSite.restrictIsoSpec_hom_app, CategoryTheory.factorThruImage_comp_imageUnopOp_inv, SSet.Subcomplex.mem_nonDegenerate_iff, CondensedSet.topCatAdjunctionUnit_val_app_apply, AugmentedSimplexCategory.equivAugmentedSimplicialObjectFunctorCompToArrowIso_hom_app_right, 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, CategoryTheory.Limits.isLimitOfCoconeLeftOpOfCone_lift, AlgebraicGeometry.Scheme.evaluation_eq_zero_iff_notMem_basicOpen, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctor_obj_pt, CommMonCat.coyonedaType_obj_obj_coe, 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, 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, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.inv_naturality_assoc, CategoryTheory.forgetEnrichmentOppositeEquivalence_unitIso, AlgebraicGeometry.Scheme.Hom.appIso_inv_app, InfiniteGalois.proj_adjoin_singleton_val, CategoryTheory.Subfunctor.Subpresheaf.IsGeneratedBy.mem, SSet.Truncated.Edge.map_associator_hom, CategoryTheory.CategoryOfElements.costructuredArrowYonedaEquivalence_inverse, CategoryTheory.Limits.isIndObject_iff, CategoryTheory.Limits.hasColimitsOfShape_op_of_hasLimitsOfShape, 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, CategoryTheory.isSeparator_iff_faithful_coyoneda_obj, PresheafOfModules.toFreeYonedaCoproduct_fromFreeYonedaCoproduct_assoc, SSet.degenerate_zero, CategoryTheory.SimplicialObject.δ_comp_σ_of_gt'_assoc, CategoryTheory.ShortComplex.RightHomologyData.unop_f', TopCat.Presheaf.app_surjective_of_stalkFunctor_map_bijective, CategoryTheory.Limits.BinaryFan.op_mk, CategoryTheory.Sieve.uliftNatTransOfLe_comm, 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, CategoryTheory.instPreservesFiniteLimitsFunctorOppositeSheafLeftAdjointSheafToPresheaf, CategoryTheory.yoneda_preservesLimitsOfShape, AlgebraicGeometry.Scheme.Hom.map_appLE_assoc, CategoryTheory.ComposableArrows.opEquivalence_functor_map_app, CategoryTheory.Equivalence.sheafCongrPrecoherent_inverse_obj_val_obj, CategoryTheory.ShortComplex.SnakeInput.op_L₀, CategoryTheory.GrothendieckTopology.map_yonedaEquiv', CategoryTheory.GrothendieckTopology.W_inverseImage_whiskeringLeft, LightCondensed.instPreservesLimitsOfShapeOppositeLightProfiniteDiscreteCarrier, 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, 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.Pretriangulated.Opposite.contractibleTriangleIso_inv_hom₁, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_toSpecΓ, AlgebraicGeometry.StructureSheaf.globalSectionsIso_hom, SheafOfModules.instPreservesFiniteLimitsFunctorOppositeAddCommGrpCatCompSheafToSheafSheafToPresheaf, CategoryTheory.uliftYoneda_obj_obj, CategoryTheory.typeEquiv_counitIso_inv_app_val_app, CategoryTheory.ShortComplex.homologyOpIso_hom_naturality_assoc, CategoryTheory.OverPresheafAux.counitAuxAux_inv, CategoryTheory.MorphismProperty.IsMultiplicative.op, AlgebraicGeometry.Scheme.Hom.map_appLE'_assoc, SSet.stdSimplex.coe_edge_down_toOrderHom, CategoryTheory.preservesFiniteColimits_preadditiveYonedaObj_of_injective, AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap_app', AlgebraicGeometry.IsAffineOpen.toSpecΓ_isoSpec_inv, CategoryTheory.ComposableArrows.opEquivalence_inverse_obj, AlgebraicGeometry.Scheme.Hom.appIso_inv_naturality, CategoryTheory.SimplicialObject.δ_comp_δ'_assoc, AlgebraicGeometry.Scheme.IdealSheafData.map_ideal, CategoryTheory.Limits.opProductIsoCoproduct'_comp_self, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_presheaf_map, CategoryTheory.instIsEquivalenceOppositeOpOp, CategoryTheory.SimplicialObject.δ_comp_σ_self', CategoryTheory.Limits.π_comp_colimitOpIsoOpLimit_inv_assoc, LightCondensed.forget_map_val_app, CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinset_obj_map, AlgebraicGeometry.Scheme.basicOpen_add_le, AlgebraicGeometry.Scheme.Hom.comp_app_assoc, AlgebraicGeometry.Scheme.comp_app, CategoryTheory.yonedaMon_map_app, CategoryTheory.shrinkYonedaEquiv_naturality, CategoryTheory.Injective.projective_iff_injective_op, AlgebraicGeometry.Scheme.zeroLocus_map, AlgebraicGeometry.Scheme.Hom.finite_app, TopCat.Presheaf.stalk_hom_ext_iff, LightCondensed.isoLocallyConstantOfIsColimit_inv, AlgebraicGeometry.Scheme.AffineZariskiSite.coequifibered_iff_forall_isLocalizationAway, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_zeroLocus, CategoryTheory.Functor.opUnopEquiv_functor, AlgebraicGeometry.PresheafedSpace.restrictStalkIso_inv_eq_germ_apply, AlgebraicGeometry.Scheme.evaluation_naturality, CategoryTheory.Limits.IsIndObject.finallySmall, 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, CategoryTheory.sheafBotEquivalence_inverse_obj_val, 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.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, SSet.horn.edge_coe, HomologicalComplex.isSupportedOutside_op_iff, HomologicalComplex.unopEquivalence_unitIso, 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.Adjunction.corepresentableBy_homEquiv, AlgebraicGeometry.Scheme.IdealSheafData.ofIdeals_mono, AlgebraicGeometry.Scheme.instIsOpenImmersionToSpecΓOfIsQuasiAffine, CategoryTheory.ObjectProperty.instSmallOppositeOp_1, AlgebraicGeometry.Scheme.homOfLE_appTop, ContinuousMap.yonedaPresheaf_obj, CategoryTheory.Equalizer.Presieve.sheaf_condition, CategoryTheory.cosimplicialSimplicialEquiv_functor_map_app, CategoryTheory.Limits.isLimitConeUnopOfCocone_lift, CategoryTheory.NatIso.unop_symm, CategoryTheory.Presheaf.isLocallyInjective_whisker_iff, CategoryTheory.Cat.opFunctor_map, 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', AlgebraicGeometry.IsAffineOpen.fromSpec_app_self, AlgebraicGeometry.Spec.toPresheafedSpace_map_op, CategoryTheory.Limits.IndObjectPresentation.yoneda_isColimit_desc, PresheafOfModules.injective_of_mono, CategoryTheory.Limits.instHasLimitOppositeDiscreteOpFunctor, 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, CategoryTheory.Limits.coyonedaOpColimitIsoLimitCoyoneda'_hom_comp_π, CategoryTheory.lan_preservesFiniteLimits_of_preservesFiniteLimits, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.app_inv_app'_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.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, TopCat.Presheaf.app_bijective_of_stalkFunctor_map_bijective, CategoryTheory.full_linearCoyoneda, CategoryTheory.Pseudofunctor.DescentData.ofObj_obj, AlgebraicGeometry.Scheme.Hom.preimage_basicOpen, CategoryTheory.RelCat.opEquivalence_functor, AlgebraicTopology.DoldKan.compatibility_N₂_N₁_karoubi, CategoryTheory.IsGrothendieckAbelian.preservesColimit_coyoneda_obj_of_mono, 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.GrothendieckTopology.Point.W_isInvertedBy_presheafFiber, CategoryTheory.Equivalence.sheafCongr.inverse_obj_val_obj, CategoryTheory.isDetector_unop_iff, CategoryTheory.Limits.coneOfCoconeLeftOp_pt, CategoryTheory.Sheaf.adjunction_counit_app_val, CategoryTheory.ObjectProperty.instIsClosedUnderLimitsOfShapeOppositeOpOfIsClosedUnderColimitsOfShape_1, CategoryTheory.cokernelOpUnop_inv, CategoryTheory.GrothendieckTopology.pseudofunctorOver_toPrelaxFunctor_toPrelaxFunctorStruct_toPrefunctor_obj_str_id_val_app, CategoryTheory.coyonedaEquiv_naturality, CategoryTheory.SimplicialObject.δ_comp_δ', AlgebraicGeometry.Scheme.isoSpec_Spec_hom, 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.coyonedaEquiv_apply, SSet.Subcomplex.iSup_ofSimplex_nonDegenerate_eq_top, PresheafOfModules.pushforward_obj_map_apply, CategoryTheory.ObjectProperty.isDetecting_unop_iff, CategoryTheory.MonoidalCategory.MonoidalRightAction.rightActionOfOppositeRightAction_actionAssocIso_unop, 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, SSet.Truncated.StrictSegal.spineInjective, AlgebraicGeometry.Scheme.image_basicOpen, CategoryTheory.Enriched.FunctorCategory.enrichedComp_π_assoc, CategoryTheory.DinatTrans.compNatTrans_app, HomologicalComplex.homologyOp_hom_naturality_assoc, AlgebraicGeometry.tilde.isIso_toOpen_top, CategoryTheory.Presheaf.isLocallyInjective_forget_iff, CategoryTheory.regularTopology.isLocallySurjective_iff, PartialOrder.mem_range_nerve_σ_iff, CategoryTheory.StructuredArrow.toCostructuredArrow_map, CategoryTheory.Pseudofunctor.DescentData.hom_comp_assoc, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.invApp_app, TopCat.Presheaf.SubmonoidPresheaf.map, CategoryTheory.Abelian.extFunctor_obj, CategoryTheory.ShortComplex.cyclesOpIso_inv_naturality, CategoryTheory.Limits.coneRightOpOfCocone_π, CategoryTheory.Equalizer.Presieve.Arrows.FirstObj.ext_iff, SimplicialObject.Splitting.πSummand_comp_cofan_inj_id_comp_PInfty_eq_PInfty, CategoryTheory.kernel.ι_unop, CategoryTheory.ShortComplex.Splitting.unop_r, CategoryTheory.CostructuredArrow.overEquivPresheafCostructuredArrow_functor_map_toOverCompYoneda, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.app_invApp_assoc, CategoryTheory.NatTrans.equifibered_op_iff, CategoryTheory.Comon.monoidal_whiskerRight_hom, CategoryTheory.Functor.IsCoverDense.Types.appHom_restrict, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.app_invApp, CategoryTheory.OverPresheafAux.counitBackward_counitForward, AlgebraicGeometry.PresheafedSpace.restrictStalkIso_hom_eq_germ_apply, CategoryTheory.Comma.unopFunctor_obj, AlgebraicGeometry.HasRingHomProperty.appTop, 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, SimplicialObject.Split.toKaroubiNondegComplexFunctorIsoN₁_hom_app_f_f, CategoryTheory.GrothendieckTopology.yonedaEquiv_symm_map, CategoryTheory.CategoryOfElements.costructuredArrowULiftYonedaEquivalence_functor_map, CategoryTheory.unop_neg, TopCat.Sheaf.existsUnique_gluing', SSet.Path.map_arrow, CategoryTheory.Pretriangulated.Opposite.commShift_natTrans_op_int, CategoryTheory.Functor.rightOp_faithful, 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.Iso.unop_hom_inv_id_app, CategoryTheory.Coyoneda.naturality, Condensed.isoLocallyConstantOfIsColimit_inv, CategoryTheory.Presheaf.isLocallySurjective_iff_range_sheafify_eq_top, CategoryTheory.MonoidalCategory.MonoidalLeftAction.oppositeLeftAction_actionHomLeft_op, 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, 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, TopCat.Presheaf.pushforwardPullbackAdjunction_unit_app_app_germToPullbackStalk_assoc, 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, 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, CategoryTheory.Functor.sheafPushforwardContinuous_obj_val_map, AlgebraicGeometry.instIsIsoSchemeAppUnitOppositeCommRingCatAdjunctionOfIsAffine, 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, PresheafOfModules.unitHomEquiv_apply_coe, CategoryTheory.Equivalence.symmEquiv_functor, 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.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.Limits.IndObjectPresentation.toCostructuredArrow_map_left, AlgebraicGeometry.germ_stalkClosedPointIso_hom, CategoryTheory.Join.opEquiv_inverse_map_inclLeft_op, TopCat.Presheaf.isLocallySurjective_iff, CategoryTheory.Pretriangulated.opShiftFunctorEquivalenceSymmHomEquiv_apply, fintypeToFinBoolAlgOp_map, CategoryTheory.Limits.multispanIndexCoend_left, AlgebraicGeometry.Scheme.toSpecΓ_isoSpec_inv, CategoryTheory.MorphismProperty.LeftFraction.op_s, AlgebraicGeometry.IsFinite.instHasAffinePropertyAndIsAffineFiniteCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTopHomAppTop, CategoryTheory.Functor.final_op_of_initial, CategoryTheory.MonoidalCategory.DayConvolutionUnit.rightUnitorCorepresentingIso_hom_app_app, SheafOfModules.pushforwardCongr_inv_app_val_app, CategoryTheory.Functor.PullbackObjObj.ofHasPullback_π, CategoryTheory.MonoidalCategory.MonoidalLeftAction.leftActionOfOppositeLeftAction_actionUnitIso, HomologicalComplex.instHasHomologyOppositeObjSymmOpFunctorOp, AlgebraicGeometry.Scheme.Modules.mapPresheaf_app, AugmentedSimplexCategory.equivAugmentedSimplicialObject_functor_map_right, CategoryTheory.preservesLimits_preadditiveYoneda_obj, CategoryTheory.Adjunction.rightOp_counit, CategoryTheory.instPreservesFiniteLimitsFunctorOppositeSheafPresheafToSheaf, SSet.StrictSegal.spineToSimplex_spine, SSet.Subcomplex.prod_obj, SimplicialObject.Split.cofan_inj_naturality_symm_assoc, CategoryTheory.Functor.IsCoverDense.Types.appIso_inv, 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.Equivalence.sheafCongr.unitIso_hom_app_val_app, CategoryTheory.monoidalOpOp_η, CategoryTheory.Comon.monoidal_associator_hom_hom, CategoryTheory.Limits.PreservesFiniteLimitsOfIsFilteredCostructuredArrowYonedaAux.isoAux_hom_app, typeToBoolAlgOp_map, AlgebraicGeometry.Scheme.Hom.preservesLocalization_normalizationDiagramMap, CategoryTheory.op_inv_leftUnitor, 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, AlgebraicGeometry.ΓSpec.isIso_locallyRingedSpaceAdjunction_counit, SSet.Truncated.HomotopyCategory.BinaryProduct.inverse_map_mkHom_homMk_id, CategoryTheory.additive_coyonedaObj', CategoryTheory.Subfunctor.range_eq_ofSection', SimplicialObject.opFunctor_obj_δ, CategoryTheory.unop_add, CategoryTheory.Functor.FullyFaithful.homNatIso'_hom_app_down, CategoryTheory.Functor.leftOpId_inv_app, CategoryTheory.Functor.PreOneHypercoverDenseData.multicospanMap_app, CategoryTheory.SimplicialObject.σ_comp_σ, CategoryTheory.Limits.pullbackIsoOpPushout_inv_snd, AlgebraicGeometry.Spec.sheafedSpaceMap_hom_c_app, CategoryTheory.typeEquiv_functor_obj_val_map, SSet.const_app, AlgebraicGeometry.PresheafedSpace.GlueData.ι_isOpenEmbedding, TopCat.Presheaf.Pushforward.comp_inv_app, RingCat.moduleCatRestrictScalarsPseudofunctor_mapId, AlgebraicGeometry.Scheme.instFullOppositeIdealSheafDataOverSubschemeFunctor, CategoryTheory.regularTopology.isSheaf_yoneda_obj, CategoryTheory.MonoidalCategory.MonoidalRightAction.oppositeRightAction_actionAssocIso_op, 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.Functor.IsCoverDense.Types.sheafIso_hom_val, CategoryTheory.Limits.coendFunctor_obj, SSet.Truncated.Edge.map_snd, CategoryTheory.ShortComplex.unop_X₃, CategoryTheory.sheafToPresheafCompCoyonedaCompWhiskeringLeftSheafToPresheaf_app_app, CategoryTheory.Limits.end_.map_id, AlgebraicGeometry.Scheme.Hom.appIso_inv_naturality_assoc, CategoryTheory.leftDualFunctor_map, AlgebraicTopology.DoldKan.MorphComponents.id_φ, SSet.stdSimplex.instFiniteObjOppositeSimplexCategory, AlgebraicGeometry.Scheme.Hom.appLE_map_assoc, SSet.StrictSegal.isPointwiseRightKanExtensionAt.fac_aux₂, CategoryTheory.Limits.walkingParallelPairOpEquiv_unitIso_inv_app_zero, CategoryTheory.ParametrizedAdjunction.homEquiv_symm_naturality_two, CategoryTheory.SimplicialObject.δ_comp_δ'', PresheafOfModules.freeAdjunctionUnit_app, CategoryTheory.Presheaf.isLocallySurjective_iff_range_sheafify_eq_top', CategoryTheory.op_rightUnitor, smoothSheaf.obj_eq, CategoryTheory.Presieve.FamilyOfElements.singletonEquiv_symm_apply, CategoryTheory.Functor.IsLocalization.op_iff, CategoryTheory.MorphismProperty.LeftFraction.unop_s, 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, CategoryTheory.Presheaf.isSheaf_iff_isLimit, CategoryTheory.Functor.ofOpSequence_map_homOfLE_succ, CategoryTheory.Under.opEquivOpOver_functor_obj, CategoryTheory.MonoidalCategory.MonoidalLeftAction.leftActionOfOppositeLeftAction_actionHomLeft_unop, SSet.Subcomplex.preimage_obj, AlgebraicGeometry.instIsRightAdjointCommRingCatOppositeSchemeΓ, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.isoRestrict_hom_hom_c_app, AlgebraicGeometry.IsAffineOpen.iSup_basicOpen_eq_self_iff, SSet.Subcomplex.image_top, AlgebraicGeometry.Scheme.Hom.comp_appIso, CategoryTheory.Limits.hasPushouts_opposite, CategoryTheory.Pretriangulated.triangleOpEquivalence_functor, CompHausLike.LocallyConstantModule.functorToPresheaves_obj_map, CategoryTheory.Enriched.Functor.natTransEquiv_symm_app_app_apply, AlgebraicGeometry.Scheme.basicOpen_one, CategoryTheory.Limits.isIndObject_limit_comp_yoneda, CategoryTheory.Limits.preservesColimits_leftOp, CategoryTheory.Square.op_X₁, CategoryTheory.MonoidalCategory.MonoidalLeftAction.oppositeLeftAction_actionObj, HomologicalComplex.extend_op_d, SSet.Truncated.HomotopyCategory.descOfTruncation_map_homMk, PresheafOfModules.presheaf_obj_coe, CategoryTheory.GrothendieckTopology.uliftYonedaIsoYoneda_hom_app_val_app, 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, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_invApp, SSet.Truncated.HomotopyCategory.BinaryProduct.inverse_map_mkHom_homMk_homMk, CategoryTheory.Functor.sheafPushforwardContinuousComp_inv_app_val_app, SSet.Truncated.Edge.map_id, CategoryTheory.Subfunctor.IsGeneratedBy.iSup_eq, AlgebraicGeometry.Scheme.Modules.Hom.id_app, CategoryTheory.Adjunction.rightOp_eq, CategoryTheory.Limits.Cone.equiv_inv_π, CategoryTheory.rightDualFunctor_map, SSet.orderEmbeddingN_apply, CategoryTheory.isCofilteredOrEmpty_op_of_isFilteredOrEmpty, CategoryTheory.ShortComplex.hasRightHomology_iff_unop, AlgebraicGeometry.targetAffineLocally_affineAnd_iff, CompHausLike.LocallyConstant.adjunction_left_triangle, AlgebraicGeometry.StructureSheaf.algebraMap_obj_top_bijective, CategoryTheory.extensiveTopology.isLocallySurjective_iff, AlgebraicGeometry.Scheme.Hom.appIso_inv_app_assoc, CategoryTheory.NatIso.unop_whiskerRight, CategoryTheory.Limits.π_comp_colimitOpIsoOpLimit_inv, CategoryTheory.Functor.leftOpRightOpEquiv_inverse_map, 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.hasFiniteCoproducts_opposite, CategoryTheory.Presheaf.isSheaf_iff_multiequalizer, HomologicalComplex.unopSymm_d, AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalHom, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_self, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_appLE_assoc, CategoryTheory.Limits.ProductsFromFiniteCofiltered.finiteSubproductsCone_pt, CategoryTheory.Sieve.uliftNatTransOfLe_app_down_coe, SimplicialObject.Split.Hom.comm, AlgebraicGeometry.Scheme.Γ_map, CategoryTheory.Equivalence.sheafCongrPreregular_counitIso_hom_app_val_app, 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, CategoryTheory.SimplicialObject.Augmented.rightOp_right_obj, 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, AlgebraicTopology.DoldKan.QInfty_f_naturality, CategoryTheory.Limits.walkingSpanOpEquiv_unitIso_inv_app, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.instIsIsoCommRingCatInvApp, CategoryTheory.NatTrans.removeRightOp_app, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_naturality_apply, TopCat.Presheaf.stalkFunctor_map_germ_apply, HomologicalComplex.isStrictlySupported_op_iff, TopCat.Presheaf.germ_res_apply', 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, CategoryTheory.Equivalence.sheafCongrPrecoherent_counitIso_inv_app_val_app, CategoryTheory.LocalizerMorphism.RightResolution.unopFunctor_obj, AlgebraicGeometry.Scheme.Modules.Hom.isIso_iff_isIso_app, CategoryTheory.SimplicialObject.Truncated.rightExtensionInclusion_right_as, Condensed.isoFinYonedaComponents_inv_comp, CategoryTheory.Sheaf.coneΓ_pt, AlgebraicGeometry.Spec_zeroLocus_eq_zeroLocus, SSet.face_le_horn, SSet.Subcomplex.image_preimage_le, CategoryTheory.Functor.PullbackObjObj.mapArrowRight_comp, CategoryTheory.Equivalence.sheafCongr.functor_map_val_app, AlgebraicGeometry.Scheme.Opens.ι_appTop, AlgebraicGeometry.Scheme.Modules.Hom.comp_app, CategoryTheory.unopHom_apply, CategoryTheory.Limits.hasLimit_of_hasColimit_leftOp, CategoryTheory.Functor.sheafPushforwardContinuousComp_hom_app_val_app, HomologicalComplex.cyclesOpIso_hom_naturality_assoc, CategoryTheory.instReflectsIsomorphismsSheafFunctorOppositeSheafToPresheaf, Condensed.instFinalOppositeDiscreteQuotientCarrierToTopTotallyDisconnectedSpaceCostructuredArrowFintypeCatProfiniteOpToProfiniteOpPtAsLimitConeFunctorOp, CategoryTheory.Adjunction.compPreadditiveYonedaIso_hom_app_app_apply, CategoryTheory.Limits.IndObjectPresentation.yoneda_ι_app, CategoryTheory.SimplicialObject.δ_comp_δ_self, CategoryTheory.ShortComplex.RightHomologyData.op_H, AddCommGrpCat.coyonedaType_obj_map, CategoryTheory.Square.opFunctor_obj, 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.Square.unop_f₁₃, AlgebraicGeometry.Proj.zero_apply, SSet.Truncated.StrictSegal.spineToSimplex_vertex, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_inv_naturality_assoc, PresheafOfModules.Derivation'.d_app, PresheafOfModules.neg_app, CategoryTheory.CosimplicialObject.Augmented.leftOpRightOpIso_hom_right_app, CategoryTheory.Functor.sheafPushforwardContinuous_obj_val_obj, 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.ShortComplex.opMap_τ₂, CategoryTheory.Pretriangulated.mem_distTriang_op_iff', CategoryTheory.cosimplicialSimplicialEquiv_unitIso_hom_app, CategoryTheory.Join.inclRightCompOpEquivInverse_hom_app_op, StalkSkyscraperPresheafAdjunctionAuxs.germ_fromStalk_assoc, CategoryTheory.Functor.RepresentableBy.isRepresentedBy, CategoryTheory.Functor.IsLeftAdjoint.leftOp, CategoryTheory.Functor.representable_preservesLimitsOfShape, AlgebraicGeometry.mono_pushoutSection_of_isCompact_of_flat_left_of_ringHomFlat, 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.NatIso.op_associator, AlgebraicGeometry.IsAffineOpen.toSpecΓ_fromSpec_assoc, 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, 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, CategoryTheory.Equivalence.inverseFunctorObj'_hom_app, CategoryTheory.Limits.IndObjectPresentation.toCostructuredArrow_obj_hom, CategoryTheory.Presheaf.hasSeparator, AlgebraicGeometry.Scheme.Modules.pseudofunctor_mapId_hom_τr, CategoryTheory.Functor.op_commShiftIso_inv_app, AlgebraicGeometry.instQuasiSeparatedToSpecΓOfQuasiSeparatedSpaceCarrierCarrierCommRingCat, CategoryTheory.Comma.unopFunctorCompSnd_inv_app, PresheafOfModules.map_comp, AlgebraicGeometry.IsAffineOpen.basicOpenSectionsToAffine_isIso, CategoryTheory.cocones_obj_map_app, PresheafOfModules.instPreservesLimitsOfShapeFunctorOppositeAbToPresheaf, CategoryTheory.coyonedaPreservesLimitsOfShape, SSet.Truncated.StrictSegal.spine_δ_vertex_ge, AlgebraicGeometry.Scheme.IdealSheafData.subschemeFunctor_obj, CategoryTheory.ShiftedHom.opEquiv'_symm_apply, AlgebraicGeometry.PresheafedSpace.map_id_c_app, SSet.StrictSegal.spineToSimplex_edge, TopCat.Presheaf.germ_exist, CategoryTheory.yonedaFunctor_preservesLimits, AlgebraicGeometry.exists_smooth_of_formallySmooth_stalk, AlgebraicGeometry.Scheme.comp_appTop_assoc, CategoryTheory.GrothendieckTopology.W_whiskerLeft_iff, 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.widePullbackShapeOp_obj, TopCat.Presheaf.SheafConditionEqualizerProducts.res_π, CategoryTheory.SimplicialObject.Augmented.whiskering_map_app_left, AlgebraicGeometry.Scheme.local_affine, TopCat.Presheaf.germ_res, HomotopicalAlgebra.trivialCofibrations_op, AlgebraicGeometry.PresheafedSpace.GlueData.f_invApp_f_app, CategoryTheory.OverPresheafAux.counitAux_hom, 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.Pretriangulated.instCommShiftOppositeOpOpEquivalenceInt, TopCat.Presheaf.instIsLocalizationCarrierObjOppositeOpensCarrierCommRingCatObjLocalizationPresheaf, CategoryTheory.Functor.leftOp_additive, SSet.Truncated.spine_arrow, AugmentedSimplexCategory.equivAugmentedSimplicialObject_counitIso_hom_app_left_app, CategoryTheory.CategoryOfElements.fromCostructuredArrow_map_coe, AlgebraicGeometry.instIsAffineObjOppositeCommRingCatSchemeSpec, AlgebraicGeometry.Scheme.Hom.etale_appLE, CategoryTheory.Equivalence.leftOp_inverse_map, PresheafOfModules.evaluation_preservesColimitsOfSize, CompHausOpToFrame.faithful, CategoryTheory.Equivalence.inverseFunctorMapIso_symm_eq_isoInverseOfIsoFunctor, SSet.stdSimplex.monotone_apply, CategoryTheory.isoSheafify_hom, CategoryTheory.Pretriangulated.Opposite.commShift_adjunction_op_int, 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.RetractArrow.unop_r_right, AlgebraicGeometry.Scheme.isoSpec_Spec, CategoryTheory.Pretriangulated.TriangleOpEquivalence.unitIso_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, CategoryTheory.yonedaGrpObjIsoOfRepresentableBy_hom, CategoryTheory.Limits.hasLimit_op_of_hasColimit, 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, AlgebraicGeometry.IsClosedImmersion.hasAffineProperty, SheafOfModules.Presentation.map_relations_I, CategoryTheory.Join.opEquiv_functor_obj_op_left, SSet.Subcomplex.image_ofSimplex, CategoryTheory.Functor.IsRightAdjoint.op, PresheafOfModules.instPreservesLimitsOfSizeFunctorOppositeAbToPresheaf, 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, CategoryTheory.Coyoneda.colimitCoconeIsColimit_desc, TopCat.Presheaf.app_injective_of_stalkFunctor_map_injective, CategoryTheory.Join.opEquiv_inverse_obj_left_op, 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, SSet.Subcomplex.le_iff_of_hasDimensionLT, CategoryTheory.ShortComplex.SnakeInput.op_L₃, topCatOpToFrm_obj_coe, 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.ParametrizedAdjunction.homEquiv_naturality_three, CategoryTheory.Presheaf.instIsLeftKanExtensionOppositeObjFunctorTypeYonedaYonedaMap, AlgebraicGeometry.Scheme.eq_zeroLocus_of_isClosed_of_isAffine, CategoryTheory.Limits.limitOpIsoOpColimit_inv_comp_π_assoc, CategoryTheory.comp_app, 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, TopCat.subpresheafToTypes_map_coe, CategoryTheory.MorphismProperty.RightFraction.op_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.toSheafification_app, SSet.nonDegenerate_iff_of_isIso, CategoryTheory.Pretriangulated.Opposite.OpOpCommShift.iso_inv_app_assoc, AlgebraicGeometry.LocallyRingedSpace.evaluation_naturality, 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.Limits.ι_comp_colimitRightOpIsoUnopLimit_hom, CategoryTheory.Comma.opFunctorCompSnd_hom_app, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_map_d, CategoryTheory.op_hom_associator, CategoryTheory.Presheaf.isSheaf_iff_preservesFiniteProducts_of_projective, topToLocale_map, CategoryTheory.opOpEquivalence_unitIso, CategoryTheory.Limits.hasProducts_opposite, CategoryTheory.WithInitial.opEquiv_counitIso_inv_app, CategoryTheory.GrothendieckTopology.OneHypercoverFamily.IsSheafIff.fac', CategoryTheory.Pretriangulated.TriangleOpEquivalence.functor_obj, CategoryTheory.OverPresheafAux.unitAuxAux_inv_app_snd_coe, CategoryTheory.Limits.widePushoutShapeOpEquiv_counitIso, 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, 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.Functor.sheafPushforwardContinuousId_hom_app_val_app, CategoryTheory.Equalizer.Presieve.Arrows.compatible_iff_of_small, SSet.Truncated.comp_app_assoc, CategoryTheory.Coyoneda.colimitCocone_pt, 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.Join.inclLeftCompOpEquivInverse_inv_app_op, 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, TopCat.Presheaf.pushforward_map_app', 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, CondensedSet.topCatAdjunctionUnit_val_app, CategoryTheory.GrothendieckTopology.OneHypercoverFamily.IsSheafIff.fac'_assoc, AlgebraicGeometry.eq_zero_of_basicOpen_eq_bot, CategoryTheory.Functor.PreOneHypercoverDenseData.multicospanMapIso_inv, AlgebraicGeometry.Scheme.homOfLE_appLE, CategoryTheory.SimplicialObject.δ_comp_σ_self, CategoryTheory.Functor.sheafPushforwardContinuousId'_inv_app_val_app, preservesFilteredColimits_coyoneda, CategoryTheory.Adjunction.Triple.leftToRight_op, LightCondensed.hom_ext_iff, CategoryTheory.uliftYoneda_map_app, CategoryTheory.Yoneda.yoneda_full, CompHausLike.LocallyConstantModule.functorToPresheaves_obj_obj_carrier, CategoryTheory.Presieve.isSheafFor_over_map_op_comp_ofArrows_iff, CategoryTheory.Functor.map_opShiftFunctorEquivalence_counitIso_hom_app_unop, lightProfiniteToLightCondSetIsoTopCatToLightCondSet_hom_app_val_app_apply, CategoryTheory.Iso.unop_inv, 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, AlgebraicGeometry.isLocalization_away_of_isAffine, CategoryTheory.Presieve.isSheafFor_over_map_op_comp_iff, CategoryTheory.Adjunction.leftOp_counit, AlgebraicGeometry.Scheme.Hom.isIso_app, CategoryTheory.equivYoneda_hom_app, CategoryTheory.Presheaf.instIsCardinalPresentableFunctorOppositeTypeObjUliftYonedaOfHasColimitsOfSize, CategoryTheory.Functor.mapTriangleOpCompTriangleOpEquivalenceFunctorApp_hom_hom₃, AlgebraicGeometry.Scheme.basicOpen_appLE, TopCat.Presheaf.presheafEquivOfIso_unitIso_inv_app_app, CategoryTheory.Limits.coend.map_comp, CategoryTheory.Limits.isLimitOfCoconeOfConeUnop_lift, CategoryTheory.Presieve.FamilyOfElements.IsAmalgamation.compPresheafMap, 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, 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, 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, AlgebraicGeometry.Scheme.IdealSheafData.ofIdealTop_ideal, AlgebraicTopology.DoldKan.HigherFacesVanish.comp_P_eq_self_assoc, CategoryTheory.MonoidalCategory.MonoidalLeftAction.leftActionOfOppositeLeftAction_actionAssocIso_unop, CategoryTheory.ShortComplex.LeftHomologyData.op_p, CategoryTheory.Limits.coyonedaOpColimitIsoLimitCoyoneda_inv_comp_π, 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, CategoryTheory.Limits.preservesColimitsOfSize_leftOp, TopCat.Presheaf.stalkPushforward_germ, SSet.Truncated.spine_vertex, CategoryTheory.instFinallySmallOppositeOfInitiallySmall, CategoryTheory.yonedaMap_app_apply, CategoryTheory.Limits.BinaryCofan.op_mk, CategoryTheory.Sheaf.ΓHomEquiv_naturality_right_symm, AlgebraicGeometry.SheafedSpace.comp_hom_c_app', CategoryTheory.Functor.IsCoverDense.Types.naturality_apply, SSet.stdSimplex.obj₀Equiv_symm_mem_face_iff, SimplicialObject.opFunctorCompOpFunctorIso_hom_app_app, CategoryTheory.Equivalence.sheafCongr.functor_obj_val_map, SSet.range_eq_iSup_sigma_ι, SSet.Subcomplex.BicartSq.isPushout, CategoryTheory.forgetEnrichmentOppositeEquivalence_inverse, AlgebraicGeometry.ΓSpecIso_hom_stalkClosedPointIso_inv, germ_skyscraperPresheafStalkOfSpecializes_hom_assoc, CategoryTheory.NatTrans.rightOpWhiskerRight_assoc, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_apply, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctor_obj_π_app, SSet.Subcomplex.PairingCore.nonDegenerate₂, CategoryTheory.Sheaf.coneΓ_π_app, CategoryTheory.SimplicialObject.Truncated.trunc_map_app, 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, CategoryTheory.PresheafOfGroups.OneCochain.ev_precomp, CategoryTheory.Limits.FormalCoproduct.instPreservesLimitOppositeDiscreteFunctorCompOpObjFunctorEvalOp, TopCat.Presheaf.whiskerIsoMapGenerateCocone_hom_hom, AlgebraicGeometry.LocallyRingedSpace.Γ_map, CategoryTheory.GrothendieckTopology.Cover.multicospanComp_hom_app, CategoryTheory.Equivalence.sheafCongrPrecoherent_unitIso_inv_app_val_app, AlgebraicGeometry.Flat.flat_of_affine_subset, AlgebraicGeometry.isCompactOpen_iff_eq_basicOpen_union, TopCat.Presheaf.presheafEquivOfIso_functor_obj_obj, AlgebraicGeometry.PresheafedSpace.componentwiseDiagram_obj, AlgebraicGeometry.AffineSpace.functor_obj_obj, CategoryTheory.SimplicialObject.whiskering_obj_obj_obj, CategoryTheory.Functor.IsRepresentedBy.uliftYonedaIso_hom, CategoryTheory.Limits.end_.map_comp, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.s_comp_σ_assoc, CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinsetLimitCone_cone_pt, CategoryTheory.Sheaf.ΓRes_map, AlgebraicTopology.DoldKan.HigherFacesVanish.comp_P_eq_self, AlgebraicGeometry.Spec.algebraMap_stalkIso_inv, CategoryTheory.Functor.PreOneHypercoverDenseData.multicospanIndex_right, CategoryTheory.Limits.ProductsFromFiniteCofiltered.liftToFinset_map_app, 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, LightCondSet.hom_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, 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, 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, AlgebraicGeometry.IsOpenImmersion.ΓIso_inv, AlgebraicGeometry.Scheme.Modules.Hom.sub_app, CategoryTheory.RelCat.opEquivalence_counitIso, CategoryTheory.ShortComplex.rightHomologyMap_op, SSet.Truncated.Edge.map_edge, CategoryTheory.instFullGrpFunctorOppositeGrpCatYonedaGrp, AlgebraicGeometry.LocallyRingedSpace.restrict_presheaf_map, AlgebraicGeometry.AffineScheme.Spec_essSurj, CategoryTheory.coyonedaEquiv_coyoneda_map, 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, CategoryTheory.Sheaf.cartesianMonoidalCategoryWhiskerRight_val, AlgebraicGeometry.Proj.add_apply, CategoryTheory.sheafifyMap_comp, AlgebraicGeometry.PresheafedSpace.Γ_obj, CategoryTheory.Limits.coneOfSectionCompYoneda_π, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctor_map_hom, CategoryTheory.Functor.instFaithfulOppositeTypeRestrictedULiftYonedaOfIsDense, CategoryTheory.op_add, CategoryTheory.Comma.opFunctorCompFst_inv_app, CategoryTheory.Limits.isIndObject_iff_preservesFiniteLimits, CategoryTheory.Functor.IsRightAdjoint.leftOp, CategoryTheory.Limits.coend.ι_map, CategoryTheory.isCodetector_op_iff, CategoryTheory.CostructuredArrow.toStructuredArrow_obj, SSet.skeletonOfMono_zero, AlgebraicGeometry.Scheme.Hom.ι_toNormalization_assoc, CategoryTheory.Pseudofunctor.toDescentData_obj, 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.SimplicialObject.δ_naturality_assoc, CategoryTheory.simplicialCosimplicialEquiv_inverse_obj, CategoryTheory.Limits.pullbackIsoOpPushout_hom_inl, CategoryTheory.ShortComplex.Homotopy.unop_h₁, CategoryTheory.Comma.unopFunctor_map, CategoryTheory.Limits.walkingCospanOpEquiv_functor_obj, TopCat.Presheaf.germ_stalkPullbackInv, CategoryTheory.Limits.IndObjectPresentation.ofCocone_isColimit, CategoryTheory.preservesHomology_preadditiveYonedaObj_of_injective, 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, CategoryTheory.Functor.leftOpRightOpEquiv_counitIso_hom_app_app, 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.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.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, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIso_hom_app_hom, CategoryTheory.OverPresheafAux.OverArrows.app_val, AlgebraicGeometry.Scheme.stalkMap_germ_apply, AlgebraicGeometry.Scheme.IdealSheafData.range_glueDataObjι, CategoryTheory.Presieve.FamilyOfElements.isAmalgamation_singleton_iff, CategoryTheory.GrothendieckTopology.pseudofunctorOver_toPrelaxFunctor_toPrelaxFunctorStruct_toPrefunctor_map_toFunctor_map_val_app, 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.ShortComplex.opEquiv_inverse, CategoryTheory.Subfunctor.range_eq_ofSection, CategoryTheory.orderDualEquivalence_counitIso, CategoryTheory.linearYoneda_obj_additive, 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, CommAlgCat.one_op_of_unop_hom, AlgebraicGeometry.Scheme.Hom.germ_stalkMap, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.fromBiprod_biprodIsoProd_inv_apply, 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, AlgebraicGeometry.Scheme.toOpen_eq, CategoryTheory.NatTrans.Coequifibered.op, CategoryTheory.Limits.has_filtered_colimits_op_of_has_cofiltered_limits, PresheafOfModules.ι_fromFreeYonedaCoproduct_apply, CategoryTheory.Limits.hasColimit_of_hasLimit_unop, SSet.Augmented.stdSimplex_obj_hom_app, AlgebraicGeometry.Spec.toSheafedSpace_map, CategoryTheory.ShortComplex.RightHomologyData.unop_K, 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.GrothendieckTopology.MayerVietorisSquare.shortComplex_g, CategoryTheory.uliftYoneda_map_app_down, CategoryTheory.orderDualEquivalence_functor_obj, CategoryTheory.Comma.opEquiv_functor, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctorObj_pt, LightProfinite.instPreservesLimitsOfShapeOppositeNatForgetContinuousMapCarrierToTopAndTotallyDisconnectedSpaceSecondCountableTopology, CategoryTheory.Functor.opId_inv_app, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom_assoc, CategoryTheory.isoOpEquiv_symm_apply, SimplicialObject.Splitting.decomposition_id, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.isPushoutAddCommGrpFreeSheaf, SSet.Truncated.Path.arrow_src, AlgebraicGeometry.PresheafedSpace.GlueData.snd_invApp_t_app, SSet.spine_map_vertex, CategoryTheory.Limits.coneLeftOpOfCocone_pt, CategoryTheory.Limits.FormalCoproduct.evalOpCompInlIsoId_hom_app_app, PresheafOfModules.free_obj, CategoryTheory.Functor.IsCoverDense.presheafIso_inv, CategoryTheory.Limits.widePullbackShapeUnop_obj, CategoryTheory.Functor.RepresentableBy.ext_iff, CategoryTheory.GrothendieckTopology.yoneda_obj_val, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.inv_naturality, CategoryTheory.Functor.const.opObjUnop_inv_app, CategoryTheory.Limits.walkingParallelPairOp_zero, CategoryTheory.Arrow.AugmentedCechNerve.ExtraDegeneracy.s_comp_π_0, CategoryTheory.flat_iff_lan_flat, CategoryTheory.IndParallelPairPresentation.hg, CategoryTheory.ShortComplex.LeftHomologyData.unop_H, CategoryTheory.Functor.IsCoverDense.Types.sheafIso_inv_val, CategoryTheory.Equivalence.rightOp_counitIso_hom_app, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_obj, AlgebraicGeometry.Scheme.Modules.restrictAdjunction_unit_app_app, CategoryTheory.Sheaf.instPreservesFiniteLimitsFunctorOppositeSheafToPresheafOfHasFiniteLimits, CategoryTheory.yonedaPairingExt_iff, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_homEquiv_apply, CategoryTheory.op_sum, CategoryTheory.OverPresheafAux.app_unitForward, AlgebraicGeometry.StructureSheaf.comapₗ_const, CategoryTheory.isCoseparator_iff_faithful_yoneda_obj, AlgebraicTopology.DoldKan.MorphComponents.postComp_b, CategoryTheory.instPresheafIsFiniteObjFunctorOppositeTypeYoneda, CategoryTheory.Limits.BinaryCofan.unop_mk, TopCat.Presheaf.restrictOpenCommRingCat_apply, CategoryTheory.unop_rightUnitor, CategoryTheory.Comma.opEquiv_unitIso, SSet.hasDimensionLT_iSup_iff, AlgebraicGeometry.Scheme.Hom.isIntegral_app, CategoryTheory.ObjectProperty.instIsClosedUnderLimitsOfShapeOppositeOpOfIsClosedUnderColimitsOfShape, CategoryTheory.GrothendieckTopology.yonedaEquiv_naturality, CategoryTheory.ShortComplex.Splitting.op_s, CategoryTheory.sheafificationAdjunction_unit_app, CategoryTheory.typeEquiv_counitIso_hom_app_val_app, CategoryTheory.Limits.Fork.op_ι_app_one, CategoryTheory.CategoryOfElements.toCostructuredArrow_obj, CategoryTheory.OverPresheafAux.yonedaCollectionFunctor_map, CategoryTheory.preadditiveYonedaObj_obj_isAddCommGroup, CategoryTheory.Functor.IsDenseSubsite.mapPreimage_comp_map_assoc, CategoryTheory.ShiftedHom.opEquiv'_symm_add, AlgebraicGeometry.ΓSpecIso_inv_ΓSpec_adjunction_homEquiv, CategoryTheory.Functor.opUnopEquiv_unitIso, SSet.Subcomplex.le_iff_contains_nonDegenerate, CategoryTheory.LocalizerMorphism.op_functor, CategoryTheory.isCodetecting_op_iff, AlgebraicGeometry.RingedSpace.zeroLocus_empty_eq_univ, CategoryTheory.Functor.FullyFaithful.compYonedaCompWhiskeringLeftMaxRight_hom_app_app_down, CategoryTheory.instFaithfulSheafFunctorOppositeSheafToPresheaf, 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, 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.instFaithfulSheafFunctorOppositeCompSheafComposeSheafToPresheaf, CategoryTheory.NatIso.op_refl, CategoryTheory.Comon.monoidal_tensorUnit_comon_comul, CategoryTheory.Limits.PullbackCone.op_inl, CategoryTheory.NatTrans.rightOp_app, CategoryTheory.Presheaf.instPreservesFiniteProductsOppositeVal, CategoryTheory.Enriched.Functor.natTransEquiv_symm_whiskerRight_functorHom_app, CategoryTheory.Adjunction.compCoyonedaIso_hom_app_app, SSet.horn_obj_zero, CategoryTheory.Iso.op_refl, InfiniteGalois.toAlgEquivAux_eq_liftNormal, CategoryTheory.Functor.closedIhom_map_app, AlgebraicTopology.DoldKan.hσ'_eq, 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, 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, AlgebraicGeometry.Scheme.Hom.map_appLE', AlgebraicGeometry.LocallyRingedSpace.GlueData.ι_isoSheafedSpace_inv_assoc, CategoryTheory.Limits.pullbackIsoUnopPushout_hom_inl_assoc, 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.ShiftedHom.opEquiv'_symm_op_opShiftFunctorEquivalence_counitIso_inv_app_op_shift, 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, AugmentedSimplexCategory.equivAugmentedSimplicialObjectFunctorCompToArrowIso_inv_app_right, CategoryTheory.Functor.map_opShiftFunctorEquivalence_unitIso_hom_app_unop_assoc, CategoryTheory.GrothendieckTopology.uliftYonedaEquiv_apply, CategoryTheory.instIsMonoidalOppositeOpOpEquivalence, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.app_invApp_assoc, CategoryTheory.Functor.IsCoverDense.sheafIso_inv_val, CategoryTheory.Square.IsPullback.op, localCohomology.ideal_powers_initial, CategoryTheory.Functor.IsRepresentedBy.map_bijective, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.c_iso', 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.Functor.PullbackObjObj.ofHasPullback_snd, 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, AlgebraicGeometry.RingedSpace.basicOpen_res_eq, LightCondensed.ihomPoints_symm_apply, CategoryTheory.sheafificationIso_hom_val, AlgebraicGeometry.Spec_zeroLocus, AlgebraicGeometry.LocallyOfFinitePresentation.finitePresentation_appLE, CategoryTheory.Pseudofunctor.IsStackFor.isEquivalence, CategoryTheory.Functor.CorepresentableBy.equivUliftCoyonedaIso_apply, CategoryTheory.nerve_obj, AlgebraicGeometry.Scheme.basicOpen_mul, LightCondensed.discrete_obj, CategoryTheory.Limits.ι_comp_colimitOpIsoOpLimit_hom, CategoryTheory.MonoidalCategory.MonoidalLeftAction.oppositeLeftAction_actionHom_op, 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, PresheafOfModules.Monoidal.tensorHom_app, SSet.Truncated.HomotopyCategory.BinaryProduct.functor_obj, CategoryTheory.Pretriangulated.shiftFunctor_op_map, CategoryTheory.Limits.limitLeftOpIsoUnopColimit_inv_comp_π, AlgebraicTopology.DoldKan.PInfty_f_naturality, AlgebraicGeometry.exists_of_res_eq_of_qcqs_of_top, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_map, CategoryTheory.ShortComplex.RightHomologyData.op_f', CategoryTheory.RelCat.opEquivalence_inverse, CategoryTheory.Limits.hasTerminal_op_of_hasInitial, CategoryTheory.Limits.preservesLimit_of_leftOp, AlgebraicTopology.DoldKan.Γ₀.Obj.map_on_summand, CategoryTheory.Sieve.uliftFunctorInclusion_is_mono, SSet.horn.spineId_arrow_coe, 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, CategoryTheory.uliftCoyonedaEquiv_naturality, CategoryTheory.Presheaf.isSheaf_iff_isSheaf_comp, AlgebraicTopology.DoldKan.HigherFacesVanish.of_comp, CategoryTheory.Pseudofunctor.CoGrothendieck.instEssSurjαCategoryObjLocallyDiscreteOppositeCatMkOpFiberForgetInducedFunctor, 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.OverPresheafAux.YonedaCollection.map₁_comp, CategoryTheory.Limits.isLimitConeLeftOpOfCocone_lift, CategoryTheory.Sheaf.isLocallyInjective_sheafToPresheaf_map_iff, PresheafOfModules.pullbackObjIsDefined_free_yoneda, RingCat.moduleCatRestrictScalarsPseudofunctor_obj, CategoryTheory.HasCoseparator.hasSeparator_op, CategoryTheory.cokernelUnopUnop_hom, CategoryTheory.Sheaf.id_val, 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, 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.const_comp, 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, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom, AlgebraicGeometry.preservesLimit_rightOp_Γ, CategoryTheory.GrothendieckTopology.pseudofunctorOver_toPrelaxFunctor_toPrelaxFunctorStruct_toPrefunctor_map_toFunctor_obj_val_map, CategoryTheory.GrothendieckTopology.overMapPullbackComp_inv_app_val_app, 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.GrothendieckTopology.diagramFunctor_map, CategoryTheory.isSeparator_op_iff, SSet.Edge.map_edge, CategoryTheory.equivYoneda_inv_app, CategoryTheory.uliftCoyonedaEquiv_symm_map_assoc, AlgebraicGeometry.StructureSheaf.instIsLocalizedModuleCarrierStalkAbPresheafOpensCarrierTopModuleStructurePresheafPrimeComplAsIdealToStalkₗ, CategoryTheory.Pretriangulated.TriangleOpEquivalence.inverse_obj, SSet.Edge.CompStruct.map_simplex, CategoryTheory.Pretriangulated.commShiftIso_opOp_hom_app, CategoryTheory.presheafToSheafCompComposeAndSheafifyIso_inv_app, CommMonCat.coyoneda_map_app, AugmentedSimplexCategory.equivAugmentedSimplicialObject_counitIso_inv_app_right, CategoryTheory.MonoidalCategory.MonoidalRightAction.rightActionOfOppositeRightAction_actionAssocIso, AlgebraicGeometry.basicOpen_eq_of_affine', 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.yoneda'_obj_val, CategoryTheory.Limits.IndObjectPresentation.extend_isColimit_desc_app, CategoryTheory.Limits.IndObjectPresentation.ofCocone_ι, CategoryTheory.Limits.opProdIsoCoprod_inv_inr, AlgebraicGeometry.Spec.coe_toTop_map_hom_apply_asIdeal, AlgebraicGeometry.Scheme.Opens.toScheme_presheaf_map, CategoryTheory.ShortComplex.leftHomologyFunctorOpNatIso_hom_app, 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.Functor.FullyFaithful.compUliftCoyonedaCompWhiskeringLeft_inv_app_app_down, LightCondensed.lanPresheafNatIso_hom_app, CategoryTheory.Limits.widePullbackShapeOpEquiv_functor, SSet.mem_nonDegenerate_iff_notMem_degenerate, CategoryTheory.Limits.pushoutIsoOpPullback_inl_hom_assoc, CategoryTheory.Subfunctor.mem_ofSection_obj, smoothSheafCommRing.ι_forgetStalk_hom_apply, CategoryTheory.Functor.IsDenseSubsite.mapPreimage_comp, TopCat.Sheaf.objSupIsoProdEqLocus_hom_snd, SSet.finite_subcomplex_top_iff, AlgebraicGeometry.stalkToFiberRingHom_germ, SSet.Truncated.StrictSegal.spine_δ_arrow_eq, AlgebraicGeometry.StructureSheaf.const_mul_cancel', AlgebraicGeometry.Scheme.isoSpec_Spec_inv, CategoryTheory.Functor.IsCoverDense.homOver_app, 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.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, CategoryTheory.Limits.isColimitCoconeOfConeRightOp_desc, CategoryTheory.Limits.desc_op_comp_opCoproductIsoProduct_hom, CategoryTheory.Equivalence.sheafCongrPrecoherent_inverse_obj_val_map, CategoryTheory.Adjunction.Quadruple.op_adj₁, lightCondSetToTopCat_obj_carrier, SSet.Subcomplex.image_le_range, CategoryTheory.ShortComplex.Exact.op, CategoryTheory.Equivalence.sheafCongrPrecoherent_unitIso_hom_app_val_app, AlgebraicGeometry.instSubsingletonCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensOfIsEmpty, AlgebraicGeometry.exists_eq_pow_mul_of_isAffineOpen, CategoryTheory.GrothendieckTopology.instIsLocalizationFunctorOppositeSheafPresheafToSheafW, CategoryTheory.SimplicialObject.Truncated.whiskering_obj_obj_map, CategoryTheory.Pseudofunctor.toDescentData_map_hom, CategoryTheory.NatIso.removeOp_inv, smoothSheafCommRing.ι_forgetStalk_hom_assoc, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo_assoc, CategoryTheory.MorphismProperty.ContainsIdentities.op, AlgebraicGeometry.Proj.awayMap_awayToSection, CategoryTheory.Groupoid.invEquivalence_counitIso, SheafOfModules.Presentation.mapRelations_mapGenerators_assoc, CategoryTheory.HasLiftingProperty.iff_unop, AlgebraicGeometry.instPreservesColimitsOfShapeOppositeCommRingCatSchemeDiscretePEmptySpec, CategoryTheory.GrothendieckTopology.uliftYonedaOpCompCoyoneda_app_app, AlgebraicGeometry.finite_appTop_of_universallyClosed, CategoryTheory.Square.opFunctor_map_τ₃, CategoryTheory.Presheaf.map_comp_uliftYonedaEquiv_down_assoc, CategoryTheory.CategoryOfElements.costructuredArrowULiftYonedaEquivalence_inverse_obj, CategoryTheory.isCoseparating_unop_iff, AlgebraicGeometry.IsLocallyNoetherian.component_noetherian, CategoryTheory.GrothendieckTopology.Point.instPreservesFiniteLimitsFunctorOppositePresheafFiberOfLocallySmallOfHasFiniteLimitsOfAB5OfSize, CategoryTheory.Join.opEquiv_inverse_map_edge_op, CategoryTheory.Coyoneda.instHasColimitObjOppositeFunctorTypeCoyoneda, CategoryTheory.Presheaf.instIsCardinalLocallyPresentableFunctorOppositeOfHasPullbacks, 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.isCoseparator_iff_faithful_preadditiveYonedaObj, CategoryTheory.yonedaGrpObj_obj_coe, CategoryTheory.LocalizerMorphism.RightResolution.unop_w, CategoryTheory.GrothendieckTopology.diagramFunctor_obj, AlgebraicTopology.DoldKan.N₂_obj_p_f, CategoryTheory.NatTrans.leftOpWhiskerRight_assoc, 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.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.GrothendieckTopology.HasSheafCompose.isSheaf, TopCat.Presheaf.SheafConditionEqualizerProducts.fork_pt, AlgebraicGeometry.PresheafedSpace.pushforwardDiagramToColimit_map, SSet.stdSimplex.ofSimplex_yonedaEquiv_δ, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.invApp_app_assoc, CategoryTheory.Sieve.forallYonedaIsSheaf_iff_colimit, CategoryTheory.Functor.PullbackObjObj.mapArrowRight_id, AlgebraicTopology.DoldKan.σ_comp_PInfty, CategoryTheory.Limits.preservesFiniteCoproducts_op, TopCat.Presheaf.SheafConditionEqualizerProducts.w_apply, 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, TopCat.Presheaf.pushforwardPullbackAdjunction_unit_pullback_map_germToPullbackStalk, AlgebraicGeometry.Scheme.IdealSheafData.zeroLocus_inter_subset_supportSet, CategoryTheory.Presheaf.isLocallySurjective_toPlus, CategoryTheory.Functor.PreOneHypercoverDenseData.multicospanIndex_fst, CategoryTheory.Limits.walkingParallelPairOpEquiv_unitIso_one, AlgebraicGeometry.PresheafedSpace.Γ_map, 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.Square.op_f₁₃, CategoryTheory.Iso.unop_hom, AlgebraicGeometry.Scheme.IdealSheafData.map_ideal', CompHausLike.LocallyConstant.instIsIsoFunctorTypeUnitSheafCoherentTopologyAdjunction, CategoryTheory.NatTrans.coequifibered_unop_iff, CategoryTheory.Functor.IsDenseSubsite.mapPreimage_of_eq, AlgebraicGeometry.StructureSheaf.toOpenₗ_top_bijective, 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.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.closedSieves_map_coe, 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, CategoryTheory.Limits.PullbackCone.unop_pt, CategoryTheory.Subpresheaf.sheafify_isSheaf, CategoryTheory.Arrow.AugmentedCechNerve.ExtraDegeneracy.s_comp_π_succ, CategoryTheory.simplicialToCosimplicialAugmented_obj, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom_apply, 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, AlgebraicGeometry.Scheme.IdealSheafData.ideal_sup, CategoryTheory.cokernelOpUnop_hom, CategoryTheory.Equivalence.symmEquiv_inverse, CategoryTheory.Equivalence.sheafCongrPreregular_inverse_obj_val_obj, AlgebraicGeometry.Scheme.evaluation_naturality_assoc, AlgebraicGeometry.AffineSpace.isoOfIsAffine_hom, CategoryTheory.Equivalence.rightOp_unitIso_inv_app, CategoryTheory.Coyoneda.coyoneda_full, AlgebraicGeometry.tilde.toOpen_res, Alexandrov.projSup_obj, AlgebraicGeometry.tilde.toOpen_res_assoc, CategoryTheory.Limits.hasZeroObject_op, CompHausLike.isIsoSigmaComparison, AlgebraicGeometry.Scheme.Hom.ι_fromNormalization_assoc, CategoryTheory.Abelian.IsGrothendieckAbelian.OppositeModuleEmbedding.faithful_embedding, CategoryTheory.Limits.IndObjectPresentation.ofCocone_F, CategoryTheory.Functor.FullyFaithful.homNatIso'_inv_app_down, HomologicalComplex.acyclic_op_iff, SSet.stdSimplex.objMk_apply, CategoryTheory.NatIso.op_hom, 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, CompHausLike.LocallyConstant.counit_app_val, CategoryTheory.Limits.widePushoutShapeUnop_map, 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, 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, Condensed.instPreservesFiniteProductsOppositeStoneanVal, 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, SheafOfModules.pushforwardNatTrans_app_val_app, CategoryTheory.Limits.IndObjectPresentation.ofCocone_ℐ, CategoryTheory.Presheaf.instIsLocallyPresentableFunctorOppositeOfHasPullbacks, Profinite.Extend.cocone_ι_app, CategoryTheory.GrothendieckTopology.W_iff, AlgebraicGeometry.HasRingHomProperty.iff_exists_appLE_locally, CategoryTheory.Limits.preservesColimit_op, CategoryTheory.StructuredArrow.functor_obj, CategoryTheory.Presheaf.isLocallySurjective_whisker_iff, 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, AlgebraicGeometry.HasRingHomProperty.iff_exists_appLE, CategoryTheory.Limits.PushoutCocone.op_pt, CategoryTheory.typeEquiv_functor_map_val_app, CategoryTheory.GrothendieckTopology.pseudofunctorOver_toPrelaxFunctor_toPrelaxFunctorStruct_toPrefunctor_obj_α, CategoryTheory.SimplicialObject.δ_comp_δ_self', 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, CategoryTheory.SimplicialObject.Augmented.rightOp_left, CategoryTheory.Limits.hasPullbacks_opposite, 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, TopCat.Presheaf.germ_res'_assoc, AlgebraicGeometry.Scheme.Opens.ι_app_self, CategoryTheory.Square.op_X₄, 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, TopCat.toSheafCompHausLike_val_map, CategoryTheory.Functor.map_shift_unop_assoc, AlgebraicGeometry.Scheme.zeroLocus_span, CategoryTheory.Functor.IsCoverDense.Types.presheafHom_app, HomotopicalAlgebra.instIsStableUnderRetractsOppositeFibrationsOfCofibrations, 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, 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.Pretriangulated.opShiftFunctorEquivalence_unitIso_hom_app_assoc, CategoryTheory.RetractArrow.op_i_left, CategoryTheory.Functor.instIsEquivalenceOppositeLeftOp, CategoryTheory.ShortComplex.op_f, AlgebraicGeometry.exists_appTop_π_eq_of_isLimit, SSet.mem_degenerate_iff, SSet.nonDegenerateEquivOfIso_apply_coe, SSet.S.le_iff_nonempty_hom, CategoryTheory.Equivalence.sheafCongr.inverse_obj_val_map, AlgebraicGeometry.Proj.stalkIso'_germ, TopCat.Presheaf.isGluing_iff_pairwise, Alexandrov.lowerCone_pt, SSet.Augmented.stdSimplex_map_left, 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, CategoryTheory.sheafification_obj, CategoryTheory.Limits.walkingParallelPairOpEquiv_unitIso_zero, AlgebraicGeometry.Scheme.restrict_presheaf_map, CategoryTheory.Comon.MonOpOpToComonObj_X, CategoryTheory.Sheaf.ΓHomEquiv_naturality_right, HomologicalComplex.quasiIso_opFunctor_map_iff, CategoryTheory.ShortComplex.unopMap_id, CategoryTheory.NatIso.op_inv, PresheafOfModules.pushforward_assoc, CategoryTheory.MonoidalCategory.DayConvolutionUnit.leftUnitorCorepresentingIso_hom_app_app, CategoryTheory.map_yonedaEquiv, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.ofRestrict_invApp, AlgebraicGeometry.formallyUnramified_iff, PresheafOfModules.map_id, AlgebraicGeometry.Spec.locallyRingedSpaceObj_presheaf_map', CategoryTheory.op_zsmul, CommGrpCat.coyonedaForget_inv_app_app, CategoryTheory.Sieve.natTransOfLe_app_coe, CategoryTheory.GrothendieckTopology.map_uliftYonedaEquiv, AlgebraicGeometry.Scheme.mem_basicOpen_top, CategoryTheory.GrothendieckTopology.uliftYonedaIsoYoneda_inv_app_val_app_down, 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, CategoryTheory.GrothendieckTopology.preservesLimits_diagramFunctor, AlgebraicGeometry.Scheme.Opens.ι_image_basicOpen', CategoryTheory.MonoidalCategory.DayConvolution.associatorCorepresentingIso_hom_app_app, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_naturality_assoc, CategoryTheory.op_epi_of_mono, AlgebraicTopology.DoldKan.N₂_obj_X_d, CategoryTheory.Limits.FormalCoproduct.evalOp_obj_map, 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, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_toSpecΓ_assoc, CategoryTheory.Join.InclRightCompRightOpOpEquivFunctor_inv_app, CategoryTheory.Subfunctor.toRangeSheafify_app_coe, CategoryTheory.RanIsSheafOfIsCocontinuous.fac, CategoryTheory.Limits.preservesColimitsOfSize_rightOp, AlgebraicGeometry.AffineSpace.homOverEquiv_apply, CategoryTheory.SimplicialObject.Augmented.toArrow_obj_hom, SimpleGraph.componentComplFunctor_finite, CategoryTheory.Limits.preservesLimitsOfShape_of_leftOp, CategoryTheory.ObjectProperty.op_isoClosure, CategoryTheory.Pretriangulated.Opposite.contractibleTriangleIso_inv_hom₂, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_presheafFiberDesc_assoc, CategoryTheory.Functor.IsCoverDense.isoOver_inv_app, CategoryTheory.Subfunctor.ofSection_le_iff, AlgebraicGeometry.Scheme.basicOpen_res, CommMonCat.coyonedaForget_hom_app_app_hom, CategoryTheory.Limits.limitOpIsoOpColimit_inv_comp_π, AlgebraicGeometry.Scheme.Opens.fromSpecStalkOfMem_toSpecΓ_assoc, TopCat.Sheaf.objSupIsoProdEqLocus_inv_fst, smoothSheaf.ι_evalHom, CategoryTheory.OverPresheafAux.YonedaCollection.map₁_id, CategoryTheory.linearYoneda_obj_obj_isModule, CategoryTheory.Limits.isColimitOfConeOfCoconeRightOp_desc, AlgebraicGeometry.Smooth.exists_isStandardSmooth, AlgebraicGeometry.SheafedSpace.Γ_def, SSet.associator_hom_app_apply, CategoryTheory.instIsIsoFunctorOppositeSheafToPresheafToSheafCompComposeAndSheafify, AlgebraicGeometry.Scheme.Modules.pseudofunctor_mapComp_inv_τl, CategoryTheory.Limits.preservesColimits_op, CategoryTheory.MorphismProperty.leftFractionRel_op_iff, AlgebraicGeometry.Scheme.comp_appTop, 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, TopCat.Presheaf.Pushforward.comp_hom_app, CategoryTheory.Functor.mapCoconeOp_hom_hom, 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, 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, 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, AlgebraicGeometry.IsAffineOpen.isLocalization_stalk, AlgebraicTopology.DoldKan.karoubi_PInfty_f, CategoryTheory.ParametrizedAdjunction.unit_whiskerRight_map, AlgebraicGeometry.Smooth.smooth_appLE, AlgebraicGeometry.StructureSheaf.res_const, AlgebraicGeometry.PresheafedSpace.GlueData.ι_jointly_surjective, CategoryTheory.Limits.coneOfSectionCompCoyoneda_π, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_comp, HomologicalComplex.evalCompCoyonedaCorepresentableBySingle_homEquiv_symm_apply, CategoryTheory.Functor.instFinalOppositeLeftOpOfInitial, CategoryTheory.Functor.op_comp_isSheaf, 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.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, SSet.boundary_eq_iSup, CategoryTheory.NatIso.removeOp_hom, CategoryTheory.Subpresheaf.isSheaf_iff, CategoryTheory.MonoidalCategory.MonoidalLeftAction.leftActionOfOppositeLeftAction_actionHom, CategoryTheory.GrothendieckTopology.Cover.index_right, Opens.mayerVietorisSquare'_toSquare, AlgebraicGeometry.Scheme.Spec.algebraMap_residueFieldIso_inv_assoc, CategoryTheory.Functor.WellOrderInductionData.map_lift, CategoryTheory.Pseudofunctor.DescentData.pullFunctorIdIso_inv_app_hom, CategoryTheory.HasLiftingProperty.transfiniteComposition.sqFunctor_obj, CategoryTheory.Limits.inr_opProdIsoCoprod_inv, smoothSheafCommRing.ι_forgetStalk_hom, CategoryTheory.Sieve.functorInclusion_is_mono, CategoryTheory.ShortComplex.HomologyData.unop_iso, CategoryTheory.GrothendieckTopology.overMapPullbackComp_hom_app_val_app, SimplicialObject.Splitting.comp_PInfty_eq_zero_iff, CategoryTheory.ShortComplex.SnakeInput.op_L₁, 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.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, CategoryTheory.Limits.coconeEquivalenceOpConeOp_inverse_obj, AlgebraicGeometry.PresheafedSpace.congr_app, 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, TopCat.Presheaf.presheafEquivOfIso_counitIso_hom_app_app, CategoryTheory.Pseudofunctor.LocallyDiscreteOpToCat.map_eq_pullHom_assoc, CategoryTheory.Idempotents.DoldKan.Γ_map_app, CategoryTheory.PresheafOfGroups.OneCocycle.ev_refl, AlgebraicGeometry.IsAffineOpen.SpecMap_appLE_fromSpec_assoc, CategoryTheory.Limits.Cocone.op_π, CategoryTheory.ShortComplex.op_X₃, CategoryTheory.Limits.preservesFiniteLimits_of_unop, CategoryTheory.Functor.rightOp_additive, HomologicalComplex.evalCompCoyonedaCorepresentableBySingle_homEquiv_apply, CategoryTheory.Limits.coend.ι_map_assoc, CategoryTheory.Limits.coconeOfConeUnop_ι, 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, HomologicalComplex.unopInverse_obj, SSet.degenerate_eq_iUnion_range_σ, AlgebraicGeometry.IsAffineOpen.fromSpec_top, CategoryTheory.WithInitial.opEquiv_functor_obj, CategoryTheory.ObjectProperty.isCodetecting_op_iff, SheafOfModules.relationsOfIsCokernelFree_s, CategoryTheory.Discrete.opposite_inverse_obj, CategoryTheory.Injective.instEnoughInjectivesOppositeOfEnoughProjectives, CategoryTheory.ShiftedHom.opEquiv_symm_apply, HomologicalComplex.opEquivalence_inverse, CategoryTheory.prodOpEquiv_counitIso_hom_app, CategoryTheory.Limits.coconeEquivalenceOpConeOp_counitIso, CategoryTheory.Limits.Fork.op_ι_app_zero, 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.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, CategoryTheory.isoOpEquiv_apply, CategoryTheory.Functor.leftOp_map, CategoryTheory.simplicialCosimplicialAugmentedEquiv_inverse, AlgebraicGeometry.IsIntegralHom.hasAffineProperty, CategoryTheory.Equivalence.leftOp_counitIso_hom_app, CategoryTheory.constantSheafAdj_counit_w, CategoryTheory.Limits.preservesColimits_of_leftOp, 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, 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, CompHausLike.LocallyConstant.unit_app, AlgebraicGeometry.Scheme.basicOpen_zero, InfiniteGalois.proj_of_le, CategoryTheory.PresheafIsGeneratedBy.of_epi, HomologicalComplex.cyclesOpNatIso_hom_app, 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.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, AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage, CategoryTheory.IsPushout.op, CategoryTheory.Subfunctor.IsGeneratedBy.image, SSet.Truncated.Path.mk₂_vertex, skyscraperPresheafCoconeOfSpecializes_ι_app, CategoryTheory.Sheaf.comp_val_assoc, HomotopicalAlgebra.cofibration_unop_iff, 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.Point.toPresheafFiber_map_injective, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_hom_eq_germ_assoc, CategoryTheory.Iso.unop_inv_hom_id_app, CategoryTheory.GrothendieckTopology.pseudofunctorOver_mapComp_hom_toNatTrans_app_val_app, AlgebraicGeometry.isFinite_iff, TopCat.Presheaf.presheafEquivOfIso_inverse_obj_obj, CategoryTheory.yoneda'_comp, SSet.prodStdSimplex.nonDegenerate_iff_strictMono_objEquiv, SheafOfModules.forgetToSheafModuleCat_map_val, CategoryTheory.Coyoneda.fullyFaithful_preimage, SSet.yonedaEquiv_comp, LightCondSet.topCatAdjunctionUnit_val_app, AlgebraicGeometry.etale_iff, SSet.Truncated.IsStrictSegal.spine_bijective, SSet.op_map, CategoryTheory.Limits.isColimitCoconeOfConeLeftOp_desc, CommGrpCat.coyonedaType_obj_map, CategoryTheory.Functor.mapConeOp_hom_hom, CategoryTheory.HasLiftingProperty.transfiniteComposition.sqFunctor_map, CategoryTheory.Subpresheaf.le_sheafify, CategoryTheory.GrothendieckTopology.yonedaULiftEquiv_symm_app_apply, HomologicalComplex.fromOpcycles_op_cyclesOpIso_inv, CategoryTheory.MorphismProperty.IsCardinalForSmallObjectArgument.preservesColimit, SSet.stdSimplex.spineId_arrow_apply_one, LightCondensed.instFinalNatCostructuredArrowOppositeFintypeCatLightProfiniteOpToLightProfiniteOpPtAsLimitConeFunctorOp, AlgebraicGeometry.Scheme.germ_residue_assoc, CategoryTheory.unop_hom_braiding, AlgebraicTopology.DoldKan.HigherFacesVanish.comp_Hσ_eq, CategoryTheory.simplicialCosimplicialAugmentedEquiv_functor, CategoryTheory.RetractArrow.unop_i_left, AlgebraicGeometry.IsAffineOpen.Spec_map_appLE_fromSpec_assoc, PresheafOfModules.instReflectsIsomorphismsSheafOfModulesFunctorOppositeAddCommGrpCatCompSheafToSheafSheafToPresheaf, CategoryTheory.cosimplicialToSimplicialAugmented_map, AlgebraicGeometry.Scheme.kerFunctor_obj, PresheafOfModules.freeAdjunction_unit_app, SSet.OneTruncation₂.nerveHomEquiv_apply, TopCat.Presheaf.Γgerm_res_apply, AlgebraicGeometry.Spec.toPresheafedSpace_map, CategoryTheory.Equivalence.unop_inverse, 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, CategoryTheory.Sheaf.sheafifyCocone_ι_app_val_assoc, commBialgCatEquivComonCommAlgCat_inverse_map_unop_hom, CategoryTheory.LocalizerMorphism.LeftResolution.opEquivalence_counitIso, CategoryTheory.unop_zsmul, CategoryTheory.Sheaf.Hom.add_app, AlgebraicGeometry.Scheme.mem_basicOpen'', CategoryTheory.PresheafIsGeneratedBy.range, CategoryTheory.Limits.Cocone.extend_ι, CategoryTheory.Presieve.extension_iff_amalgamation, SSet.StrictSegalCore.map_mkOfSucc_zero_spineToSimplex, SimpleGraph.componentComplFunctor_nonempty_of_infinite, 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.GrothendieckTopology.diagramNatTrans_id, AlgebraicGeometry.Scheme.Spec_obj, AlgebraicGeometry.tilde.toOpen_map_app_assoc, CategoryTheory.Equivalence.sheafCongr.inverse_map_val_app, CategoryTheory.Limits.isLimitOfCoconeUnopOfCone_lift, SSet.horn.primitiveEdge_coe_down, CategoryTheory.CategoryOfElements.fromCostructuredArrow_obj_mk, CategoryTheory.Functor.PullbackObjObj.mapArrowLeft_right, 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, CategoryTheory.Functor.PreOneHypercoverDenseData.multicospanIndex_snd, CategoryTheory.PreGaloisCategory.PointedGaloisObject.instHasColimitOppositeFunctorTypeCompOpInclCoyoneda, CategoryTheory.MonoidalCategory.DayConvolutionUnit.rightUnitorCorepresentingIso_inv_app_app, CategoryTheory.Limits.IndObjectPresentation.cocone_pt, 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, AlgebraicGeometry.Scheme.ΓSpecIso_naturality_assoc, SSet.StrictSegal.spineToSimplex_arrow, 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, 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, CategoryTheory.equivYoneda'_hom_val, 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, CategoryTheory.Limits.PushoutCocone.unop_pt, CategoryTheory.GrothendieckTopology.overMapPullbackCongr_hom_app_val_app, CategoryTheory.ObjectProperty.colimitsOfShape_eq_unop_limitsOfShape, LightCondensed.id_val, InfiniteGalois.algEquivToLimit_continuous, TopCat.Presheaf.germ_stalkSpecializes, CategoryTheory.Limits.SequentialProduct.cone_π_app, PresheafOfModules.freeYonedaEquiv_comp, 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, CategoryTheory.Presheaf.restrictedULiftYonedaHomEquiv'_symm_naturality_right_assoc, SSet.Truncated.HomotopyCategory.homToNerveMk_app_zero, AlgebraicGeometry.IsAffineOpen.isOpenImmersion_fromSpec, 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, CompHausLike.LocallyConstant.incl_comap, SSet.Truncated.Edge.map_whiskerRight, CategoryTheory.isDetecting_unop_iff, CategoryTheory.Presieve.FamilyOfElements.compPresheafMap_id, SSet.Truncated.comp_app, CategoryTheory.Functor.sheafPushforwardContinuousId_inv_app_val_app, 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.Pseudofunctor.DescentData.isoMk_hom_hom, AlgebraicGeometry.Scheme.Spec_map, CategoryTheory.RelCat.opFunctor_comp_unopFunctor_eq, CategoryTheory.evalEquiv_symm_apply, CategoryTheory.Functor.map_opShiftFunctorEquivalence_unitIso_inv_app_unop_assoc, SSet.Subcomplex.preimage_iSup, CategoryTheory.NatTrans.Equifibered.op, CategoryTheory.Limits.Cone.op_pt, CategoryTheory.Equivalence.unop_unitIso, CategoryTheory.instFullFunctorOppositeTypeShrinkYoneda, CategoryTheory.cosimplicialSimplicialEquiv_inverse_map, CategoryTheory.Adjunction.representableBy_homEquiv, CategoryTheory.Functor.Elements.initialOfRepresentableBy_snd, CategoryTheory.Equivalence.inverseFunctorObjIso_hom, AlgebraicGeometry.PresheafedSpace.restrict_presheaf, CategoryTheory.Limits.preservesColimits_of_rightOp, CategoryTheory.ParametrizedAdjunction.homEquiv_naturality_one, CategoryTheory.SimplicialObject.δ_comp_σ_of_le_assoc, CategoryTheory.SimplicialObject.δ_comp_δ, HomologicalComplex.instIsStrictlySupportedOppositeOpOp, SSet.Path.map_vertex, CategoryTheory.Limits.preservesLimitsOfSize_of_leftOp, SSet.Truncated.Edge.CompStruct.map_simplex, CategoryTheory.Pseudofunctor.DescentData.pullFunctorObjHom_eq, CategoryTheory.GrothendieckTopology.W_eq_isLocal_range_sheafToPresheaf_obj, AlgebraicGeometry.Scheme.germ_residue, CategoryTheory.MonoidalCategory.MonoidalLeftAction.oppositeLeftAction_actionHomLeft, HomologicalComplex.unopEquivalence_inverse, CategoryTheory.Limits.coconeRightOpOfCone_pt, CategoryTheory.Limits.walkingSpanOpEquiv_unitIso_hom_app, 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, 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, AlgebraicGeometry.Scheme.Hom.ι_fromNormalization, AlgebraicGeometry.LocallyRingedSpace.toΓSpec_preimage_zeroLocus_eq, AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_inv_eq_germ, 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, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_hom, CategoryTheory.Pseudofunctor.DescentData.comp_hom_assoc, AlgebraicGeometry.Scheme.Hom.map_appLE, SSet.PtSimplex.RelStruct.δ_succ_map, CategoryTheory.Functor.PullbackObjObj.ofHasPullback_pt, CategoryTheory.Limits.op_zero, CategoryTheory.Iso.unop_inv_hom_id_app_assoc, 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.π_comp_colimitUnopIsoOpLimit_inv_assoc, AlgebraicGeometry.basicOpen_eq_bot_iff, SSet.σ_naturality_apply, 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.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.whiskering_linearYoneda₂, SSet.Truncated.HomotopyCategory.BinaryProduct.functor_map, CategoryTheory.Pseudofunctor.CoGrothendieck.categoryStruct_comp_fiber, CategoryTheory.op_neg, CategoryTheory.isCoseparator_op_iff, AddCommMonCat.coyonedaForget_hom_app_app_hom, AlgebraicTopology.DoldKan.decomposition_Q, PresheafOfModules.Finite.evaluation_preservesFiniteColimits, CategoryTheory.ShortComplex.RightHomologyData.unop_π, CategoryTheory.ShortComplex.leftHomologyFunctorOpNatIso_inv_app, CategoryTheory.Presheaf.restrictedULiftYonedaHomEquiv'_symm_app_naturality_left_assoc, CategoryTheory.LocalizerMorphism.RightResolution.op_w, CategoryTheory.Functor.op_commShiftIso_hom_app, CategoryTheory.instSmallHomFunctorOppositeTypeColimitCompYoneda, AlgebraicTopology.DoldKan.σ_comp_P_eq_zero, CategoryTheory.Equivalence.op_unitIso, CategoryTheory.Groupoid.invEquivalence_unitIso, AlgebraicGeometry.Spec.fromSpecStalk_eq, CategoryTheory.Limits.coconeEquivalenceOpConeOp_inverse_map_hom, skyscraperPresheafCoconeOfSpecializes_pt, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_over, CategoryTheory.instPreservesColimitFunctorOppositeTypeObjCoyonedaOpYoneda, CategoryTheory.Over.opEquivOpUnder_functor_map, CategoryTheory.op_sub, CategoryTheory.Subfunctor.to_sheafifyLift, SSet.S.mk_map_le, CategoryTheory.ShortComplex.Homotopy.unop_h₀, SSet.S.IsUniquelyCodimOneFace.existsUnique_δ_cast_simplex, AlgebraicGeometry.PresheafedSpace.GlueData.π_ιInvApp_π, TopCat.Presheaf.stalkFunctor_map_germ, CategoryTheory.PresheafHom.isAmalgamation_iff, CategoryTheory.Limits.SequentialProduct.functorMap_commSq_succ, LightCondensed.instHasColimitsOfShapeCostructuredArrowOppositeFintypeCatLightProfiniteOpToLightProfiniteType, AlgebraicGeometry.Scheme.IdealSheafData.ideal_biInf, 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, 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', CondensedSet.toTopCatMap_hom_apply, CategoryTheory.Functor.IsRepresentedBy.of_isoObj, 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, 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.uliftYonedaMap_app_apply, CategoryTheory.Functor.WellOrderInductionData.Extension.map_limit, 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, AlgebraicGeometry.StructureSheaf.instAwayObjOppositeOpensCarrierTopValStructureSheafInTypeOpBasicOpen, 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.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, AlgebraicGeometry.IsAffineOpen.ΓSpecIso_hom_fromSpec_app, lightProfiniteToLightCondSetIsoTopCatToLightCondSet_inv_app_val_app_hom_hom, CategoryTheory.sheafBotEquivalence_counitIso, CategoryTheory.kernelOpOp_hom, CategoryTheory.Limits.coconeUnopOfCone_ι, AlgebraicGeometry.Scheme.exists_le_and_germ_injective, AlgebraicGeometry.Scheme.Modules.pseudofunctor_map_r, CategoryTheory.cones_obj_obj, 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, 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, CategoryTheory.MorphismProperty.instIsStableUnderRetractsOppositeOp, CategoryTheory.ShortComplex.cyclesOpIso_inv_op_iCycles, 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.Equivalence.sheafCongrPrecoherent_functor_map_val_app, CategoryTheory.SimplicialObject.augmentedCechNerve_obj_left_map, AlgebraicGeometry.AffineSpace.comp_homOfVector_assoc, CategoryTheory.Functor.leftOpRightOpEquiv_functor_map_app, CategoryTheory.sheafification_reflective, CategoryTheory.yonedaGrp_map_app, AlgebraicGeometry.Scheme.Hom.appLE_comp_appLE, CategoryTheory.ShortComplex.HomologyData.unop_right, PresheafOfModules.germ_ringCat_smul, AlgebraicTopology.DoldKan.Q_f_naturality, 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.Presheaf.compULiftYonedaIsoULiftYonedaCompLan.coconeApp_naturality, CategoryTheory.Under.opEquivOpOver_unitIso, 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, CategoryTheory.MorphismProperty.presheaf_monomorphisms_le_monomorphisms, CategoryTheory.PreOneHypercover.Hom.mapMultiforkOfIsLimit_ι_assoc, AlgebraicGeometry.isIso_pushoutSection_of_isQuasiSeparated_of_flat_right, CategoryTheory.Limits.limitLeftOpIsoUnopColimit_inv_comp_π_assoc, LightCondensed.underlying_map, PresheafOfModules.naturality_apply, CategoryTheory.ShortComplex.RightHomologyData.unop_H, AlgebraicGeometry.IsAffineOpen.isoSpec_hom, PresheafOfModules.toPresheaf_map_sheafificationHomEquiv_def, AlgebraicGeometry.Scheme.Modules.Hom.app_smul, AlgebraicTopology.DoldKan.Γ₀.Obj.map_epi_on_summand_id, SSet.Truncated.Edge.CompStruct.d₁, 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, AlgebraicGeometry.Scheme.Γ_obj_op, CategoryTheory.Join.opEquiv_functor_map_op_inclLeft, 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, TopCat.Presheaf.IsSheaf.isSheafOpensLeCover, CategoryTheory.WithInitial.opEquiv_inverse_map, CategoryTheory.Equivalence.sheafCongr.functor_obj_val_obj, CategoryTheory.Presieve.FamilyOfElements.map_comp, CategoryTheory.Under.opEquivOpOver_functor_map, CategoryTheory.Functor.IsRepresentedBy.iff_natIso, TopCat.Presheaf.IsSheaf.isSheafUniqueGluing, CategoryTheory.Limits.preservesColimitsOfShape_op, CategoryTheory.Equivalence.sheafCongrPreregular_unitIso_inv_app_val_app, SSet.Subcomplex.PairingCore.notMem₂, PresheafOfModules.instFullRestrictScalarsIdFunctorOppositeRingCat, SSet.Subcomplex.topIso_hom, SSet.OneTruncation₂.id_edge, AlgebraicTopology.inclusionOfMooreComplexMap_f, CategoryTheory.yonedaEquiv_symm_map, CategoryTheory.Pretriangulated.Opposite.instIsTriangulatedOpposite, AlgebraicGeometry.Scheme.image_zeroLocus, AlgebraicGeometry.Scheme.Hom.appLE_map, CategoryTheory.HasCodetector.hasDetector_op, AlgebraicGeometry.Scheme.basicOpen_eq_bot_iff_forall_evaluation_eq_zero, AlgebraicGeometry.HasRingHomProperty.iff_of_source_openCover, AlgebraicGeometry.LocallyRingedSpace.toΓSpecCBasicOpens_app, CategoryTheory.ShortComplex.cyclesOpIso_inv_naturality_assoc, CategoryTheory.Limits.coconeEquivalenceOpConeOp_functor_map, AlgebraicGeometry.LocallyRingedSpace.evaluation_eq_zero_iff_notMem_basicOpen, AlgebraicGeometry.isIso_toSpecΓ, CategoryTheory.SimplicialObject.augmentedCechNerve_obj_left_obj, CategoryTheory.Limits.pullbackIsoUnopPushout_inv_snd_assoc, 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, CategoryTheory.Equivalence.sheafCongr.unitIso_inv_app_val_app, SheafOfModules.pushforwardSections_coe, AlgebraicGeometry.Scheme.Opens.topIso_hom, 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, PresheafOfModules.Sheafify.map_smul, CategoryTheory.ComposableArrows.opEquivalence_counitIso_hom_app_app, CategoryTheory.uliftCoyonedaEquiv_symm_map, 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.presheafIsGeneratedBy_of_isFinite, AlgebraicGeometry.Scheme.ofRestrict_app, AlgebraicGeometry.structurePresheafInModuleCat_obj_carrier, 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, AlgebraicTopology.DoldKan.Γ₀.Obj.map_on_summand', TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv_functor, CategoryTheory.Limits.coneUnopOfCocone_π, CategoryTheory.ShortComplex.Splitting.unop_s, CategoryTheory.ObjectProperty.IsCodetecting.isIso_iff_of_epi, CategoryTheory.NatIso.unop_whiskerLeft, HomotopicalAlgebra.instHasTwoOutOfThreePropertyOppositeWeakEquivalences, CategoryTheory.Limits.isColimitCoconeOfConeUnop_desc, LightCondSet.toTopCatMap_hom_apply, CategoryTheory.Presieve.isSheaf_comp_uliftFunctor_iff, CategoryTheory.Limits.PreservesFiniteLimitsOfIsFilteredCostructuredArrowYonedaAux.isIso_post, 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, AlgebraicGeometry.SheafedSpace.instEpiTopCatBaseHomPresheafedSpaceπ, AlgebraicGeometry.StructureSheaf.const_mul, AlgebraicGeometry.AffineSpace.map_appTop_coord, CategoryTheory.Comma.unopFunctorCompSnd_hom_app, CategoryTheory.Pretriangulated.preadditiveYoneda_map_distinguished, CategoryTheory.coyonedaFunctor_reflectsLimits, AlgebraicGeometry.Scheme.toSpecΓ_preimage_basicOpen, CategoryTheory.Presheaf.instIsLocallyInjectiveHomιOpposite, LightCondensed.lanPresheafExt_hom, CategoryTheory.yonedaEquiv_symm_naturality_right, CategoryTheory.PreGaloisCategory.PointedGaloisObject.cocone_app, CategoryTheory.Limits.FormalCoproduct.cech_obj, 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, AlgebraicGeometry.Scheme.Hom.app_appIso_inv, AlgebraicGeometry.isIso_fromTildeΓ_iff, CategoryTheory.isCodetecting_unop_iff, TopCat.Sheaf.interUnionPullbackCone_pt, CategoryTheory.NatTrans.removeRightOp_id, 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.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, CategoryTheory.Presheaf.isSheaf_iff_isSheaf_forget, CategoryTheory.ShortComplex.fromOpcycles_op_cyclesOpIso_inv, SSet.horn₃₂.desc.multicofork_π_zero, CategoryTheory.Limits.FormalCoproduct.cosimplicialObjectFunctor_map_app, AlgebraicGeometry.Scheme.Hom.normalizationObjIso_hom_val, AlgebraicGeometry.Scheme.openCoverBasicOpenTop_f, CategoryTheory.Functor.unopId_hom_app, 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, 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.hasCodetector_op_iff, 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, CategoryTheory.ShortComplex.LeftHomologyData.unop_g', 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.GrothendieckTopology.uliftYoneda_obj_val_obj, CategoryTheory.Injective.injective_iff_preservesEpimorphisms_preadditiveYoneda_obj, CategoryTheory.uliftYonedaIsoYoneda_inv_app_app_down, AlgebraicGeometry.isField_of_isIntegral_of_subsingleton, CategoryTheory.NatTrans.leftOp_comp, 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, CategoryTheory.Limits.pullbackIsoUnopPushout_hom_inr, HomologicalComplex.opcyclesOpIso_hom_toCycles_op_assoc, CategoryTheory.Functor.instEssSurjOppositeLeftOp, CategoryTheory.Limits.opSpan_inv_app, 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, CategoryTheory.Functor.sheafPushforwardContinuousId'_hom_app_val_app, 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.Functor.IsCoverDense.sheafIso_hom_val, TopCat.Presheaf.stalkFunctor_map_germ_apply', CondensedMod.LocallyConstant.instFaithfulModuleCatSheafCompHausCoherentTopologyConstantSheaf, CategoryTheory.Functor.PullbackObjObj.π_iso_of_iso_left_hom, AlgebraicTopology.DoldKan.P_f_naturality, fintypeToFinBoolAlgOp_obj, PresheafOfModules.freeAdjunction_homEquiv, CategoryTheory.GrothendieckTopology.pseudofunctorOver_mapId_inv_toNatTrans_app_val_app, CategoryTheory.Pretriangulated.Opposite.rotate_distinguished_triangle, 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.instFullSheafFunctorOppositeSheafToPresheaf, AlgebraicGeometry.isLocalization_basicOpen_of_qcqs, CategoryTheory.unop_sub, AlgebraicGeometry.Scheme.IdealSheafData.strictMono_ideal, SimplicialObject.Split.cofan_inj_naturality_symm, CommGrpCat.coyoneda_map_app, 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.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, SheafOfModules.forgetToSheafModuleCat_obj_val, CategoryTheory.yoneda'_map_val, AlgebraicGeometry.LocallyRingedSpace.evaluation_naturality_apply, AlgebraicTopology.DoldKan.factors_normalizedMooreComplex_PInfty, SimplexCategory.II_obj, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_appLE, AlgebraicGeometry.IsAffineOpen.fromSpec_toSpecΓ, SSet.associator_inv_app_apply, AlgebraicGeometry.map_injective_of_isIntegral, CategoryTheory.instIsLeftAdjointFunctorOppositeSheafPresheafToSheaf, CategoryTheory.Functor.map_distinguished_op_exact, AlgebraicGeometry.exists_eq_pow_mul_of_isCompact_of_isQuasiSeparated, AlgebraicGeometry.AffineSpace.homOfVector_appTop_coord, CategoryTheory.GrothendieckTopology.uliftYoneda_obj_val_map_down, SSet.stdSimplex.nonDegenerateEquiv_symm_apply_coe, AlgebraicGeometry.sourceAffineLocally_morphismRestrict, PresheafOfModules.evaluation_obj, CategoryTheory.RanIsSheafOfIsCocontinuous.liftAux_map, CategoryTheory.OverPresheafAux.map_mkPrecomp_eqToHom, CategoryTheory.Limits.end_.condition_assoc, CategoryTheory.Presheaf.subsingleton_iff_isSeparatedFor, CategoryTheory.Sheaf.cartesianMonoidalCategoryWhiskerLeft_val, AlgebraicGeometry.Scheme.IsQuasiAffine.isBasis_basicOpen, AlgebraicGeometry.Proj.stalkIso'_symm_mk, CategoryTheory.Sheaf.ΓRes_naturality, 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, CategoryTheory.Limits.preservesFiniteCoproducts_leftOp, AlgebraicGeometry.structureSheafInType.smul_apply, CategoryTheory.prodOpEquiv_counitIso_inv_app, CategoryTheory.unop_associator, CategoryTheory.GrothendieckTopology.diagramNatTrans_comp, CategoryTheory.uliftYonedaFunctor_preservesLimits, AlgebraicGeometry.Scheme.isoSpec_image_zeroLocus, CategoryTheory.Equalizer.Presieve.isSheafFor_singleton_iff_of_hasPullback, 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.CategoryOfElements.costructuredArrowULiftYonedaEquivalence_inverse_map, CategoryTheory.Functor.rightOp_map_unop, CategoryTheory.sheafifyLift_id_toSheafify_assoc, AlgebraicGeometry.Scheme.IdealSheafData.ideal_map_of_isAffineHom, CategoryTheory.Comon.MonOpOpToComonObj_comon_counit, CategoryTheory.GrothendieckTopology.pseudofunctorOver_mapId_hom_toNatTrans_app_val_app, TopCat.Presheaf.pushforwardPullbackAdjunction_unit_pullback_map_germToPullbackStalk_assoc, AlgebraicGeometry.isCompact_and_isOpen_iff_finite_and_eq_biUnion_basicOpen, 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, AlgebraicGeometry.LocallyRingedSpace.SpecΓIdentity_inv_app, CategoryTheory.GrothendieckTopology.OneHypercover.multiforkLift_map_assoc, CategoryTheory.representablyCoflat_op_iff, CategoryTheory.Functor.leftOpRightOpIso_inv_app, HomologicalComplex.unopFunctor_additive, CategoryTheory.Meq.condition, AlgebraicGeometry.Scheme.AffineZariskiSite.presieveOfSections_eq_ofArrows, CategoryTheory.Sheaf.sheafifyCocone_ι_app_val, AlgebraicGeometry.Spec.toTop_obj_carrier, CategoryTheory.SimplicialObject.δ_comp_σ_succ_assoc, AlgebraicGeometry.Scheme.Hom.inv_app, CategoryTheory.NonemptyParallelPairPresentationAux.hg, SSet.RelativeMorphism.map_coe, 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, AlgebraicGeometry.RingedSpace.mem_basicOpen, CategoryTheory.Functor.IsLocalization.op, CategoryTheory.Limits.PushoutCocone.unop_snd, CategoryTheory.Square.op_f₃₄, CategoryTheory.isCoseparator_unop_iff, SSet.Truncated.HomotopyCategory.BinaryProduct.inverse_obj, AlgebraicGeometry.Scheme.Hom.app_injective, CategoryTheory.Functor.mapPresheaf_obj_presheaf, CategoryTheory.ShortComplex.opcyclesOpIso_inv_naturality_assoc, CategoryTheory.OverPresheafAux.counitAuxAux_hom, CategoryTheory.NatTrans.removeOp_app, CategoryTheory.MorphismProperty.RightFraction.unop_Y', HomologicalComplex.unopEquivalence_counitIso, SSet.Subcomplex.N.notMem, AlgebraicGeometry.Scheme.IdealSheafData.subschemeFunctor_map, CategoryTheory.Limits.colimitCoyonedaHomIsoLimitLeftOp_π_apply, CategoryTheory.Functor.IsCoverDense.restrictHomEquivHom_naturality_right, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_inv_app, Condensed.id_val, AlgebraicGeometry.Scheme.Modules.inv_app, 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.Functor.sheafPushforwardContinuousComp'_hom_app_val_app, SSet.horn₃₁.desc.multicofork_π_three, CategoryTheory.MonoidalCategory.MonoidalLeftAction.oppositeLeftAction_actionObj_op, CategoryTheory.Pretriangulated.instIsTriangulatedOppositeUnopUnop, AlgebraicGeometry.AffineScheme.Spec_full, AlgebraicGeometry.nonempty_isColimit_Γ_mapCocone, AlgebraicTopology.DoldKan.hσ'_eq', CategoryTheory.GrothendieckTopology.map_yonedaULiftEquiv, HomologicalComplex.opInverse_map, CategoryTheory.GrothendieckTopology.overMapPullbackCongr_inv_app_val_app, CategoryTheory.MorphismProperty.RightFraction.op_s, SSet.StrictSegal.isRightKanExtension, CategoryTheory.Limits.limitUnopIsoUnopColimit_inv_comp_π, CategoryTheory.Subfunctor.Subpresheaf.ofSection_eq_range, CommRingCat.preservesFilteredColimits_coyoneda, CategoryTheory.Limits.preservesColimitsOfShape_of_rightOp, CategoryTheory.Limits.walkingParallelPairOpEquiv_counitIso_inv_app_op_one, CategoryTheory.Equivalence.sheafCongr.counitIso_inv_app_val_app, 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, 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, TopCat.Presheaf.presheafEquivOfIso_functor_obj_map, CategoryTheory.NatTrans.unop_app, AlgebraicGeometry.LocallyRingedSpace.toΓSpec_preimage_basicOpen_eq, CategoryTheory.Subpresheaf.to_sheafifyLift, CategoryTheory.Limits.preservesLimits_leftOp, AlgebraicGeometry.StructureSheaf.algebraMap_germ_apply, 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.MonoidalCategory.MonoidalLeftAction.oppositeLeftAction_actionUnitIso, CategoryTheory.Presieve.isSheafFor_ofArrows_iff_bijective_toCompabible, AlgebraicGeometry.Scheme.AffineZariskiSite.mem_grothendieckTopology_iff_sectionsOfPresieve, CategoryTheory.Equivalence.symmEquivInverse_obj_unitIso_hom, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_ι, CategoryTheory.Functor.instInitialOppositeRightOpOfFinal, CategoryTheory.Under.opEquivOpOver_inverse_map, SimplicialObject.Splitting.cofan_inj_eq_assoc, PresheafOfModules.Sheafify.mul_smul, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_appTop, CategoryTheory.ObjectProperty.isClosedUnderLimitsOfShape_iff_unop, AlgebraicGeometry.PresheafedSpace.restrictStalkIso_inv_eq_germ_assoc, CategoryTheory.GrothendieckTopology.Point.sheafFiberCompIso_inv_app, CategoryTheory.GrothendieckTopology.uliftYoneda_map_val_app_down, CategoryTheory.Limits.FormalCoproduct.evalOpCompInlIsoId_inv_app_app, SimplexCategory.II_δ, CategoryTheory.Limits.Cofork.op_ι, AlgebraicGeometry.Scheme.IdealSheafData.map_ideal_basicOpen, CommGrpCat.coyonedaType_obj_obj_coe, CategoryTheory.MorphismProperty.MapFactorizationData.op_p, TopCat.Presheaf.presheafEquivOfIso_inverse_map_app, 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.Comon.monoidal_tensorObj_comon_counit, CategoryTheory.Functor.PullbackObjObj.ofHasPullback_fst, CategoryTheory.Functor.IsRepresentedBy.representableBy_homEquiv_apply, PresheafOfModules.instAdditiveFunctorOppositeAbToPresheaf, CategoryTheory.Sheaf.tensorUnit_isSheaf, 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.orderDualEquivalence_inverse_map, AlgebraicGeometry.essImage_Spec, CategoryTheory.Limits.widePullbackShapeOpEquiv_counitIso, SSet.horn.primitiveTriangle_coe, CategoryTheory.Limits.endFunctor_map, CompHausLike.LocallyConstant.functorToPresheaves_obj_map, CategoryTheory.sheafCompose_obj_val, CategoryTheory.Limits.walkingParallelPairOp_right, CategoryTheory.WithTerminal.opEquiv_functor_map, CategoryTheory.Join.InclRightCompRightOpOpEquivFunctor_hom_app, PresheafOfModules.zero_app, AlgebraicGeometry.Scheme.Modules.pseudofunctor_mapId_inv_τl, 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, AlgebraicGeometry.Scheme.Hom.fromNormalization_app, CategoryTheory.Functor.mapTriangleOpCompTriangleOpEquivalenceFunctorApp_hom_hom₂, CategoryTheory.Limits.coneOfCoconeUnop_pt, CategoryTheory.Presheaf.restrictedULiftYonedaHomEquiv'_symm_naturality_right, AlgebraicTopology.DoldKan.Γ₀.Obj.map_on_summand'_assoc, 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, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_presheafFiberCompIso_hom_app, CategoryTheory.Limits.preservesLimits_rightOp, TopCat.Presheaf.isSheaf_iff_isTerminal_of_indiscrete, CategoryTheory.OverPresheafAux.counitForward_val_fst, CategoryTheory.Limits.instHasCoproductOppositeOp, CategoryTheory.MonoidalCategory.MonoidalRightAction.rightActionOfOppositeRightAction_actionHomLeft_unop, CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily_apply, CategoryTheory.GrothendieckTopology.yonedaEquiv_yoneda_map, commBialgCatEquivComonCommAlgCat_counitIso_inv_app, CategoryTheory.Pseudofunctor.CoGrothendieck.compIso_hom_app, AlgebraicGeometry.Scheme.id_app, CategoryTheory.Pseudofunctor.DescentData.hom_comp, CategoryTheory.Pseudofunctor.IsPrestackFor.nonempty_fullyFaithful, Alexandrov.projSup_map, CategoryTheory.LocalizerMorphism.instHasRightResolutionsOppositeOpOpOfHasLeftResolutions, 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, PresheafOfModules.toPresheaf_map_sheafificationHomEquiv, AlgebraicGeometry.Scheme.Modules.pseudofunctor_mapId_inv_τr, CategoryTheory.presheafHom_map_app_op_mk_id, CategoryTheory.Limits.preservesColimit_of_unop, Condensed.lanPresheafIso_hom, TopCat.Presheaf.pushforwardPullbackAdjunction_unit_app_app_germToPullbackStalk, 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.stdSimplex.face_singleton_compl, PresheafOfModules.instIsRightAdjointPushforwardIdFunctorOppositeRingCat, CategoryTheory.CategoryOfElements.fromCostructuredArrow_obj_fst, SSet.ι₀_app_fst, CategoryTheory.Functor.IsRightAdjoint.rightOp, SSet.Truncated.spine_injective, 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.GrothendieckTopology.yonedaEquiv_apply, 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.SimplicialObject.augmentedCechNerve_obj_hom_app, SimplicialObject.Splitting.cofan_inj_epi_naturality, CategoryTheory.map_coyonedaEquiv, 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, CategoryTheory.ObjectProperty.small_op_iff, CategoryTheory.instIsCardinalAccessibleObjOppositeFunctorTypeUliftCoyonedaOpOfIsCardinalPresentable, SSet.Truncated.Edge.CompStruct.idCompId_simplex, CategoryTheory.Yoneda.obj_map_id, TopCat.Presheaf.isSheaf_of_isOpenEmbedding, AlgebraicGeometry.Scheme.AffineZariskiSite.restrictIsoSpec_inv_app, CategoryTheory.Groupoid.invFunctor_obj, CategoryTheory.GrothendieckTopology.uliftYonedaOpCompCoyoneda_inv_app_app_val_app, AlgebraicGeometry.ΓSpec.toSpecΓ_unop, CategoryTheory.Limits.widePullbackShapeOp_map, CategoryTheory.map_yonedaEquiv', CategoryTheory.opOpEquivalence_functor, CategoryTheory.ShortComplex.quasiIso_opMap_iff, CategoryTheory.Limits.Cowedge.condition, CommMonCat.coyonedaType_map_app, AlgebraicGeometry.isIso_pushoutSection_of_isAffineOpen, SSet.range_eq_iSup_of_isColimit, LightCondensed.instPreservesFiniteProductsOppositeLightProfiniteVal, CategoryTheory.CategoryOfElements.toCostructuredArrow_map, CategoryTheory.MonoidalCategory.MonoidalRightAction.oppositeRightAction_actionUnitIso, CategoryTheory.ObjectProperty.isClosedUnderLimitsOfShape_iff_op, 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, AlgebraicGeometry.AffineScheme.ΓIsEquiv, CategoryTheory.GrothendieckTopology.yonedaEquiv_symm_naturality_right, CategoryTheory.Limits.π_comp_opProductIsoCoproduct_hom, CategoryTheory.Limits.walkingCospanOpEquiv_inverse_map, SSet.Subcomplex.preimage_iInf, 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, AlgebraicGeometry.tilde.instIsLocalizedModuleCarrierCarrierObjOppositeOpensCarrierCarrierCommRingCatSpecModuleCatPresheafModulesSheafModulesSpecToSheafOpBasicOpenPowersHomToOpen, CategoryTheory.StructuredArrow.toCostructuredArrow_obj, AlgebraicGeometry.Scheme.isNilpotent_iff_basicOpen_eq_bot_of_isCompact, classifyingSpaceUniversalCover_obj, AlgebraicGeometry.SheafedSpace.GlueData.ι_jointly_surjective, CategoryTheory.Comon.ComonToMonOpOpObj_X, CategoryTheory.Presheaf.isLocallyInjective_comp, 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, 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.PresheafOfGroups.Cochain₀.mul_apply, AlgebraicGeometry.RingedSpace.mem_basicOpen', CategoryTheory.ShortComplex.op_X₂, CategoryTheory.Presheaf.isLocallySurjective_presheafToSheaf_map_iff, CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan.coconeApp_naturality_assoc, 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.WithTerminal.opEquiv_inverse_map, CategoryTheory.Functor.PullbackObjObj.isPullback, Subobject.presheaf_map, CategoryTheory.constantPresheafAdj_unit_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.Presheaf.instIsLeftKanExtensionOppositeObjFunctorTypeUliftYonedaUliftYonedaMap, AlgebraicGeometry.SpecMap_ΓSpecIso_inv_toSpecΓ_assoc, CategoryTheory.ParametrizedAdjunction.whiskerLeft_map_counit_assoc, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv_apply, AlgebraicGeometry.PresheafedSpace.ofRestrict_c_app, CategoryTheory.Limits.SequentialProduct.functorMap_commSq, CategoryTheory.Sheaf.isSheaf_yoneda_obj, AlgebraicGeometry.LocallyRingedSpace.toΓSpecCApp_iff, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.ofRestrict_invApp_apply, 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, AlgebraicGeometry.LocallyRingedSpace.Γ_obj, SSet.Truncated.StrictSegal.spineToSimplex_map, CategoryTheory.MorphismProperty.ContainsIdentities.of_unop, AlgebraicGeometry.Spec.map_app, CategoryTheory.preservesColimitsOfShape_of_isCardinalPresentable, SSet.PtSimplex.MulStruct.δ_castSucc_castSucc_map, CategoryTheory.instIsClosedUnderColimitsOfShapeFunctorOppositeTypeIsIndObjectOfIsFiltered, ContinuousMap.instPreservesFiniteProductsOppositeTopCatYonedaPresheaf', CategoryTheory.Limits.ι_comp_colimitUnopIsoOpLimit_hom, AlgebraicGeometry.Scheme.Spec_fromSpecStalk, CategoryTheory.Functor.op_map, PresheafOfModules.pushforward₀_obj_map, CategoryTheory.NatTrans.unop_comp, SheafOfModules.unitHomEquiv_apply_coe, CategoryTheory.Abelian.IsGrothendieckAbelian.OppositeModuleEmbedding.preservesFiniteLimits_embedding, CategoryTheory.Limits.widePullbackShapeOpEquiv_unitIso, CategoryTheory.preadditiveYonedaObj_map, 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.Sheaf.image_val, CategoryTheory.whiskering_preadditiveCoyoneda, CategoryTheory.ParametrizedAdjunction.arrowHomEquiv_apply_right_snd_assoc, typeToBoolAlgOp_obj, CategoryTheory.ShiftedHom.opEquiv_symm_comp, CategoryTheory.Coyoneda.ULiftCoyoneda.instFaithfulOppositeFunctorTypeUliftCoyoneda, 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, 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	231 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.Pretriangulated.opShiftFunctorEquivalence_add_unitIso_hom_app_eq, CategoryTheory.ShortComplex.homologyOpIso_hom_naturality, CategoryTheory.Pseudofunctor.DescentData.pullFunctorCompIso_inv_app_hom, CategoryTheory.op_comp, CategoryTheory.eqToHom_comp_homOfLE_op, CategoryTheory.WithInitial.opEquiv_unitIso_inv_app, AlgebraicGeometry.Scheme.restrictFunctorΓ_inv_app, CategoryTheory.Limits.parallelPairOpIso_inv_app_zero, 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.WithInitial.opEquiv_counitIso_hom_app, CategoryTheory.Pretriangulated.shiftFunctorAdd'_op_hom_app, 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.Limits.opCoproductIsoProduct'_hom_comp_proj, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_hom_naturality, CategoryTheory.ShortComplex.homologyOpIso_inv_naturality, 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.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.Limits.opParallelPairIso_hom_app_zero, CategoryTheory.ObjectProperty.op_singleton, CategoryTheory.Pseudofunctor.DescentData.instIsIsoαCategoryObjLocallyDiscreteOppositeCatMkOpHom, CategoryTheory.OverPresheafAux.counitForward_val_snd, CategoryTheory.Limits.opParallelPairIso_hom_app_one, CategoryTheory.Limits.opCoproductIsoProduct'_inv_comp_inj, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_unitIso_hom_app, 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.Pseudofunctor.DescentData.ofObj_obj, CategoryTheory.op_comp_assoc, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_zero_unitIso_hom_app, 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.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, CategoryTheory.Join.inclRightCompOpEquivInverse_hom_app_op, CategoryTheory.Functor.op_commShiftIso_inv_app, PresheafOfModules.map_comp, AlgebraicGeometry.PresheafedSpace.GlueData.f_invApp_f_app, AugmentedSimplexCategory.equivAugmentedSimplicialObject_functor_obj_hom_app, CategoryTheory.Functor.const.opObjOp_hom_app, CategoryTheory.ShortComplex.opcyclesOpIso_inv_naturality, 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, 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.yonedaPairing_map, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_inv_app, 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.ShiftedHom.opEquiv'_symm_op_opShiftFunctorEquivalence_counitIso_inv_app_op_shift, CategoryTheory.Functor.shift_map_op, 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.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.pullFunctorObjHom_eq_assoc, TopCat.Presheaf.presheafEquivOfIso_counitIso_inv_app_app, CategoryTheory.ShiftedHom.opEquiv'_symm_comp, CategoryTheory.MorphismProperty.instRespectsLeftOppositeOpOfRespectsRight, PresheafOfModules.map_id, CategoryTheory.ShortComplex.homologyMap_op, CategoryTheory.Limits.limitOpIsoOpColimit_inv_comp_π, 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, 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, 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.Functor.op_commShiftIso_hom_app, CategoryTheory.Pretriangulated.Opposite.OpOpCommShift.iso_hom_app, 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, 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.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.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.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.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	38 math math: leftOp_unitIso_hom_app, sheafCongrPrecoherent_counitIso_hom_app_val_app, CategoryTheory.ComposableArrows.opEquivalence_counitIso_inv_app_app, sheafCongr.counitIso_hom_app_val_app, sheafCongrPreregular_counitIso_inv_app_val_app, rightOp_counitIso_inv_app, leftOp_unitIso_inv_app, leftOp_counitIso_inv_app, sheafCongrPreregular_unitIso_hom_app_val_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, sheafCongr.unitIso_hom_app_val_app, sheafCongrPreregular_counitIso_hom_app_val_app, sheafCongrPrecoherent_counitIso_inv_app_val_app, TopCat.Presheaf.presheafEquivOfIso_unitIso_inv_app_app, op_functor, TopCat.Presheaf.whiskerIsoMapGenerateCocone_hom_hom, sheafCongrPrecoherent_unitIso_inv_app_val_app, rightOp_counitIso_hom_app, sheafCongrPrecoherent_unitIso_hom_app_val_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, sheafCongrPreregular_unitIso_inv_app_val_app, sheafCongr.unitIso_inv_app_val_app, CategoryTheory.ComposableArrows.opEquivalence_counitIso_hom_app_app, op_counitIso, sheafCongr.counitIso_inv_app_val_app, 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	106 math math: CategoryTheory.Adjunction.leftOp_unit, CategoryTheory.CategoryOfElements.costructuredArrowYonedaEquivalenceInverseπ_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.hasLimit_leftOp_of_hasColimit, CategoryTheory.NatTrans.leftOp_app, CategoryTheory.Comma.opEquiv_counitIso, CategoryTheory.Limits.isColimitCoconeLeftOpOfCone_desc, CategoryTheory.Limits.coconeOfConeLeftOp_pt, CategoryTheory.Limits.colimitYonedaHomIsoLimit'_π_apply, CategoryTheory.Limits.π_comp_colimitLeftOpIsoUnopLimit_inv, leftOp_obj, rightOpLeftOpIso_hom_app, CategoryTheory.Limits.isLimitOfCoconeOfConeLeftOp_lift, leftOp_faithful, 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.simplicialCosimplicialEquiv_unitIso_hom_app, CategoryTheory.Limits.preservesLimitsOfSize_leftOp, CategoryTheory.Limits.coneOfCoconeLeftOp_π_app, CategoryTheory.Limits.isColimitOfConeOfCoconeLeftOp_desc, leftOpRightOpIso_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.preservesFiniteProducts_leftOp, CategoryTheory.Limits.coneOfCoconeLeftOp_pt, rightOpLeftOpIso_inv_app, 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.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, 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.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.preservesColimitsOfShape_leftOp, CategoryTheory.ComposableArrows.opEquivalence_inverse_map, CategoryTheory.Comma.opFunctorCompSnd_inv_app, instIsEquivalenceOppositeLeftOp, CategoryTheory.Comma.opEquiv_inverse, instFinalOppositeLeftOpOfInitial, CategoryTheory.Limits.coconeOfConeLeftOp_ι_app, 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.ι_comp_colimitLeftOpIsoUnopLimit_hom_assoc, CategoryTheory.Limits.limitLeftOpIsoUnopColimit_inv_comp_π_assoc, CategoryTheory.Limits.ι_comp_colimitLeftOpIsoUnopLimit_hom, CategoryTheory.Limits.limitLeftOpIsoUnopColimit_hom_comp_ι, CategoryTheory.NatTrans.leftOp_comp, instEssSurjOppositeLeftOp, leftOpRightOpEquiv_inverse_obj, CategoryTheory.Limits.preservesFiniteCoproducts_leftOp, leftOpRightOpIso_inv_app, CategoryTheory.Limits.colimitCoyonedaHomIsoLimitLeftOp_π_apply, CategoryTheory.Limits.preservesLimits_leftOp, AlgebraicGeometry.PresheafedSpace.colimitCocone_ι_app_c, 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	644 math math: CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence.functor_obj_hom_right, FullyFaithful.homNatIsoMaxRight_inv_app, PresheafOfModules.instIsRightAdjointPushforwardCompFunctorOppositeRingCatWhiskerLeftOp, CategoryTheory.Adjunction.compUliftCoyonedaIso_hom_app_app_down, smoothSheafCommRing.ι_forgetStalk_inv, CategoryTheory.Limits.IndizationClosedUnderFilteredColimitsAux.exists_nonempty_limit_obj_of_isColimit, 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, CategoryTheory.Limits.coconeEquivalenceOpConeOp_unitIso, AlgebraicGeometry.LocallyRingedSpace.Γ_def, CategoryTheory.GrothendieckTopology.yonedaOpCompCoyoneda_hom_app_app_down, CategoryTheory.Comma.opFunctor_obj, opUnopIso_hom_app, CategoryTheory.Localization.isoOfHom_unop, CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan.uliftYonedaEquiv_ι_presheafHom, CategoryTheory.Limits.Cone.unop_ι, Profinite.Extend.cocone_pt, mapTriangleOpCompTriangleOpEquivalenceFunctorApp_inv_hom₁, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverse_obj_π_app, CategoryTheory.sheafToPresheafCompCoyonedaCompWhiskeringLeftSheafToPresheaf_hom_app_app_val, CategoryTheory.MorphismProperty.LeftFraction.op_map, AlgebraicGeometry.Scheme.restrictFunctorΓ_inv_app, 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, CategoryTheory.Pretriangulated.shiftFunctorZero_op_inv_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, CategoryTheory.Limits.limitOpIsoOpColimit_hom_comp_ι_assoc, CategoryTheory.sheafToPresheafCompCoyonedaCompWhiskeringLeftSheafToPresheaf_inv_app_app, CategoryTheory.IsFiltered.iff_nonempty_limit, CategoryTheory.Comma.opEquiv_counitIso, 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, TopCat.Presheaf.pushforwardToOfIso_app, 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.TwoSquare.structuredArrowRightwardsOpEquivalence.functor_obj_left_right, CategoryTheory.Sheaf.isPullback_square_op_map_yoneda_presheafToSheaf_yoneda_iff, CategoryTheory.NatIso.op_trans, CategoryTheory.sheafToPresheafCompYonedaCompWhiskeringLeftSheafToPresheaf_hom_app_app_val, CategoryTheory.Limits.parallelPairOpIso_inv_app_one, LightProfinite.Extend.cocone_ι_app, 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, mapTriangleOpCompTriangleOpEquivalenceFunctorApp_inv_hom₃, CategoryTheory.Pretriangulated.Opposite.UnopUnopCommShift.iso_inv_app_assoc, FullyFaithful.homNatIso_inv_app_down, opId_hom_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, 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, sheafPushforwardContinuousNatTrans_app_val, CategoryTheory.Adjunction.Quadruple.op_leftTriple, CategoryTheory.NatTrans.op_id, instFaithfulOppositeOp, 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, sheafPushforwardContinuous_map_val_app, CategoryTheory.Limits.preservesLimitsOfSize_op, Condensed.finYoneda_map, mapCoconeOp_inv_hom, CategoryTheory.Limits.Cocone.op_pt, CategoryTheory.NatTrans.op_whiskerRight_assoc, CategoryTheory.regularTopology.equalizerConditionMap_iff_nonempty_isLimit, FullyFaithful.compUliftYonedaCompWhiskeringLeft_inv_app_app_down, 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.Adjunction.Triple.op_rightToLeft, CategoryTheory.SimplicialObject.Truncated.rightExtensionInclusion_left, CategoryTheory.Limits.coyonedaOpColimitIsoLimitCoyoneda_inv_comp_π_assoc, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.invApp_app_apply, CategoryTheory.instRepresentablyFlatOppositeOpOfRepresentablyCoflat, CategoryTheory.Equivalence.sheafCongrPreregular_functor_map_val_app, 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.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, CategoryTheory.Equivalence.sheafCongrPreregular_inverse_map_val_app, AlgebraicGeometry.Scheme.Γ_def, CategoryTheory.presheafHom_map_app, CategoryTheory.Limits.opParallelPairIso_hom_app_one, 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, 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.Equivalence.sheafCongrPrecoherent_inverse_map_val_app, CategoryTheory.Presheaf.restrictedULiftYoneda_map_app, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.inv_naturality_assoc, 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, PresheafOfModules.pullback_assoc, CategoryTheory.Limits.opSpan_hom_app, CategoryTheory.Limits.coconeEquivalenceOpConeOp_functor_obj, CategoryTheory.NatTrans.op_whiskerRight, 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, CategoryTheory.lan_preservesFiniteLimits_of_preservesFiniteLimits, unopOpIso_hom_app, AlgebraicGeometry.ΓSpec.unop_locallyRingedSpaceAdjunction_counit_app', op_comp_isSheaf_of_types, 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, 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, AlgebraicGeometry.Spec.sheafedSpaceMap_hom_c_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, CategoryTheory.Equivalence.sheafCongr.functor_map_val_app, 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.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, 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.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.Presieve.isSheafFor_over_map_op_comp_ofArrows_iff, map_opShiftFunctorEquivalence_counitIso_hom_app_unop, CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence_unitIso, CategoryTheory.NatIso.op_leftUnitor, 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, IsCoverDense.Types.naturality_apply, CategoryTheory.NatTrans.rightOpWhiskerRight_assoc, 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.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, CategoryTheory.GrothendieckTopology.pseudofunctorOver_toPrelaxFunctor_toPrelaxFunctorStruct_toPrefunctor_map_toFunctor_map_val_app, opHom_obj, AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap_app_assoc, CategoryTheory.Pretriangulated.opShiftFunctorEquivalence_counitIso_hom_app_assoc, 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.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.NatIso.op_refl, CategoryTheory.Adjunction.compCoyonedaIso_hom_app_app, closedIhom_map_app, CategoryTheory.NatTrans.leftOpWhiskerRight, SSet.horn.spineId_vertex_coe, CategoryTheory.Presieve.FamilyOfElements.Compatible.functorPullback, map_opShiftFunctorEquivalence_unitIso_hom_app_unop_assoc, CategoryTheory.GrothendieckTopology.uliftYonedaOpCompCoyoneda_hom_app_app_down, CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan.uliftYonedaEquiv_presheafHom_uliftYoneda_obj, shift_map_op, CategoryTheory.Limits.ι_comp_colimitOpIsoOpLimit_hom, CategoryTheory.Pretriangulated.shiftFunctor_op_map, SSet.horn.spineId_arrow_coe, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.invApp_app, CategoryTheory.SimplicialObject.IsCoskeletal.isRightKanExtension, TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverse_map_hom, Profinite.Extend.functorOp_final, FullyFaithful.compUliftYonedaCompWhiskeringLeft_hom_app_app_down, 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, 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, TopCat.Presheaf.pushforwardPullbackAdjunction_unit_pullback_map_germToPullbackStalk, TopologicalSpace.Opens.op_map_comp_obj, 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, 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.NatIso.op_inv, PresheafOfModules.pushforward_assoc, Condensed.isoFinYoneda_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.Limits.coconeEquivalenceOpConeOp_inverse_obj, AlgebraicGeometry.PresheafedSpace.congr_app, CategoryTheory.Limits.Cocone.op_π, CategoryTheory.HasLiftingProperty.transfiniteComposition.wellOrderInductionData.map_lift, CategoryTheory.Limits.colimitYonedaHomIsoLimitOp_π_apply, LightCondensed.isColimitLocallyConstantPresheaf_desc_apply, CategoryTheory.Limits.coconeEquivalenceOpConeOp_counitIso, 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, mapConeOp_hom_hom, LightCondensed.instFinalNatCostructuredArrowOppositeFintypeCatLightProfiniteOpToLightProfiniteOpPtAsLimitConeFunctorOp, SSet.OneTruncation₂.nerveHomEquiv_apply, CategoryTheory.Equivalence.sheafCongr.inverse_map_val_app, 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.GrothendieckTopology.overMapPullbackCongr_hom_app_val_app, 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.Adjunction.representableBy_homEquiv, AlgebraicGeometry.PresheafedSpace.restrict_presheaf, ContinuousMap.comp_yonedaPresheaf', CategoryTheory.Enriched.FunctorCategory.enrichedComp_π, CategoryTheory.Presheaf.isLimit_iff_isSheafFor, CategoryTheory.Pretriangulated.commShiftIso_unopUnop_hom_app, CategoryTheory.ShortComplex.leftHomologyFunctorOpNatIso_inv_app, op_commShiftIso_hom_app, CategoryTheory.Equivalence.op_unitIso, CategoryTheory.Limits.coconeEquivalenceOpConeOp_inverse_map_hom, 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, CategoryTheory.Equivalence.sheafCongrPrecoherent_functor_map_val_app, SmallCategories.instPreservesFiniteLimitsSheafSheafPullbackOfRepresentablyFlat, CategoryTheory.Pretriangulated.commShiftIso_unopUnop_hom_app_assoc, CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan.coconeApp_naturality, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.inv_invApp, TopCat.Presheaf.IsSheaf.isSheafOpensLeCover, CategoryTheory.Limits.preservesColimitsOfShape_op, AlgebraicGeometry.LocallyRingedSpace.toΓSpecCBasicOpens_app, CategoryTheory.Limits.coconeEquivalenceOpConeOp_functor_map, CategoryTheory.Limits.opParallelPairIso_inv_app_zero, SheafOfModules.pushforwardSections_coe, 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, 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, TopCat.Presheaf.pushforwardPullbackAdjunction_unit_pullback_map_germToPullbackStalk_assoc, CategoryTheory.representablyCoflat_op_iff, AlgebraicGeometry.PresheafedSpace.GlueData.snd_invApp_t_app', 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, CategoryTheory.GrothendieckTopology.overMapPullbackCongr_inv_app_val_app, SSet.StrictSegal.isRightKanExtension, CategoryTheory.Limits.opHomCompWhiskeringLimYonedaIsoCocones_inv_app_app, AlgebraicGeometry.LocallyRingedSpace.toΓSpec_preimage_basicOpen_eq, map_opShiftFunctorEquivalence_unitIso_hom_app_unop, op_obj, CategoryTheory.Limits.FormalCoproduct.evalOpCompInlIsoId_inv_app_app, CategoryTheory.Adjunction.Quadruple.op_adj₃, mapTriangleOpCompTriangleOpEquivalenceFunctorApp_hom_hom₂, CategoryTheory.CostructuredArrow.toStructuredArrow_map, IsCoverDense.Types.pushforwardFamily_apply, LightProfinite.Extend.functorOp_final, CategoryTheory.presheafHom_map_app_op_mk_id, Condensed.lanPresheafIso_hom, CategoryTheory.TwoSquare.structuredArrowRightwardsOpEquivalence.functor_obj_left_left_as, CategoryTheory.Limits.limitOpIsoOpColimit_hom_comp_ι, TopCat.Presheaf.isSheaf_of_isOpenEmbedding, CategoryTheory.GrothendieckTopology.uliftYonedaOpCompCoyoneda_inv_app_app_val_app, AlgebraicGeometry.ΓSpec.toSpecΓ_unop, 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.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	158 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, AlgebraicGeometry.ΓSpec.isIso_adjunction_counit, CategoryTheory.FunctorToTypes.functorHomEquiv_symm_apply_app_app, AlgebraicGeometry.instPreservesLimitSchemeOppositeCommRingCatRightOpΓOfIsAffineHomMapOfCompactSpaceOfQuasiSeparatedSpaceCarrierCarrierObj, 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.ι_comp_colimitRightOpIsoUnopLimit_hom_assoc, AlgebraicGeometry.Scheme.kerAdjunction_unit_app, 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, AlgebraicGeometry.AffineScheme.Γ_preservesLimits, natTransEquiv_symm_apply_app, 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, 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.Adjunction.rightOp_counit, 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.isLimitOfCoconeRightOpOfCone_lift, CategoryTheory.Limits.coconeOfConeRightOp_pt, CategoryTheory.Limits.isLimitConeOfCoconeRightOp_lift, CategoryTheory.Limits.coneOfCoconeRightOp_pt, CategoryTheory.Limits.ι_comp_colimitRightOpIsoUnopLimit_hom, 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, 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.coconeRightOpOfCone_ι, AlgebraicGeometry.ΓSpec.right_triangle, CategoryTheory.Join.InclLeftCompRightOpOpEquivFunctor_inv_app, CategoryTheory.SimplicialObject.Augmented.rightOp_hom_app, 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, leftOpRightOpEquiv_functor_map_app, 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.Join.InclRightCompRightOpOpEquivFunctor_hom_app, CategoryTheory.Limits.preservesLimits_rightOp, AlgebraicGeometry.Scheme.AffineZariskiSite.PreservesLocalization.isOpenImmersion, IsRightAdjoint.rightOp, AlgebraicGeometry.Scheme.AffineZariskiSite.restrictIsoSpec_inv_app, 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	90 math math: CategoryTheory.Limits.limitUnopIsoUnopColimit_hom_comp_ι, CategoryTheory.NatTrans.unop_whiskerLeft, opUnopIso_hom_app, unopOpIso_inv_app, CategoryTheory.NatIso.unop_rightUnitor, CategoryTheory.NatTrans.unop_comp_assoc, CategoryTheory.NatTrans.unop_whiskerLeft_assoc, opInv_obj, 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.NatIso.unop_trans, unop_additive, CategoryTheory.NatTrans.removeUnop_app, CategoryTheory.Limits.preservesColimitsOfSize_unop, CategoryTheory.Limits.colimitYonedaHomIsoLimit_π_apply, CategoryTheory.Limits.coneOfCoconeUnop_π, CategoryTheory.Limits.isLimitConeOfCoconeUnop_lift, CategoryTheory.Limits.isColimitOfConeUnopOfCocone_desc, CategoryTheory.cosimplicialSimplicialEquiv_inverse_obj, 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.preservesLimitsOfShape_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.preservesColimits_unop, unop_obj, CategoryTheory.NatIso.unop_whiskerRight, CategoryTheory.Limits.hasLimit_unop_of_hasColimit, CategoryTheory.cosimplicialSimplicialEquiv_unitIso_hom_app, unop_map, opInv_map, CategoryTheory.NatIso.unop_leftUnitor, CategoryTheory.NatIso.unop_associator, CategoryTheory.NatTrans.unop_id, CategoryTheory.Limits.preservesFiniteProducts_unop, CategoryTheory.Limits.isLimitOfCoconeOfConeUnop_lift, CategoryTheory.cosimplicialSimplicialEquiv_unitIso_inv_app, CategoryTheory.Limits.preservesLimitsOfSize_unop, CategoryTheory.Limits.coconeUnopOfCone_pt, IsLocalization.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.coconeOfConeUnop_ι, CategoryTheory.NatTrans.removeUnop_id, CategoryTheory.Equivalence.unop_inverse, 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.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.limitUnopIsoUnopColimit_inv_comp_π, CategoryTheory.NatTrans.unop_app, CategoryTheory.Limits.coneOfCoconeUnop_pt, unopComp_hom_app, 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	32 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.Limits.coconeEquivalenceOpConeOp_counitIso, 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.TwoSquare.structuredArrowRightwardsOpEquivalence.functor_obj_left_right, CategoryTheory.Functor.sheafPushforwardContinuousNatTrans_app_val, 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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