hom 📖 | CompOp | 1589 mathmath: TopCat.isInducing_pullback_to_prod, AlgebraicGeometry.Scheme.toSpecΓ_apply, Sequential.homeoOfIso_apply, TopCat.binaryCofan_isColimit_iff, CategoryTheory.ShortComplex.pOpcycles_comp_moduleCatOpcyclesIso_hom_apply, Representation.repOfTprodIso_inv_apply, AlgCat.hom_inv_apply, smoothSheafCommRing.ι_forgetStalk_inv, CategoryTheory.GrothendieckTopology.Plus.toPlus_mk, AddMagmaCat.coe_id, AlgebraicGeometry.QuasiCompact.isCompact_preimage_singleton, DistLat.hom_inv_apply, AlgebraicGeometry.Scheme.fromSpecStalk_closedPoint, BoolAlg.coe_id, CommAlgCat.comp_apply, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_injective, CategoryTheory.PreGaloisCategory.mulAction_def, CommRingCat.KaehlerDifferential.map_d, ModuleCat.MonoidalCategory.braiding_hom_apply, AlgebraicGeometry.LocallyRingedSpace.stalkMap_hom_inv_assoc, ModuleCat.restrictScalars.map_apply, GrpWithZero.forget_map, CommSemiRingCat.hom_inv_apply, TopCat.pullbackIsoProdSubtype_hom_fst, CommRingCat.HomTopology.mvPolynomialHomeomorph_apply_snd, AlgebraicGeometry.LocallyRingedSpace.residueFieldMap_comp, TopologicalSpace.Opens.map_coe, AlgebraicGeometry.Scheme.Spec_map_stalkMap_fromSpecStalk, AddGrpCat.forget₂_map, AlgebraicGeometry.Scheme.Cover.LocallyDirected.directed, groupCohomology.isoCocycles₁_hom_comp_i_apply, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.stalk_iso, AddCommGrpCat.biproductIsoPi_inv_comp_π_apply, AddCommGrpCat.comp_apply, AddGrpCat.id_apply, AlgebraicGeometry.QuasiCompact.isCompact_preimage, Lat.ext_iff, TopCat.GlueData.preimage_image_eq_image, AlgebraicGeometry.RingedSpace.basicOpen_res, AlgebraicGeometry.Scheme.Hom.denseRange, AlgebraicGeometry.Scheme.isEmpty_pullback_iff, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv, TopCat.GlueData.π_surjective, UniformSpaceCat.completionHom_val, CommRingCat.closure_range_union_range_eq_top_of_isPushout, groupHomology.d₁₀_single_one, DeltaGenerated.topToDeltaGenerated_map_hom_hom_apply, AlgebraicGeometry.Scheme.Hom.closePoints_subset_preimage_closedPoints, TopCat.isOpenEmbedding_iff_comp_isIso, LinOrd.ofHom_apply, groupHomology.coinvariantsMk_comp_H0Iso_inv_apply, HomologicalComplex.Hom.comm_from_apply, AlgebraicGeometry.coprodSpec_apply, AlgebraicGeometry.morphismRestrict_base_coe, AlgebraicGeometry.Scheme.Hom.fiberHomeo_apply, AlgebraicGeometry.surjectiveOnStalks_iff, AddMonCat.comp_apply, TopCat.Presheaf.locally_surjective_iff_surjective_on_stalks, Rep.diagonalSuccIsoFree_inv_hom_single, Frm.coe_comp, Representation.repOfTprodIso_apply, AlgebraicGeometry.Scheme.affineBasisCover_is_basis, ModuleCat.freeHomEquiv_apply, CompactlyGenerated.homeoOfIso_apply, CategoryTheory.ShortComplex.moduleCatCyclesIso_hom_i_apply, AlgebraicGeometry.range_eq_range_of_surjective, AlgebraicGeometry.LocallyRingedSpace.notMem_prime_iff_unit_in_stalk, groupCohomology.toCocycles_comp_isoCocycles₂_hom_apply, AlgebraicGeometry.Scheme.Cover.isOpenMap_fromGlued, TopCat.Presheaf.stalkSpecializes_stalkPushforward, AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_specStalkEquiv, groupHomology.chainsMap_f_3_comp_chainsIso₃_apply, CategoryTheory.NatTrans.naturality_apply, AlgebraicGeometry.Scheme.Pullback.Triplet.ofPoint_x, TopCat.GlueData.open_image_open, LightCondensed.forget_obj_val_map, MonCat.forget_map, groupHomology.d₃₂_single, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_eq_iff', ModuleCat.mkOfSMul'_smul, GrpCat.surjective_of_epi, ModuleCat.extendScalarsId_hom_app_one_tmul, AlgebraicGeometry.Scheme.Hom.stalkMap_congr_point, AlgebraicGeometry.ValuativeCriterion.Existence.specializingMap, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_fromSpec, groupHomology.mapCycles₁_comp_apply, groupHomology.cyclesIso₀_inv_comp_iCycles_apply, SemiRingCat.inv_hom_apply, AlgebraicGeometry.LocallyRingedSpace.basicOpen_eq_bot_iff_forall_evaluation_eq_zero, AlgebraicGeometry.SheafedSpace.colimit_exists_rep, SemiRingCat.comp_apply, CategoryTheory.PreGaloisCategory.endEquivSectionsFibers_π, AlgebraicGeometry.Scheme.Hom.range_fiberι, TopologicalSpace.Opens.coe_inclusion', ModuleCat.ι_coprodIsoDirectSum_hom_apply, surjective_of_epi_of_preservesPushout, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.app_invApp, TopologicalSpace.OpenNhds.inclusionMapIso_hom, ModuleCat.restrictScalarsComp'App_hom_apply, AlgebraicGeometry.Scheme.Cover.iUnion_range, PresheafOfModules.epi_iff_surjective, AlgebraicGeometry.Scheme.mem_basicOpen', CommAlgCat.inv_hom_apply, LightCondensed.ihomPoints_symm_comp, AlgebraicGeometry.Scheme.Hom.mem_opensRange, ModuleCat.cokernel_π_cokernelIsoRangeQuotient_hom_apply, Sequential.homeoOfIso_symm_apply, CategoryTheory.ShortComplex.ShortExact.ab_surjective_g, AlgebraicGeometry.isCompact_iff_exists, CommRingCat.id_apply, ModuleCat.AlternatingMap.postcomp_apply, AlgebraicGeometry.HasRingHomProperty.stalkMap, CommGrpCat.epi_iff_surjective, ModuleCat.linearIndependent_shortExact, Rep.diagonalSuccIsoTensorTrivial_inv_hom_single_left, MonCat.ext_iff, ModuleCat.monoidalClosed_uncurry, TopCat.uliftFunctorObjHomeo_symm_naturality_apply, AddCommMonCat.coe_id, Rep.diagonalHomEquiv_symm_apply, SemiNormedGrp.explicitCokernelπ_desc_apply, CategoryTheory.Abelian.Ext.preadditiveYoneda_homologySequenceδ_singleTriangle_apply, CategoryTheory.Presheaf.imageSieve_app, smoothSheafCommRing.ι_evalHom_apply, TopCat.Presheaf.germ_exist_of_isBasis, SemimoduleCat.MonoidalCategory.rightUnitor_hom_apply, RingCat.Colimits.cocone_naturality_components, CompHausLike.coe_id, CategoryTheory.Limits.kernelSubobject_arrow'_apply, groupCohomology.coe_mapCocycles₁, AlgebraicGeometry.Scheme.OpenCover.finiteSubcover_f, AlgebraicGeometry.Scheme.stalkSpecializes_stalkMap, AlgebraicGeometry.Scheme.Modules.germ_restrictStalkNatIso_inv_app, AddCommMonCat.ofHom_apply, FunctorToFintypeCat.naturality, AlgebraicGeometry.SpecMap_preimage_basicOpen, CommRingCat.mkUnder_ext_iff, CommGrpCat.forget_map, HomologicalComplex.Hom.comm_to_apply, AlgebraicGeometry.Scheme.stalkMap_congr_assoc, smoothSheafCommRing.eval_germ, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_surjective, PresheafOfModules.pushforward_map_app_apply, AlgebraicGeometry.instIsGermInjectiveAtCoeContinuousMapCarrierCarrierCommRingCatHomTopCatBaseOfIsOpenImmersion, LightCondensed.finYoneda_map, BoolAlg.ext_iff, AlgebraicGeometry.Scheme.Pullback.SpecTensorTo_SpecOfPoint, AlgebraicGeometry.Scheme.hom_inv_apply, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.invApp_app_apply, AlgebraicGeometry.Scheme.Hom.range_asFiberHom, PresheafOfModules.sections_property, TopologicalSpace.Opens.map_obj, AlgCat.inv_hom_apply, AlgebraicGeometry.Scheme.Hom.tendsto_cofinite_cofinite, AlgebraicGeometry.localRingHom_comp_stalkIso_apply, PresheafOfModules.toSheafify_app_apply', AlgebraicGeometry.Scheme.Hom.stalkSpecializes_stalkMap, AlgebraicGeometry.Scheme.Hom.coe_resLE_base, AlgebraicGeometry.IsAffineOpen.fromSpecStalk_closedPoint, TopCat.isOpenEmbedding_iff, AlgebraicGeometry.IsClosedImmersion.base_closed, AlgebraicGeometry.quasiCompact_iff, Profinite.effectiveEpi_tfae, PresheafOfModules.Derivation.d_map, TopologicalSpace.OpenNhds.inclusionMapIso_inv, TopCat.Presheaf.germ_stalkSpecializes_apply, CommRingCat.free_map_coe, TopCat.range_prod_map, Rep.diagonalSuccIsoTensorTrivial_inv_hom_single_right, AddMonCat.ext_iff, groupCohomology.cocycles₂.d₂₃_apply, SemiNormedGrp.coe_comp, ModuleCat.extendRestrictScalarsAdj_homEquiv_apply, Lat.coe_id, groupHomology.d₃₂_single_one_thd, AlgebraicGeometry.Scheme.Hom.stalkMap_congr_hom_assoc, CategoryTheory.Limits.Concrete.colimit_rep_eq_iff_exists, AlgebraicGeometry.Scheme.Pullback.Triplet.isPullback_SpecMap_tensor, AlgebraicGeometry.Scheme.forget_map, TopCat.GlueData.isOpen_iff, AlgebraicGeometry.sigmaMk_mk, groupHomology.isoCycles₁_inv_comp_iCycles_apply, FinBddDistLat.ofHom_apply, AlgebraicGeometry.Scheme.Pullback.Triplet.specTensorTo_base_snd, RingCat.forget₂_map, CommBialgCat.id_apply, CategoryTheory.Limits.kernel.condition_apply, Preord.coe_id, groupCohomology.map_H0Iso_hom_f_apply, AlgebraicGeometry.LocallyRingedSpace.stalkMap_congr_point_assoc, groupCohomology.d₁₂_comp_d₂₃_apply, groupCohomology.H0IsoOfIsTrivial_inv_apply, CategoryTheory.Limits.Concrete.multiequalizerEquiv_apply, AlgebraicGeometry.Scheme.isoSpec_inv_preimage_zeroLocus, PresheafOfModules.congr_map_apply, PresheafOfModules.freeYonedaEquiv_symm_app, groupCohomology.eq_d₂₃_comp_inv_apply, groupCohomology.eq_d₁₂_comp_inv_apply, PresheafOfModules.restrictScalarsObj_map, AlgebraicGeometry.IsPreimmersion.isEmbedding, AddCommGrpCat.neg_hom_apply, groupHomology.chains₁ToCoinvariantsKer_surjective, TopCat.coequalizer_isOpen_iff, Preord.ext_iff, RingCat.inv_hom_apply, TopCat.GlueData.ι_fromOpenSubsetsGlue_apply, CommGrpCat.ext_iff, AlgebraicGeometry.Scheme.Hom.SpecMap_residueFieldMap_fromSpecResidueField_assoc, comp_apply, Profinite.NobelingProof.GoodProducts.linearIndependent_comp_of_eval, Rep.diagonalSuccIsoTensorTrivial_inv_hom_single_single, AlgebraicGeometry.Scheme.exists_germ_injective, Preord.inv_hom_apply, CommGrpCat.coe_comp, SemiRingCat.FilteredColimits.colimitCoconeIsColimit.descMonoidHom_quotMk, CategoryTheory.Limits.Concrete.productEquiv_symm_apply_π, AlgebraicGeometry.IsOpenImmersion.iff_isIso_stalkMap, AlgebraicGeometry.Scheme.Cover.instIsIsoCommRingCatStalkMapFromGlued, ModuleCat.ExtendRestrictScalarsAdj.HomEquiv.toRestrictScalars_hom_apply, LightProfinite.proj_comp_transitionMapLE', Semigrp.hom_inv_apply, TopCat.Presheaf.germ_res_apply, AlgebraicGeometry.Scheme.Pullback.carrierEquiv_eq_iff, Condensed.epi_iff_surjective_on_stonean, AlgebraicGeometry.Scheme.Hom.toImage_app_injective, CommGrpCat.hom_inv_apply, BddOrd.coe_id, TopCat.pullback_topology, TopCat.Presheaf.germ_stalkPullbackInv_assoc, AlgebraicGeometry.SpecToEquivOfLocalRing_apply_snd_coe, Preord.hom_inv_apply, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.biprodAddEquiv_symm_biprodIsoProd_hom_toBiprod_apply, Semigrp.inv_hom_apply, AlgebraicGeometry.Scheme.exists_affine_mem_range_and_range_subset, Rep.FiniteCyclicGroup.groupHomologyπOdd_eq_zero_iff, AddCommGrpCat.coe_comp, SemimoduleCat.MonoidalCategory.tensorHom_tmul, groupCohomology.cochainsMap_f_1_comp_cochainsIso₁_apply, groupCohomology.mapCocycles₂_comp_i_apply, NonemptyFinLinOrd.epi_iff_surjective, CommAlgCat.ofHom_apply, groupCohomology.cocyclesMap_comp_isoCocycles₂_hom_apply, CommMonCat.coe_comp, AlgebraicGeometry.IsOpenImmersion.isOpen_range, AlgebraicGeometry.Scheme.affineOpenCover_idx, ModuleCat.MonoidalCategory.associator_hom_apply, groupCohomology.cochainsMap_f_3_comp_cochainsIso₃_apply, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.stalk_iso, Rep.norm_comm_apply, AlgebraicGeometry.LocallyRingedSpace.iso_inv_base_hom_base_apply, AlgebraicGeometry.Scheme.Pullback.Triplet.snd_SpecTensorTo_apply, LinOrd.hom_inv_apply, groupHomology.cycles₁IsoOfIsTrivial_hom_apply, BddDistLat.coe_id, AddCommGrpCat.forget₂_map, GrpWithZero.hom_comp, CategoryTheory.hom_id, TopCat.pullbackFst_apply, TwoP.swapEquiv_functor_map_hom_toFun, groupCohomology.cocyclesMap_cocyclesIso₀_hom_f_apply, AddSemigrp.coe_id, Lat.forget_map, AlgebraicGeometry.PresheafedSpace.stalkMap.stalkSpecializes_stalkMap, ModuleCat.MonoidalCategory.tensorHom_tmul, GrpWithZero.coe_id, groupHomology.d₂₁_single_inv_mul_ρ_add_single, FinBddDistLat.hom_inv_apply, AlgCat.forget_map, TopCat.colimit_topology, Profinite.epi_iff_surjective, AlgebraicGeometry.Scheme.presieve₀_mem_precoverage_iff, RingCat.forget_map_apply, TopologicalSpace.Opens.isOpenEmbedding_obj_top, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_appTop_coord, PresheafOfModules.sectionsMap_coe, AddMonCat.FilteredColimits.colimit_add_mk_eq, CategoryTheory.ShortComplex.abToCycles_apply_coe, groupHomology.d₁₀_comp_coinvariantsMk_apply, AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToΓ_ΓToStalk, AlgebraicGeometry.isIso_iff_isIso_stalkMap, AlgebraicGeometry.LocallyRingedSpace.stalkMap_inv_hom, Rep.diagonalSuccIsoFree_inv_hom_single_single, BddOrd.ext_iff, TopCat.isOpenEmbedding_iff_isIso_comp', groupCohomology.H1IsoOfIsTrivial_inv_apply, AddGrpCat.FilteredColimits.colimit_add_mk_eq, PresheafOfModules.map_comp_apply, SemiNormedGrp.ext_iff, AlgebraicGeometry.Scheme.Pullback.residueFieldCongr_inv_residueFieldMap_ofPoint, ModuleCat.biprodIsoProd_inv_comp_snd_apply, AlgebraicGeometry.IsOpenImmersion.range_pullbackSnd, PresheafOfModules.pushforward_map_app_apply', CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_jointly_surjective, UniformSpaceCat.hom_ext_iff, AlgebraicGeometry.Scheme.Hom.isDiscrete_preimage_singleton, TopCat.pullbackSnd_apply, AlgebraicGeometry.Scheme.IdealSheafData.range_glueDataObjι_ι_eq_support_inter, LightProfinite.isClosedMap, AlgebraicGeometry.LocallyRingedSpace.stalkMap_congr, AlgebraicGeometry.IsOpenImmersion.range_pullback_fst_of_right, groupCohomology.H2π_comp_map_apply, CommSemiRingCat.ofHom_apply, AlgebraicGeometry.Scheme.Hom.coe_resLE_apply, PartOrd.coe_id, Preord.ofHom_apply, SemimoduleCat.MonoidalCategory.whiskerRight_apply, AlgebraicGeometry.pointEquivClosedPoint_apply_coe, CategoryTheory.congr_arg, CategoryTheory.GrothendieckTopology.Plus.toPlus_eq_mk, FinBddDistLat.ext_iff, CategoryTheory.GrothendieckTopology.Plus.res_mk_eq_mk_pullback, TopCat.isIso_iff_isHomeomorph, AlgebraicGeometry.RingedSpace.mem_top_basicOpen, Lat.comp_apply, CategoryTheory.ShortComplex.moduleCat_zero_apply, AlgebraicGeometry.Scheme.Opens.germ_stalkIso_inv, AlgebraicGeometry.Scheme.range_fromSpecStalk, TopCat.prod_topology, HeytAlg.coe_comp, AlgebraicGeometry.stalkToFiberRingHom_homogeneousLocalizationToStalk, CondensedSet.topCatAdjunctionCounit_bijective, Frm.ext_iff, GrpWithZero.coe_comp, CategoryTheory.Abelian.Ext.preadditiveCoyoneda_homologySequenceδ_singleTriangle_apply, AlgebraicGeometry.coprodMk_inl, PartOrdEmb.comp_apply, AlgebraicGeometry.Scheme.Pullback.Triplet.hy, AlgebraicGeometry.LocallyRingedSpace.stalkMap_inv_hom_assoc, AlgCat.id_apply, TopCat.sigmaIsoSigma_inv_apply, ModuleCat.exteriorPower.desc_mk, PresheafOfModules.unit_map_one, CompHausLike.homeoOfIso_symm_apply, Condensed.finYoneda_map, AlgebraicGeometry.Scheme.Pullback.carrierEquiv_symm_fst, MonCat.FilteredColimits.M.map_mk, SemimoduleCat.MonoidalCategory.rightUnitor_inv_apply, AddMonCat.forget_map, CommAlgCat.algEquivOfIso_apply, PartOrdEmb.hom_inv_apply, AlgebraicGeometry.Scheme.Hom.asFiberHom_fiberι_assoc, DistLat.ext_iff, CategoryTheory.PreGaloisCategory.toAut_surjective_isGalois_finite_family, AlgebraicGeometry.Scheme.preimage_basicOpen_top, CompHausLike.isClosedMap, AddCommGrpCat.ext_iff, CategoryTheory.PreGaloisCategory.toAut_hom_app_apply, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_naturality_apply, LinOrd.inv_hom_apply, AlgebraicGeometry.PresheafedSpace.GlueData.pullback_base, CategoryTheory.Limits.Concrete.isColimit_rep_eq_iff_exists, AlgebraicGeometry.Scheme.Hom.app_appIso_inv_assoc, CategoryTheory.Adjunction.compPreadditiveYonedaIso_inv_app_app_apply, AlgebraicGeometry.IsAffineOpen.basicOpen_fromSpec_app, FintypeCat.toProfinite_map_hom_hom_apply, TopCat.pullbackIsoProdSubtype_inv_fst_assoc, Rep.coindVEquiv_symm_apply_coe, AlgebraicGeometry.Scheme.Pullback.range_fst_comp, TopologicalSpace.OpenNhds.op_map_id_obj, FinBddDistLat.coe_id, CategoryTheory.Limits.Concrete.prodEquiv_apply_fst, CategoryTheory.Limits.kernel.lift_ι_apply, AlgebraicGeometry.Scheme.isoSpec_inv_image_zeroLocus, AlgebraicGeometry.Scheme.IdealSheafData.range_glueDataObjι_ι, AlgebraicGeometry.Scheme.germToFunctionField_injective, CategoryTheory.Limits.FintypeCat.productEquiv_apply, groupCohomology.H1IsoOfIsTrivial_H1π_apply_apply, AlgebraicGeometry.Scheme.Modules.germ_restrictStalkNatIso_hom_app, ModuleCat.imageIsoRange_inv_image_ι_apply, CategoryTheory.Iso.toAlgEquiv_symm_apply, FintypeCat.comp_apply, PresheafOfModules.map_smul, FGModuleCat.FGModuleCatEvaluation_apply, ModuleCat.epi_iff_surjective, AddCommGrpCat.free_map_coe, FintypeCat.uSwitchEquiv_naturality, AlgebraicGeometry.morphismRestrict_base, CategoryTheory.ShortComplex.ShortExact.ab_injective_f, AlgebraicGeometry.Scheme.Opens.ι_image_basicOpen_topIso_inv, PresheafOfModules.Monoidal.tensorObj_map_tmul, CommAlgCat.algEquivOfIso_symm_apply, ModuleCat.exteriorPower.map_mk, FintypeCat.equivEquivIso_symm_apply_symm_apply, CategoryTheory.PreGaloisCategory.evaluation_injective_of_isConnected, FinBddDistLat.inv_hom_apply, AlgebraicGeometry.Scheme.toSpecΓ_preimage_zeroLocus, LightProfinite.effectiveEpi_iff_surjective, AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open, SimplexCategory.toTop_map, ModuleCat.id_apply, AlgebraicGeometry.Scheme.Hom.quasiFiniteAt_comp_iff_of_isOpenImmersion, Rep.FiniteCyclicGroup.groupCohomologyπOdd_eq_iff, TopologicalSpace.Opens.inclusion'_top_functor, AlgebraicGeometry.Scheme.Cover.isOpenEmbedding_fromGlued, AlgebraicGeometry.Scheme.Hom.app_surjective, SemiNormedGrp.zero_apply, AlgebraicGeometry.coe_opensRestrict_symm_apply, groupCohomology.d₁₂_apply_mem_cocycles₂, AddMonCat.coe_comp, ModuleCat.hom_inv_apply, ModuleCat.monoidalClosed_curry, groupCohomology.d₀₁_apply_mem_cocycles₁, AlgebraicGeometry.Scheme.Hom.isCompact_preimage_singleton, AddMagmaCat.forget_map, AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal_ideal, ModuleCat.MonoidalCategory.leftUnitor_hom_apply, AlgebraicGeometry.Scheme.Pullback.Triplet.Spec_map_tensor_isPullback, ModuleCat.exteriorPower.iso₀_hom_apply, FintypeCat.hom_ext_iff, groupHomology.cyclesMap_comp_cyclesIso₀_hom_apply, Rep.applyAsHom_comm_apply, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.invApp_app_apply, groupHomology.d₂₁_single_inv_self_ρ_sub_self_inv, AlgebraicGeometry.Scheme.Cover.covers, epi_iff_surjective_of_preservesPushout, groupHomology.chainsMap_f_single, CategoryTheory.yonedaMon_naturality_assoc, ModuleCat.restrictScalarsId'App_hom_apply, groupCohomology.subtype_comp_d₀₁_apply, SheafOfModules.pushforwardComp_inv_app_val_app, AlgebraicGeometry.Scheme.Hom.stalkMap_congr_point_assoc, AddMagmaCat.ofHom_apply, groupCohomology.H2π_eq_iff, TopologicalSpace.Opens.map_functor_eq, AlgebraicGeometry.Scheme.stalkMap_congr_point_assoc, AlgebraicGeometry.Scheme.GlueData.isOpen_iff, AlgebraicGeometry.LocallyRingedSpace.stalkMap_congr_assoc, PartOrdEmb.Limits.CoconePt.fac_apply, AlgebraicGeometry.QuasiCompactCover.isCompactOpenCovered_of_isCompact, HeytAlg.id_apply, AddCommGrpCat.biprodIsoProd_inv_comp_snd_apply, groupHomology.toCycles_comp_isoCycles₁_hom_apply, AddCommMonCat.comp_apply, FinBddDistLat.forget_map, CategoryTheory.ShortComplex.exact_iff_of_hasForget, FintypeCat.id_apply, AlgebraicGeometry.PresheafedSpace.stalkMap.stalkSpecializes_stalkMap_apply, SimplexCategory.toTop₀_map, AlgebraicGeometry.Scheme.Hom.mem_preimage, AlgebraicGeometry.Scheme.Spec_map_stalkMap_fromSpecStalk_assoc, CommBialgCat.bialgEquivOfIso_apply, AlgebraicGeometry.Scheme.Hom.isConstructible_preimage, AlgebraicGeometry.LocallyRingedSpace.isLocalHomStalkMap, AlgebraicGeometry.IsImmersion.isLocallyClosed_range, AlgebraicGeometry.Scheme.Γevaluation_naturality_apply, SemiNormedGrp.id_apply, CompHausLike.finiteCoproduct.isOpenEmbedding_ι, CategoryTheory.ShortComplex.moduleCat_pOpcycles_eq_zero_iff, PresheafOfModules.mono_iff_surjective, SemimoduleCat.MonoidalCategory.tensorLift_tmul, CategoryTheory.ShortComplex.ab_exact_iff_function_exact, CompHausLike.Sigma.isOpenEmbedding_ι, AlgebraicGeometry.ExistsHomHomCompEqCompAux.exists_index, Semigrp.id_apply, DistLat.forget_map, Rep.indToCoindAux_comm, TopCat.range_pullback_map, CategoryTheory.yonedaMon_naturality, CategoryTheory.Limits.FintypeCat.productEquiv_symm_comp_π_apply, AlgebraicGeometry.Scheme.Cover.exists_lift_trans_eq, Rep.FiniteCyclicGroup.groupCohomologyπEven_eq_zero_iff, AlgebraicGeometry.Spec_map_localization_isIso, LinOrd.id_apply, AlgebraicGeometry.Scheme.stalkClosedPointTo_fromSpecStalk, BddDistLat.coe_comp, AlgebraicGeometry.Scheme.coe_homeoOfIso_symm, AlgebraicGeometry.ProjectiveSpectrum.Proj.isLocalization_atPrime, AlgebraicGeometry.Scheme.stalkMap_hom_inv_assoc, AddGrpCat.zero_apply, groupHomology.cyclesIso₀_comp_H0π_apply, CategoryTheory.ShortComplex.moduleCatCyclesIso_inv_π_assoc_apply, groupHomology.eq_d₃₂_comp_inv_apply, AlgebraicGeometry.Scheme.IdealSheafData.range_subschemeι, CondensedMod.epi_iff_surjective_on_stonean, AlgebraicGeometry.PresheafedSpace.GlueData.ι_image_preimage_eq, AlgebraicGeometry.PresheafedSpace.stalkMap.congr, AlgebraicGeometry.Scheme.Hom.isClosedMap, AlgebraicGeometry.Scheme.Hom.isLocallyClosed_range, AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_stalkMap_toSpec_assoc, TopCat.uliftFunctorObjHomeo_naturality_apply, CategoryTheory.GlueData.glue_condition_apply, TopCat.homeoOfIso_apply, AlgebraicGeometry.morphismRestrict_appLE, Stonean.epi_iff_surjective, AlgebraicGeometry.IsClosedImmersion.iff_isPreimmersion, AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_stalkMap_toSpec, CategoryTheory.PreGaloisCategory.fiberBinaryProductEquiv_symm_fst_apply, AlgebraicGeometry.Scheme.Hom.apply_mem_image_iff, AlgebraicGeometry.Scheme.Hom.asFiberHom_apply, CategoryTheory.Limits.Concrete.isColimit_exists_rep, ModuleCat.inv_hom_apply, SemiNormedGrp.completion.norm_incl_eq, TopCat.Presheaf.germToPullbackStalk_stalkPullbackHom, groupHomology.mapCycles₁_id_comp_apply, AlgebraicGeometry.genericPoint_eq_of_isOpenImmersion, AlgebraicGeometry.Scheme.homeoOfIso_apply, CommGrpCat.mono_iff_injective, PresheafOfModules.presheaf_map_apply_coe, AlgebraicGeometry.Scheme.Hom.preimage_basicOpen_top, Preord.id_apply, CommRingCat.forget_map, AlgebraicGeometry.Scheme.Opens.germ_stalkIso_inv_assoc, SemiRingCat.FilteredColimits.colimitCoconeIsColimit.descAddMonoidHom_quotMk, AlgebraicGeometry.formallySmooth_stalkMap_iff, TopCat.Sheaf.objSupIsoProdEqLocus_hom_fst, TopCat.nonempty_isLimit_iff_eq_induced, AlgebraicGeometry.isIso_stalkMap_coprodSpec, PartOrdEmb.ofHom_apply, smoothSheafCommRing.evalHom_germ, AlgebraicGeometry.LocallyRingedSpace.residue_comp_residueFieldMap_eq_stalkMap_comp_residue, CategoryTheory.PreGaloisCategory.toAut_surjective_isGalois, AlgebraicGeometry.exists_appTop_π_eq_of_isAffine_of_isLimit, AlgebraicGeometry.Scheme.instIsIsoCommRingCatResidueFieldMapOfIsOpenImmersion, AlgebraicGeometry.Scheme.instIsOverMapResidueFieldMapOverInferInstanceOverClass, CategoryTheory.congr_fun, ModuleCat.mono_iff_injective, CategoryTheory.Presheaf.app_localPreimage, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.mk₀_f_comp_biprodAddEquiv_symm_biprodIsoProd_hom, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.image_preimage_is_empty, HeytAlg.hom_inv_apply, Action.FintypeCat.toEndHom_apply, SemimoduleCat.hom_inv_apply, groupHomology.pOpcycles_comp_opcyclesIso_hom_apply, PresheafOfModules.forgetToPresheafModuleCatObjMap_apply, groupHomology.mapCycles₁_comp_i_apply, MagmaCat.coe_comp, SheafOfModules.pushforwardCongr_hom_app_val_app, Rep.FiniteCyclicGroup.groupHomologyπEven_eq_iff, ModuleCat.comp_apply, TopCat.id_app, AddMonCat.coe_id, ModuleCat.restrictScalarsCongr_hom_app, AlgebraicGeometry.Scheme.Γevaluation_naturality_assoc, AlgebraicGeometry.Scheme.Pullback.carrierEquiv_symm_snd, ModuleCat.kernelIsoKer_inv_kernel_ι_apply, HomologicalComplex.alternatingConst_iCycles_even_comp_apply, groupHomology.cyclesIso₀_inv_comp_cyclesMap_apply, AddCommMonCat.neg_hom_apply, AlgebraicGeometry.isCompl_range_inl_inr, ModuleCat.MonoidalCategory.rightUnitor_hom_apply, LightCondensed.isoFinYonedaComponents_inv_comp, AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap_app, ModuleCat.FilteredColimits.M.mk_map, groupCohomology.π_comp_H0Iso_hom_apply, CommMonCat.forget_map, AddMonCat.FilteredColimits.M.map_mk, groupHomology.coe_mapCycles₂, TopCat.piIsoPi_inv_π_apply, AlgebraicGeometry.Scheme.Pullback.range_map, DistLat.id_apply, injective_of_mono_of_preservesPullback, AlgebraicGeometry.ProjIsoSpecTopComponent.fromSpec_toSpec, groupCohomology.isoCocycles₁_inv_comp_iCocycles_apply, AlgebraicGeometry.SurjectiveOnStalks.isEmbedding_pullback, AlgebraicGeometry.AffineSpace.SpecIso_inv_appTop_coord, groupHomology.H1π_comp_map_apply, CommMonCat.comp_apply, AlgebraicGeometry.Scheme.Hom.isSpectralMap, AddMagmaCat.coe_comp, AlgebraicGeometry.mem_range_iff_of_surjective, FGModuleCat.FGModuleCatCoevaluation_apply_one, smoothSheafCommRing.forgetStalk_hom_comp_evalHom_assoc, TwoP.swap_map, TopCat.GlueData.f_open, AlgebraicGeometry.Scheme.Hom.isLocallyConstructible_image, BddDistLat.forget_map, AlgebraicGeometry.Scheme.evaluation_eq_zero_iff_notMem_basicOpen, AddMonCat.neg_hom_apply, AddGrpCat.ext_iff, AddSemigrp.comp_apply, CategoryTheory.PreGaloisCategory.evaluation_aut_bijective_of_isGalois, CategoryTheory.Pretriangulated.preadditiveYoneda_homologySequenceδ_apply, AlgebraicGeometry.LocallyRingedSpace.stalkMap_inv_hom_apply, AlgebraicGeometry.Scheme.stalkMap_congr_hom_assoc, AlgebraicGeometry.Scheme.Opens.germ_stalkIso_hom, PartOrd.coe_comp, groupHomology.π_comp_H0Iso_hom_apply, AlgebraicGeometry.Scheme.stalkSpecializes_stalkMap_apply, groupCohomology.eq_d₀₁_comp_inv_apply, ModuleCat.MonoidalCategory.tensorLift_tmul, CategoryTheory.Preadditive.epi_iff_surjective, AddCommMonCat.ext_iff, CommGrpCat.ofHom_apply, TopologicalSpace.OpenNhds.map_id_obj_unop, Rep.freeLiftLEquiv_apply, PresheafOfModules.surjective_of_epi, CategoryTheory.GlueData.t_inv_apply, SemimoduleCat.MonoidalCategory.tensorμ_apply, CommBialgCat.inv_hom_apply, GrpCat.hom_inv_apply, CategoryTheory.Preadditive.mono_iff_injective, AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap_app', CommRingCat.hom_inv_apply, TopCat.isOpenEmbedding_iff_comp_isIso', LightCondensed.forget_map_val_app, TopCat.isOpenEmbedding_f_zeroHypercover, TopologicalSpace.Opens.isOpenEmbedding, groupHomology.π_comp_H0IsoOfIsTrivial_hom_apply, AlgebraicGeometry.Scheme.AffineZariskiSite.coequifibered_iff_forall_isLocalizationAway, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_zeroLocus, ModuleCat.piIsoPi_inv_kernel_ι_apply, ModuleCat.MonModuleEquivalenceAlgebra.functor_map_hom_apply, CategoryTheory.Presheaf.imageSieve_apply, TopCat.GlueData.range_fromOpenSubsetsGlue, TopCat.coe_id, AlgebraicGeometry.Scheme.GlueData.ι_eq_iff, AlgebraicGeometry.FormallyUnramified.instIsSeparableCarrierResidueFieldCoeContinuousMapCarrierCarrierCommRingCatHomTopCatBaseOfLocallyOfFiniteType, AlgebraicGeometry.AffineSpace.comp_homOfVector, CondensedMod.hom_naturality_apply, FintypeCat.toLightProfinite_map_hom_hom_apply, FiniteAddGrp.ofHom_apply, AlgebraicGeometry.AffineSpace.isOpenMap_over, AlgebraicGeometry.surjective_iff, ModuleCat.lof_coprodIsoDirectSum_inv_apply, AlgebraicGeometry.germ_injective_of_isIntegral, HeytAlg.inv_hom_apply, ModuleCat.Derivation.desc_d, PresheafOfModules.injective_of_mono, AlgebraicGeometry.isClosedImmersion_iff_isAffineHom, Semigrp.ext_iff, ModuleCat.free_ε_one, Profinite.exists_isClopen_of_cofiltered, AlgebraicGeometry.AffineSpace.hom_ext_iff, groupCohomology.norm_ofAlgebraAutOnUnits_eq, UniformSpaceCat.coe_comp, AlgebraicGeometry.Scheme.SpecMap_stalkMap_fromSpecStalk_assoc, AlgebraicGeometry.Scheme.Hom.preimage_basicOpen, Semigrp.comp_apply, AlgebraicGeometry.Scheme.singleton_mem_precoverage_iff, Condensed.epi_iff_locallySurjective_on_compHaus, AlgebraicGeometry.Scheme.Hom.isOpenMap, CategoryTheory.PreGaloisCategory.autMulEquivAutGalois_symm_app, AlgEquiv.toUnder_inv_right_apply, LightCondensed.isLocallySurjective_iff_locallySurjective_on_lightProfinite, AlgebraicGeometry.Scheme.basicOpen_res_eq, PresheafOfModules.pushforward_obj_map_apply, AlgebraicGeometry.Scheme.Pullback.Triplet.hx, TopCat.GlueData.MkCore.t_inv, AlgebraicGeometry.Scheme.Hom.stalkFunctor_toImage_injective, AlgebraicGeometry.isOpenImmersion_iff_stalk, AlgebraicGeometry.Scheme.image_basicOpen, groupHomology.d₂₁_comp_d₁₀_apply, CategoryTheory.regularTopology.isLocallySurjective_iff, CategoryTheory.ShortComplex.moduleCatLeftHomologyData_liftK_hom, groupHomology.mapCycles₂_comp_apply, AlgebraicGeometry.LocallyRingedSpace.Hom.prop, AlgebraicGeometry.SurjectiveOnStalks.surj_on_stalks, AddCommMonCat.coe_comp, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.app_invApp_assoc, BddOrd.forget_map, Rep.coindFunctorIso_inv_app_hom_hom_apply_coe, CommMonCat.inv_hom_apply, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.app_invApp, FinBddDistLat.comp_apply, Rep.linearization_η_hom_apply, CommBialgCat.hom_inv_apply, TopModuleCat.hom_id, TopCat.isClosed_iff_of_isColimit, PartOrd.comp_apply, CommRingCat.toAlgHom_apply, AddSemigrp.neg_hom_apply, MonCat.comp_apply, AlgebraicGeometry.SurjectiveOnStalks.stalkMap_surjective, CategoryTheory.Sheaf.isLocallyInjective_iff_injective, CategoryTheory.Limits.imageSubobject_arrow_comp_apply, TopCat.pullbackIsoProdSubtype_inv_snd_apply, SemiNormedGrp.inv_hom_apply, AlgebraicGeometry.PresheafedSpace.stalkMap.isIso, CategoryTheory.Iso.toAlgEquiv_apply, CategoryTheory.Limits.colimit.w_apply, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_map_bijective, CategoryTheory.yonedaGrp_naturality_assoc, MagmaCat.id_apply, TopCat.GlueData.fromOpenSubsetsGlue_isOpenEmbedding, AlgebraicGeometry.Scheme.Hom.stalkMap_hom_inv, groupHomology.H2π_eq_iff, Frm.id_apply, AddSemigrp.hom_neg_apply, AlgebraicGeometry.Scheme.IdealSheafData.support_comap, TopologicalSpace.Opens.set_range_inclusion', AddCommGrpCat.zero_apply, AlgebraicGeometry.Scheme.stalkMap_congr_hom, groupHomology.H1AddEquivOfIsTrivial_single, AlgebraicGeometry.Scheme.Pullback.range_snd_comp, AddGrpCat.mono_iff_injective, CategoryTheory.Limits.kernelSubobject_arrow_comp_apply, CategoryTheory.Limits.Concrete.prodEquiv_symm_apply_snd, AddCommGrpCat.id_apply, RingCat.ofHom_apply, smoothSheafCommRing.ι_forgetStalk_inv_apply, PresheafOfModules.unitHomEquiv_apply_coe, PartOrdEmb.coe_comp, AlgebraicGeometry.LocallyRingedSpace.GlueData.ι_jointly_surjective, HeytAlg.comp_apply, ModuleCat.FreeMonoidal.εIso_inv_freeMk, groupHomology.isoCycles₂_hom_comp_i_apply, AddCommGrpCat.int_hom_ext_iff, MonCat.coe_comp, TopCat.Presheaf.isLocallySurjective_iff, groupCohomology.π_comp_H0IsoOfIsTrivial_hom_apply, AlgebraicGeometry.range_eq_univ, AddCommGrpCat.Colimits.Quot.desc_toCocone_desc_app, GrpCat.coe_comp, CommMonCat.coe_id, AlgebraicGeometry.SpecToEquivOfLocalRing_apply_fst, AlgebraicGeometry.Scheme.Hom.comp_apply, SheafOfModules.pushforwardCongr_inv_app_val_app, Frm.coe_id, groupCohomology.isoCocycles₂_inv_comp_iCocycles_apply, AlgebraicGeometry.Scheme.Hom.isOpenEmbedding, LightProfinite.surjective_transitionMapLE, TopCat.pullbackIsoProdSubtype_inv_fst, congr_arg, AlgebraicGeometry.Spec.map_apply, CommRingCat.Colimits.cocone_naturality_components, AlgebraicGeometry.pointOfClosedPoint_apply, AlgebraicGeometry.PresheafedSpace.GlueData.ι_isOpenEmbedding, CommAlgCat.hom_inv_apply, AlgebraicGeometry.Scheme.Hom.isEmbedding, ModuleCat.free_η_freeMk, AlgebraicGeometry.sigmaι_eq_iff, groupCohomology.cochainsMap_f_0_comp_cochainsIso₀_apply, AddCommGrpCat.epi_iff_surjective, FintypeCat.equivEquivIso_symm_apply_apply, CategoryTheory.Types.hom_eq_coe, SemimoduleCat.MonoidalCategory.braiding_hom_apply, groupHomology.inhomogeneousChains.d_single, CompHausLike.mono_iff_injective, ModuleCat.exteriorPower.iso₁_hom_apply, AlgebraicGeometry.Scheme.stalkMap_comp, CategoryTheory.id_apply, hom_injective, TopModuleCat.freeMap_map, SemimoduleCat.MonoidalCategory.braiding_inv_apply, AlgebraicGeometry.Scheme.residue_surjective, Preord.forget_map, SheafOfModules.pushforwardPushforwardAdj_unit_app_val_app, AlgebraicGeometry.IsOpenImmersion.range_pullback_to_base_of_left, SemiNormedGrp.explicitCokernelπ_apply_dom_eq_zero, TopCat.coinduced_of_isColimit, Lat.ofHom_apply, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_invApp, CategoryTheory.PreGaloisCategory.evaluationEquivOfIsGalois_apply, CategoryTheory.comp_apply', AlgebraicGeometry.Scheme.IdealSheafData.map_vanishingIdeal, Action.FintypeCat.quotientToEndHom_mk, AlgebraicGeometry.Scheme.Hom.stalkMap_inv_hom_assoc, CategoryTheory.extensiveTopology.isLocallySurjective_iff, smoothSheafCommRing.forgetStalk_hom_comp_evalHom, SemiNormedGrp.explicitCokernelπ_surjective, AddSemigrp.ofHom_apply, AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalHom, PartOrdEmb.inv_hom_apply, AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec_base_apply_eq, CategoryTheory.PreGaloisCategory.fiberBinaryProductEquiv_symm_snd_apply, groupCohomology.cochainsMap_f_2_comp_cochainsIso₂_apply, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_naturality_apply, TopCat.Presheaf.stalkFunctor_map_germ_apply, TopCat.Presheaf.germ_res_apply', MagmaCat.coe_id, AlgebraicGeometry.StructureSheaf.toStalk_stalkSpecializes_apply, BddDistLat.inv_hom_apply, Condensed.isoFinYonedaComponents_inv_comp, groupHomology.π_comp_H1Iso_hom_apply, AlgebraicGeometry.Scheme.Hom.residueFieldMap_congr, AlgebraicGeometry.Spec_zeroLocus_eq_zeroLocus, CommGrpCat.inv_hom_apply, TopCat.pullbackIsoProdSubtype_hom_snd, AddMonCat.id_apply, DistLat.ofHom_apply, AlgebraicGeometry.Scheme.Hom.map_mem_image_iff, CategoryTheory.Adjunction.compPreadditiveYonedaIso_hom_app_app_apply, groupCohomology.map_id_comp_H0Iso_hom_apply, SemimoduleCat.id_apply, forget₂_comp_apply, PresheafOfModules.toPresheaf_map_app_apply, RingHom.surjective_iff_epi_and_finite, AlgebraicGeometry.Scheme.map_basicOpen, CategoryTheory.ShortComplex.ShortExact.moduleCat_exact_iff_function_exact, TopologicalSpace.Opens.adjunction_counit_map_functor, AlgebraicGeometry.coprodSpec_coprodMk, AddCommGrpCat.mono_iff_injective, groupHomology.cyclesMap_comp_isoCycles₂_hom_apply, PresheafOfModules.Derivation'.d_app, CategoryTheory.ShortComplex.π_moduleCatCyclesIso_hom_assoc_apply, hom_ext_iff, AddMonCat.ofHom_apply, CategoryTheory.Iso.hom_inv_id_apply, AlgebraicGeometry.Scheme.Hom.injective, AddCommGrpCat.Colimits.Quot.ι_desc, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_w_apply, TopCat.Sheaf.objSupIsoProdEqLocus_inv_snd, CategoryTheory.Limits.Concrete.to_product_injective_of_isLimit, TopCat.pullbackIsoProdSubtype_inv_snd, AddCommGrpCat.Colimits.Quot.map_ι, ModuleCat.ExtendScalars.map_tmul, AddMagmaCat.ext_iff, ext_iff, smoothSheafCommRing.ι_forgetStalk_inv_assoc, PartOrd.inv_hom_apply, PartOrdEmb.id_apply, AlgebraicGeometry.Scheme.Hom.asFiberHom_fiberι, AddCommMonCat.hom_neg_apply, TopCat.Presheaf.germ_exist, hom_bijective, MagmaCat.ofHom_apply, TopCat.GlueData.fromOpenSubsetsGlue_injective, ModuleCat.free_μ_freeMk_tmul_freeMk, CategoryTheory.Iso.toLinearEquiv_apply, Frm.inv_hom_apply, TopologicalSpace.Opens.adjunction_counit_app_self, AlgebraicGeometry.LocallyOfFiniteType.stalkMap, ModuleCat.Derivation.d_map, AlgebraicGeometry.locallyQuasiFinite_iff_isDiscrete_preimage_singleton, AlgebraicGeometry.Scheme.IsLocallyDirected.ι_eq_ι_iff, CategoryTheory.forget₂_comp_apply, CategoryTheory.extensiveTopology.presheafIsLocallySurjective_iff, CommRingCat.ofHom_apply, SheafOfModules.pushforwardComp_hom_app_val_app, BddDistLat.ofHom_apply, BddOrd.id_apply, CategoryTheory.Limits.Concrete.prodEquiv_symm_apply_fst, LightCondSet.topCatAdjunctionCounit_bijective, CompactlyGenerated.homeoOfIso_symm_apply, CpltSepUniformSpace.hom_comp, AlgebraicGeometry.Scheme.Pullback.Triplet.specTensorTo_base_fst, AlgebraicGeometry.compactSpace_iff_exists, AddCommGrpCat.injective_of_mono, CategoryTheory.ShortComplex.ShortExact.injective_f, groupHomology.isoCycles₁_hom_comp_i_apply, GrpCat.forget₂_map, LinOrd.forget_map, AlgebraicGeometry.IsClosedImmersion.hasAffineProperty, ModuleCat.FreeMonoidal.μIso_hom_freeMk_tmul_freeMk, groupHomology.coinvariantsMk_comp_opcyclesIso₀_inv_apply, Mathlib.Tactic.Elementwise.hom_elementwise, BoolAlg.coe_comp, AlgebraicGeometry.Scheme.stalkMap_congr_point, AlgHom.toUnder_right, AlgebraicGeometry.ProjectiveSpectrum.Proj.mk_mem_toSpec_base_apply, ModuleCat.MonoidalCategory.rightUnitor_inv_apply, groupCohomology.H1π_eq_zero_iff, LinOrd.ext_iff, TopCat.range_pullback_to_prod, AlgebraicGeometry.IsAffineOpen.comap_primeIdealOf_appLE, TopCat.GlueData.ι_isOpenEmbedding, AlgebraicGeometry.Scheme.stalkMap_hom_inv, Profinite.NobelingProof.GoodProducts.square_commutes, Frm.hom_inv_apply, AlgebraicGeometry.Scheme.toSpecΓ_image_zeroLocus, groupHomology.d₃₂_single_one_fst, AlgebraicGeometry.LocallyRingedSpace.evaluation_naturality, CategoryTheory.Limits.colimit.ι_desc_apply, Rep.coind'_ext_iff, AlgebraicGeometry.isImmersion_iff, PresheafOfModules.DifferentialsConstruction.relativeDifferentials'_map_d, groupCohomology.cocyclesMap_comp_isoCocycles₁_hom_apply, CommGrpCat.coe_id, CommGrpCat.id_apply, TopCat.pullback_fst_image_snd_preimage, PresheafOfModules.Elements.fromFreeYoneda_app_apply, AlgEquiv.toUnder_hom_right_apply, groupHomology.d₂₁_single_self_inv_ρ_sub_inv_self, AlgebraicGeometry.IsAffineOpen.fromSpec_app_self_apply, AlgebraicGeometry.Scheme.stalkMap_hom_inv_apply, TopCat.continuous_iff_of_isColimit, CategoryTheory.ShortComplex.ab_exact_iff, groupHomology.H1ToTensorOfIsTrivial_H1π_single, CategoryTheory.ShortComplex.moduleCatCyclesIso_hom_i_assoc_apply, Rep.FiniteCyclicGroup.groupHomologyπEven_eq_zero_iff, groupCohomology.cocyclesMk₁_eq, lightProfiniteToLightCondSetIsoTopCatToLightCondSet_hom_app_val_app_apply, CategoryTheory.Limits.limit.lift_π_apply, AddCommGrpCat.coe_id, CommSemiRingCat.comp_apply, CommGrpCat.comp_apply, ModuleCat.MonoidalCategory.leftUnitor_inv_apply, AlgebraicGeometry.Scheme.basicOpen_appLE, HeytAlg.ofHom_apply, ModuleCat.MonModuleEquivalenceAlgebra.algebraMap, Compactum.continuous_of_hom, AlgebraicGeometry.Scheme.Hom.stalkSpecializes_stalkMap_apply, CommSemiRingCat.id_apply, PartOrdEmb.forget_map, AlgebraicGeometry.LocallyRingedSpace.isLocalHomStalkMap', AlgebraicGeometry.LocallyRingedSpace.stalkMap_hom_inv, ModuleCat.MonoidalCategory.tensorμ_apply, CommRingCat.forget_map_apply, BoolAlg.hom_inv_apply, TopCat.pullbackIsoProdSubtype_inv_fst_apply, BddDistLat.id_apply, AlgebraicGeometry.Scheme.Hom.exists_mem_and_isIso_morphismRestrict_toNormalization, CompHausLike.const_comp, TopCat.piIsoPi_hom_apply, SemimoduleCat.MonoidalCategory.associator_inv_apply, GrpCat.ofHom_apply, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_apply, CategoryTheory.PreGaloisCategory.fiber_in_connected_component, AlgebraicGeometry.Scheme.descResidueField_stalkClosedPointTo_fromSpecResidueField, groupHomology.d₂₁_apply_mem_cycles₁, CategoryTheory.ShortComplex.ShortExact.surjective_g, isIso_iff_bijective, CategoryTheory.PresheafOfGroups.OneCochain.ev_precomp, GrpCat.mono_iff_injective, CategoryTheory.comp_apply, ModuleCat.ihom_map_apply, CategoryTheory.ShortComplex.zero_apply, groupCohomology.H1π_comp_H1IsoOfIsTrivial_hom_apply, groupHomology.cyclesMap_comp_isoCycles₁_hom_apply, CompHausLike.finiteCoproduct.ι_injective, SemimoduleCat.MonoidalCategory.whiskerLeft_apply, groupHomology.toCycles_comp_isoCycles₂_hom_apply, groupCohomology.H2π_eq_zero_iff, CategoryTheory.Limits.Concrete.productEquiv_apply_apply, MagmaCat.hom_inv_apply, AlgebraicGeometry.isDominant_iff, AlgebraicGeometry.Flat.generalizingMap, MagmaCat.ext_iff, AlgebraicGeometry.Scheme.Hom.fiberι_asFiber, CommSemiRingCat.forget_map, CommAlgCat.forget_map, TopCat.GlueData.MkCore.t_id, groupCohomology.H1π_comp_map_apply, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.c_iso, Rep.leftRegularHom_hom_single, AlgebraicGeometry.IsOpenImmersion.range_pullback_snd_of_left, AlgebraicGeometry.IsOpenImmersion.instIsIsoCommRingCatStalkMap, AlgebraicGeometry.isPreimmersion_iff, AlgebraicGeometry.Scheme.AffineCover.covers, coe_id, CategoryTheory.Limits.Concrete.widePushout_exists_rep', AlgebraicGeometry.Scheme.Hom.Spec_map_residueFieldMap_fromSpecResidueField_assoc, AlgebraicGeometry.Scheme.isLocalHom_stalkClosedPointTo', ofHom_hom, AlgebraicGeometry.IsFinite.finite_preimage_singleton, TopCat.Presheaf.germ_stalkPullbackInv, GrpCat.FilteredColimits.colimit_mul_mk_eq, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.invApp_app_apply, AlgebraicGeometry.stalkMap_toStalk_apply, AlgebraicGeometry.IsAffineOpen.primeIdealOf_eq_map_closedPoint, CategoryTheory.PreGaloisCategory.exists_hom_from_galois_of_fiber, AlgebraicGeometry.Scheme.IdealSheafData.range_glueDataObjι, SemiRingCat.ofHom_apply, AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap_app_assoc, SemiRingCat.hom_inv_apply, id_apply, RingHom.surjective_of_epi_of_finite, CategoryTheory.Limits.Cocone.w_apply, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.fromBiprod_biprodIsoProd_inv_apply, TopCat.GlueData.preimage_range, FintypeCat.hom_apply, PresheafOfModules.ι_fromFreeYonedaCoproduct_apply, AlgebraicGeometry.IsOpenImmersion.range_pullback_to_base_of_right, TopCat.isOpenEmbedding_iff_isIso_comp, CategoryTheory.PreGaloisCategory.autMulEquivAutGalois_π, CategoryTheory.ShortComplex.moduleCat_exact_iff, TopCat.Presheaf.map_restrict, GrpCat.inv_hom_apply, CompHaus.effectiveEpi_tfae, groupHomology.isoCycles₂_inv_comp_iCycles_apply, CategoryTheory.PreGaloisCategory.surjective_on_fiber_of_epi, ModuleCat.range_mkQ_cokernelIsoRangeQuotient_inv_apply, BoolAlg.comp_apply, TopologicalSpace.Opens.functor_map_eq_inf, TopCat.Presheaf.restrictOpenCommRingCat_apply, ModuleCat.kernelIsoKer_hom_ker_subtype_apply, NonemptyFinLinOrd.id_apply, AlgebraicGeometry.Scheme.Pullback.Triplet.fst_SpecTensorTo_apply, mono_iff_injective_of_preservesPullback, groupHomology.cyclesMk₂_eq, PresheafOfModules.Derivation.d_app, CommBialgCat.comp_apply, Preord.coe_comp, AddGrpCat.neg_hom_apply, groupHomology.H1π_eq_zero_iff, SemiNormedGrp.ofHom_apply, TopologicalSpace.OpenNhds.isOpenEmbedding, LightCondMod.hom_naturality_apply, groupHomology.d₂₁_single_one_fst, SheafOfModules.unitToPushforwardObjUnit_val_app_apply, MagmaCat.forget_map, AlgebraicGeometry.IsClosedImmersion.isClosedEmbedding, CategoryTheory.ShortComplex.π_moduleCatCyclesIso_hom_apply, CompHausLike.homeoOfIso_apply, AlgebraicGeometry.Scheme.AffineZariskiSite.opensRange_relativeGluingData_map, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.app_invApp_assoc, AddGrpCat.hom_neg_apply, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.stalk_iso, CategoryTheory.Limits.Concrete.widePushout_exists_rep, Rep.FiniteCyclicGroup.groupCohomologyπEven_eq_iff, AlgebraicGeometry.RingedSpace.basicOpen_res_eq, AlgebraicGeometry.Spec_zeroLocus, TopCat.GlueData.image_inter, MonCat.Colimits.cocone_naturality_components, HeytAlg.coe_id, ModuleCat.piIsoPi_hom_ker_subtype_apply, CommBialgCat.bialgEquivOfIso_symm_apply, TopCat.isEmbedding_pullback_to_prod, ModuleCat.MonoidalCategory.whiskerRight_apply, FinPartOrd.comp_apply, CommRingCat.inv_hom_apply, ModuleCat.homMk_hom_apply, AlgebraicGeometry.Flat.stalkMap, ModuleCat.free_δ_freeMk, AlgebraicGeometry.Scheme.Pullback.range_snd, AlgebraicGeometry.Scheme.Opens.germ_stalkIso_hom_assoc, AlgebraicGeometry.Scheme.Hom.stalkMap_comp, TopCat.pullbackIsoProdSubtype_hom_apply, AlgebraicGeometry.Scheme.Cover.exists_eq, imageToKernel_arrow_apply, AlgebraicGeometry.Scheme.ofArrows_mem_precoverage_iff, CommRingCat.comp_apply, LinOrd.coe_id, PartOrdEmb.Hom.injective, AddCommGrpCat.biprodIsoProd_inv_comp_desc_apply, Rep.leftRegularHomEquiv_symm_single, CategoryTheory.PreGaloisCategory.comp_autMap_apply, AlgebraicGeometry.Scheme.IsLocallyDirected.ι_jointly_surjective, AlgebraicGeometry.basicOpen_eq_of_affine', AlgebraicGeometry.IsPreimmersion.base_embedding, AlgebraicGeometry.Scheme.Hom.coe_image, AlgebraicGeometry.Scheme.Hom.coe_preimage, TopCat.ofHom_apply, AlgebraicGeometry.LocallyRingedSpace.stalkSpecializes_stalkMap_apply, AddCommGrpCat.ofHom_apply, groupCohomology.cocyclesMk₂_eq, AddCommGrpCat.biprodIsoProd_inv_comp_fst_apply, TopCat.Sheaf.objSupIsoProdEqLocus_hom_snd, AlgebraicGeometry.stalkToFiberRingHom_germ, AlgebraicGeometry.Scheme.Hom.isClosedEmbedding, AlgebraicGeometry.Scheme.Hom.quasiFiniteAt_iff_isOpen_singleton_asFiber, TopCat.Sheaf.objSupIsoProdEqLocus_inv_eq_iff, groupHomology.d₃₂_apply_mem_cycles₂, AddSemigrp.coe_comp, NonemptyFinLinOrd.comp_apply, PartOrdEmb.Hom.le_iff_le, ModuleCat.ExtendRestrictScalarsAdj.HomEquiv.evalAt_apply, TwoP.swapEquiv_inverse_map_hom_toFun, AlgebraicGeometry.Scheme.Hom.finite_preimage_singleton, TopCat.induced_of_isLimit, AlgebraicGeometry.Scheme.inv_hom_apply, groupCohomology.toCocycles_comp_isoCocycles₁_hom_apply, groupHomology.map_id_comp_H0Iso_hom_apply, AddMonCat.hom_neg_apply, Action.FintypeCat.quotientToQuotientOfLE_hom_mk, AlgebraicGeometry.isIso_SpecMap_stakMap_localization, groupHomology.cyclesMk₁_eq, AlgebraicGeometry.Scheme.residue_residueFieldMap_assoc, LightCondMod.epi_iff_locallySurjective_on_lightProfinite, AlgebraicGeometry.Scheme.IdealSheafData.support_map, SemimoduleCat.inv_hom_apply, TopCat.Presheaf.SheafConditionEqualizerProducts.w_apply, AlgebraicGeometry.FormallyUnramified.stalkMap, AlgebraicGeometry.specTargetImageFactorization_app_injective, LightCondensed.ihom_map_val_app, AlgebraicGeometry.Scheme.IsGermInjectiveAt.cond, TopCat.Presheaf.pushforwardPullbackAdjunction_unit_pullback_map_germToPullbackStalk, AlgebraicGeometry.PresheafedSpace.stalkMap.congr_hom, GrpWithZero.hom_id, AddGrpCat.comp_apply, MagmaCat.inv_hom_apply, ModuleCat.FreeMonoidal.μIso_inv_freeMk, TopCat.prodIsoProd_hom_apply, AlgebraicGeometry.Scheme.OpenCover.finiteSubcover_X, AlgebraicGeometry.Scheme.Pullback.range_fst, FintypeCat.inv_hom_id_apply, SemiNormedGrp.coe_id, groupCohomology.mapCocycles₁_comp_i_apply, TopCat.comp_app, AlgebraicGeometry.IsOpenImmersion.iff_stalk_iso, AlgebraicGeometry.Scheme.Cover.fromGlued_isOpenEmbedding, AlgebraicGeometry.IsOpenImmersion.app_ΓIso_hom_apply, SemimoduleCat.forget_map, InfiniteGalois.finGaloisGroupFunctor_map_proj_eq_proj, groupHomology.mapCycles₂_id_comp_apply, AlgebraicGeometry.Scheme.residue_residueFieldMap, AlgCat.comp_apply, Compactum.str_hom_commute, AlgebraicGeometry.StructureSheaf.stalkAlgebra_map, AlgebraicGeometry.PresheafedSpace.stalkMap.comp, CategoryTheory.ShortComplex.ab_zero_apply, DistLat.coe_id, FinBddDistLat.coe_comp, TopCat.pullbackIsoProdSubtype_inv_snd_assoc, Lat.hom_inv_apply, TopCat.epi_iff_surjective, AlgebraicGeometry.Scheme.stalkMap_inv_hom_apply, MonCat.coe_id, CategoryTheory.Meq.mk_apply, AlgebraicGeometry.Scheme.SpecMap_residue_apply, RingCat.comp_apply, Rep.hom_comm_apply, SheafOfModules.pushforwardNatTrans_app_val_app, AddCommGrpCat.HasLimit.lift_hom_apply, groupHomology.H2π_comp_map_apply, CategoryTheory.PreGaloisCategory.evaluation_aut_surjective_of_isGalois, BddLat.comp_apply, AlgebraicGeometry.Scheme.Pullback.range_diagonal_subset_diagonalCoverDiagonalRange, AlgebraicGeometry.AffineSpace.reindex_appTop_coord, CondensedMod.epi_iff_locallySurjective_on_compHaus, AlgebraicGeometry.Scheme.Hom.mem_smoothLocus, MonCat.FilteredColimits.colimit_mul_mk_eq, Rep.FiniteCyclicGroup.homResolutionIso_hom_f_hom_apply, AlgebraicGeometry.Scheme.Hom.appLE_appIso_inv_apply, TopCat.GlueData.fromOpenSubsetsGlue_isOpenMap, FiniteGrp.ofHom_apply, AlgebraicGeometry.RingedSpace.isUnit_res_basicOpen, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_apply, SemimoduleCat.MonoidalCategory.leftUnitor_inv_apply, GrpCat.ext_iff, AlgebraicGeometry.exists_appTop_π_eq_of_isLimit, AlgebraicGeometry.Proj.stalkIso'_germ, AlgebraicGeometry.Scheme.Pullback.Triplet.Spec_ofPointTensor_SpecTensorTo, Rep.FiniteCyclicGroup.groupHomologyπOdd_eq_iff, CommMonCat.id_apply, SimplexCategory.concreteCategoryHom_id, CategoryTheory.PreGaloisCategory.PointedGaloisObject.Hom.comp, BddOrd.hom_inv_apply, AlgebraicGeometry.Scheme.Pullback.Triplet.ofPoint_s, AlgebraicGeometry.Scheme.mem_basicOpen_top, AlgebraicGeometry.QuasiCompactCover.isCompactOpenCovered_of_isAffineOpen, CategoryTheory.PreGaloisCategory.evaluationEquivOfIsGalois_symm_fiber, groupCohomology.π_comp_H2Iso_hom_apply, AlgebraicGeometry.Scheme.Opens.range_ι, AlgebraicGeometry.IsAffineOpen.range_fromSpec, ModuleCat.HasLimit.lift_hom_apply, AddGrpCat.epi_iff_surjective, TopCat.Presheaf.stalkSpecializes_comp_apply, AlgebraicGeometry.isClosedImmersion_iff, CategoryTheory.PreGaloisCategory.autEmbedding_range, AlgebraicGeometry.Scheme.Opens.ι_image_basicOpen', CategoryTheory.GrothendieckTopology.Plus.inj_of_sep, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_naturality_assoc, AlgebraicGeometry.Spec_closedPoint, AlgebraicGeometry.Scheme.Hom.range_subset_ker_support, AlgebraicGeometry.LocallyRingedSpace.preimage_basicOpen, AlgebraicGeometry.ProjectiveSpectrum.Proj.stalkMap_toSpec, Frm.ofHom_apply, TopModuleCat.kerι_apply, TopCat.hom_inv_id_apply, TopCat.nonempty_isColimit_iff_eq_coinduced, AlgebraicGeometry.AffineSpace.homOverEquiv_apply, AlgebraicGeometry.Scheme.Hom.finite_preimage, AlgebraicGeometry.Scheme.Opens.ι_apply, LightProfinite.surjective_transitionMap, CompHaus.epi_iff_surjective, TopCat.mono_iff_injective, CommBialgCat.ofHom_apply, AlgebraicGeometry.Scheme.basicOpen_res, AlgebraicGeometry.Scheme.iso_inv_base_hom_base_apply, AlgebraicGeometry.Scheme.Hom.continuous, AlgebraicGeometry.Scheme.Hom.asFiberHom_fiberToSpecResidueField, TopCat.Sheaf.objSupIsoProdEqLocus_inv_fst, TopCat.homeoOfIso_symm_apply, AlgebraicGeometry.Scheme.Pullback.Triplet.exists_preimage, ModuleCat.extendScalarsComp_hom_app_one_tmul, AlgebraicGeometry.Scheme.Hom.isConstructible_image, AlgebraicGeometry.Scheme.Hom.stalkMap_hom_inv_apply, AlgebraicGeometry.Surjective.surj, CategoryTheory.PreGaloisCategory.surjective_of_nonempty_fiber_of_isConnected, AlgebraicGeometry.IsClosedImmersion.isAffine_surjective_of_isAffine, AlgebraicGeometry.Scheme.stalkMap_inv_hom, AlgebraicGeometry.germ_eq_zero_of_pow_mul_eq_zero, ContinuousMap.yonedaPresheaf'_map, CategoryTheory.PreGaloisCategory.mulAction_naturality, LightProfinite.epi_iff_surjective, AlgebraicGeometry.Scheme.comp_base_apply, groupCohomology.π_comp_H1Iso_hom_apply, AlgebraicGeometry.isGermInjectiveAt_iff_of_isOpenImmersion, TopCat.GlueData.MkCore.t_inter, AlgebraicGeometry.PresheafedSpace.GlueData.ι_jointly_surjective, Lat.id_apply, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_comp, BddLat.id_apply, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_base_apply, AddCommGrpCat.forget_map, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_eq_zero_iff_notMem_basicOpen, AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec_base_apply_eq_comap, groupHomology.d₂₁_single_one_snd, SheafOfModules.pushforwardPushforwardEquivalence_unit_app_val_app, AlgebraicGeometry.IsDominant.denseRange, AlgebraicGeometry.Flat.iff_flat_stalkMap, AlgebraicGeometry.Scheme.IdealSheafData.subschemeι_app_surjective, AlgebraicGeometry.Scheme.preimage_zeroLocus, groupHomology.π_map_apply, AlgebraicGeometry.isIso_iff_stalk_iso, AlgebraicGeometry.Scheme.instIsOverMapStalkMapOverInferInstanceOverClass, CommGrpCat.one_apply, PartOrdEmb.ext_iff, groupHomology.d₃₂_single_one_snd, DistLat.coe_comp, TopCat.GlueData.ι_jointly_surjective, HomologicalComplex.alternatingConst_iCycles_odd_comp_apply, BddDistLat.ext_iff, CategoryTheory.ShortComplex.toCycles_moduleCatCyclesIso_hom_apply, groupHomology.π_comp_H2Iso_hom_apply, AlgebraicGeometry.Scheme.Hom.support_ker, CategoryTheory.PreGaloisCategory.IsNaturalSMul.naturality, BddOrd.comp_apply, CategoryTheory.Limits.Concrete.prodEquiv_apply_snd, CategoryTheory.Limits.FintypeCat.jointly_surjective, AlgebraicGeometry.Scheme.Hom.homeomorph_apply, AlgebraicGeometry.Scheme.Hom.stalkMap_congr, AlgebraicGeometry.Scheme.map_basicOpen_map, SemiNormedGrp.comp_apply, Rep.coinvariantsTensorIndHom_mk_tmul_indVMk, ModuleCat.biproductIsoPi_inv_comp_π_apply, CategoryTheory.Presheaf.equalizerSieve_apply, CommSemiRingCat.inv_hom_apply, Rep.leftRegularHomEquiv_apply, PresheafOfModules.pushforward_obj_map_apply', CategoryTheory.Preadditive.mono_iff_injective', CompHausLike.hom_ofHom, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo_Spec, AlgebraicGeometry.LocallyRingedSpace.stalkSpecializes_stalkMap, AlgebraicGeometry.Scheme.Hom.surjective, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sigma_ι_isOpenEmbedding, UniformSpaceCat.coe_id, AlgebraicGeometry.Scheme.IsLocallyDirected.exists_of_pullback_V_V, GrpCat.one_apply, AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage, AddGrpCat.coe_comp, groupHomology.coe_mapCycles₁, TopCat.isOpen_iff_of_isColimit_cofork, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_map_injective, BoolAlg.inv_hom_apply, GrpCat.forget_map, groupCohomology.d₀₁_comp_d₁₂_apply, Frm.comp_apply, Lat.inv_hom_apply, ModuleCat.free_map_apply, TopCat.isQuotientMap_of_isColimit_cofork, BddOrd.ofHom_apply, AlgebraicGeometry.Scheme.AffineZariskiSite.PreservesLocalization.opensRange_map, FintypeCat.uSwitchEquiv_symm_naturality, TopCat.Presheaf.Γgerm_res_apply, groupHomology.mapCycles₂_comp_i_apply, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_jointly_surjective₂, AlgebraicGeometry.Scheme.Hom.stalkMap_congr_assoc, groupCohomology.cocyclesIso₀_hom_comp_f_apply, SheafOfModules.pushforwardPushforwardAdj_counit_app_val_app, AlgebraicGeometry.Scheme.mem_basicOpen'', congr_hom, CategoryTheory.PreGaloisCategory.endEquivAutGalois_π, CategoryTheory.Meq.equiv_symm_eq_apply, AlgebraicGeometry.specTargetImageRingHom_surjective, Action.isContinuous_def, PartOrd.hom_inv_apply, SemiRingCat.id_apply, AlgebraicGeometry.Scheme.stalkMap_inv_hom_assoc, AlgebraicGeometry.Scheme.Cover.RelativeGluingData.preimage_toBase_eq_range_ι, CategoryTheory.Limits.Cone.w_apply, TopCat.isOpen_iff_of_isColimit, AlgebraicGeometry.locallyQuasiFinite_iff_finite_preimage_singleton, CategoryTheory.ShortComplex.moduleCatCyclesIso_inv_iCycles_assoc_apply, FinBddDistLat.id_apply, AddMagmaCat.comp_apply, CategoryTheory.ShortComplex.moduleCatCyclesIso_inv_π_apply, groupHomology.d₁₀_single_inv, AlgebraicGeometry.basicOpen_eq_of_affine, groupHomology.mkH1OfIsTrivial_apply, groupCohomology.π_map_apply, AddCommMonCat.id_apply, TopologicalSpace.OpenNhds.map_id_obj, AlgebraicGeometry.Scheme.Opens.ι_appLE, CategoryTheory.Limits.Concrete.colimit_exists_rep, AddMagmaCat.id_apply, AlgebraicGeometry.IsOpenImmersion.range_pullbackFst, AlgebraicGeometry.IsAffineOpen.fromSpec_image_zeroLocus, AlgebraicGeometry.LocallyRingedSpace.isUnit_res_toΓSpecMapBasicOpen, AlgebraicGeometry.Scheme.affineBasisCover_map_range, Frm.forget_map, BddDistLat.comp_apply, PresheafOfModules.freeYonedaEquiv_comp, AlgebraicGeometry.Flat.isQuotientMap_of_surjective, AlgebraicGeometry.Scheme.Opens.ι_app, AlgebraicGeometry.Scheme.Hom.QuasiFiniteAt.isClopen_singleton_asFiber, AlgebraicGeometry.instIsPreimmersionAsFiberHom, Semigrp.coe_id, AlgebraicGeometry.LocallyRingedSpace.stalkMap_congr_hom_assoc, BddLat.ext_iff, CategoryTheory.ShortComplex.moduleCat_pOpcycles_eq_iff, FintypeCat.hom_inv_id_apply, CompHausLike.LocallyConstant.incl_comap, CategoryTheory.Limits.kernelSubobject_arrow_apply, ModuleCat.FreeMonoidal.εIso_hom_one, AlgebraicGeometry.quasiCompact_iff_spectral, AlgebraicGeometry.Scheme.iso_hom_base_inv_base_apply, TopCat.prodIsoProd_inv_fst_apply, DistLat.comp_apply, CategoryTheory.Pretriangulated.preadditiveYoneda_shiftMap_apply, AlgebraicGeometry.quasiCompact_iff_isSpectralMap, Rep.freeLift_hom_single_single, AddSemigrp.id_apply, CategoryTheory.IsIso.hom_inv_id_apply, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_preimage_basicOpen, TopCat.effectiveEpi_iff_isQuotientMap, CategoryTheory.hom_isIso, TopCat.colimit_isOpen_iff, CommBialgCat.forget_map, groupHomology.π_comp_H1Iso_inv_apply, LightProfinite.proj_surjective, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_map_surjective, coe_comp, AlgebraicGeometry.LocallyRingedSpace.toΓSpec_preimage_zeroLocus_eq, groupCohomology.isoCocycles₂_hom_comp_i_apply, ModuleCat.extendRestrictScalarsAdj_unit_app_apply, Rep.diagonalHomEquiv_apply, AlgebraicGeometry.IsAffineOpen.isCompactOpenCovered, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.base_open, CategoryTheory.Limits.cokernel.π_desc_apply, groupHomology.d₂₁_single, CategoryTheory.ShortComplex.toCycles_moduleCatCyclesIso_hom_assoc_apply, AlgebraicGeometry.Scheme.Γevaluation_naturality, CategoryTheory.hom_comp, TopCat.GlueData.ι_eq_iff_rel, groupHomology.cycles₁IsoOfIsTrivial_inv_apply, groupHomology.eq_d₂₁_comp_inv_apply, Rep.epi_iff_surjective, SheafOfModules.pushforwardNatTrans_app_val_app_apply, AlgebraicGeometry.Scheme.Hom.appIso_inv_app_apply, PartOrd.forget_map, AlgebraicGeometry.HasRingHomProperty.stalkMap_of_respectsIso, ModuleCat.MonoidalCategory.whiskerLeft_apply, AlgebraicGeometry.ExistsHomHomCompEqCompAux.range_g_subset, TopologicalSpace.Opens.toTopCat_map, groupHomology.cyclesMk₀_eq, groupCohomology.cocyclesIso₀_inv_comp_iCocycles_apply, CategoryTheory.Preadditive.epi_iff_surjective', ModuleCat.forget_map, TopCat.Presheaf.stalkSpecializes_stalkFunctor_map_apply, groupHomology.chainsMap_f_0_comp_chainsIso₀_apply, AlgebraicGeometry.Scheme.isLocalHom_stalkClosedPointTo, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_naturality, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_basicOpen', TopCat.Presheaf.germToPullbackStalk_stalkPullbackHom_assoc, AddCommGrpCat.hom_neg_apply, AlgebraicGeometry.IsAffineOpen.app_basicOpen_eq_away_map, AlgebraicGeometry.Scheme.preimage_basicOpen, AlgebraicGeometry.Scheme.exists_le_and_germ_injective, PartOrd.ofHom_apply, RingCat.hom_inv_apply, Preord.comp_apply, AlgebraicGeometry.Scheme.Hom.SpecMap_residueFieldMap_fromSpecResidueField, TopCat.coe_comp, HeytAlg.forget_map, ModuleCat.span_rightExact, AlgebraicGeometry.Scheme.Pullback.Triplet.ofPoint_y, CommRingCat.KaehlerDifferential.ext_iff, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_invApp_assoc, AlgebraicGeometry.LocallyRingedSpace.evaluation_naturality_assoc, AlgebraicGeometry.AffineSpace.comp_homOfVector_assoc, PresheafOfModules.germ_ringCat_smul, groupCohomology.cocyclesMk₀_eq, AlgebraicGeometry.LocallyRingedSpace.toStalk_stalkMap_toΓSpec, SemiRingCat.forget₂_monCat_map, AlgebraicGeometry.Scheme.range_fromSpecResidueField, TopCat.Sheaf.eq_of_locally_eq_iff, AlgebraicGeometry.Scheme.SpecMap_stalkMap_fromSpecStalk, PresheafOfModules.naturality_apply, TopCat.exists_mem_zeroHypercover_range, AlgebraicGeometry.Scheme.Modules.Hom.app_smul, BddOrd.coe_comp, TopCat.sigmaIsoSigma_hom_ι_apply, AlgebraicGeometry.Scheme.Hom.stalkMap_inv_hom_apply, AddCommGrpCat.hom_add_apply, AlgebraicGeometry.quasiCompactCover_iff, AlgebraicGeometry.Scheme.image_zeroLocus, AlgebraicGeometry.Scheme.basicOpen_eq_bot_iff_forall_evaluation_eq_zero, UniformSpaceCat.extensionHom_val, ModuleCat.extendScalarsId_inv_app_apply, AlgebraicGeometry.LocallyRingedSpace.evaluation_eq_zero_iff_notMem_basicOpen, ModuleCat.semilinearMapAddEquiv_symm_apply_apply, AlgebraicGeometry.Proj.res_apply, TopCat.Presheaf.app_injective_iff_stalkFunctor_map_injective, AlgebraicGeometry.Scheme.Hom.fiberToSpecResidueField_apply, AlgebraicGeometry.Scheme.Hom.stalkMap_hom_inv_assoc, PresheafOfModules.fromFreeYonedaCoproduct_app_mk, ModuleCat.MonoidalCategory.braiding_inv_apply, PresheafOfModules.Sheafify.map_smul, Rep.diagonalSuccIsoFree_hom_hom_single, SemiNormedGrp.iso_isometry_of_normNoninc, NonemptyFinLinOrd.mono_iff_injective, AlgebraicGeometry.Scheme.Pullback.Triplet.ofPoint_SpecTensorTo, SemimoduleCat.ofHom_apply, PresheafOfModules.germ_smul, CategoryTheory.PreGaloisCategory.exists_galois_representative, AlgebraicGeometry.Scheme.JointlySurjective.exists_eq, DistLat.inv_hom_apply, GrpCat.epi_iff_surjective, AlgebraicGeometry.AffineSpace.map_appTop_coord, AlgebraicGeometry.LocallyRingedSpace.isLocalHom_stalkMap_congr, CategoryTheory.PreGaloisCategory.PointedGaloisObject.cocone_app, Semigrp.ofHom_apply, TopCat.ext_iff, Rep.free_ext_iff, PresheafOfModules.toSheafify_app_apply, hom_ofHom, HeytAlg.ext_iff, AlgebraicGeometry.Scheme.Hom.app_appIso_inv, TopCat.limit_topology, Semigrp.forget_map, AlgebraicGeometry.Scheme.coe_homeoOfIso, TopCat.Presheaf.stalkPushforward.comp, MonCat.id_apply, CategoryTheory.IsIso.inv_hom_id_apply, ModuleCat.ofHom_apply, AddMagmaCat.neg_hom_apply, GrpCat.id_apply, Rep.coinvariantsTensorIndInv_mk_tmul_indMk, CommAlgCat.id_apply, AlgCat.ofHom_apply, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec.image_basicOpen_eq_basicOpen, LinOrd.comp_apply, smoothSheafCommRing.forgetStalk_hom_comp_evalHom_apply, CategoryTheory.Subobject.underlyingIso_arrow_apply, AlgebraicGeometry.isClosedMap_iff_specializingMap, LinOrd.coe_comp, TopCat.pullback_snd_range, ModuleCat.restrictScalarsCongr_inv_app, AlgebraicGeometry.Scheme.image_preimage_eq_of_isPullback, AlgebraicGeometry.Scheme.Hom.instIsLocalHomCarrierStalkCommRingCatPresheafCoeContinuousMapCarrierCarrierHomTopCatBaseRingHomHomStalkMap, AlgebraicGeometry.LocallyRingedSpace.stalkMap_comp, AlgebraicGeometry.Scheme.GlueData.ι_jointly_surjective, CommMonCat.ofHom_apply, AlgebraicGeometry.SheafedSpace.isColimit_exists_rep, ModuleCat.imageIsoRange_hom_subtype_apply, TopCat.Presheaf.stalkFunctor_map_germ_apply', RingCat.id_apply, SemimoduleCat.MonoidalCategory.leftUnitor_hom_apply, PartOrdEmb.coe_id, TopCat.GlueData.MkCore.cocycle, AlgebraicGeometry.LocallyRingedSpace.stalkMap_hom_inv_apply, TopologicalSpace.OpenNhds.inclusionMapIso_inv_app, AlgebraicGeometry.Scheme.stalkClosedPointTo_comp, BoolAlg.id_apply, groupHomology.H1AddEquivOfIsTrivial_symm_tmul, AlgebraicGeometry.LocallyRingedSpace.evaluation_naturality_apply, AlgebraicGeometry.map_injective_of_isIntegral, groupHomology.d₂₁_single_ρ_add_single_inv_mul, AlgebraicGeometry.AffineSpace.homOfVector_appTop_coord, AlgebraicGeometry.Scheme.Spec_map_residue_apply, AlgebraicGeometry.Proj.stalkIso'_symm_mk, TopCat.GlueData.ι_injective, AlgebraicGeometry.Scheme.residueFieldMap_comp, CategoryTheory.Iso.inv_hom_id_apply, AlgebraicGeometry.Scheme.isoSpec_image_zeroLocus, AlgebraicGeometry.Scheme.Hom.fiberι_fiberHomeo_symm, Rep.linearization_map_hom_single, BddOrd.inv_hom_apply, AlgebraicGeometry.Scheme.Spec_stalkClosedPointTo_fromSpecStalk, TopCat.Presheaf.pushforwardPullbackAdjunction_unit_pullback_map_germToPullbackStalk_assoc, CategoryTheory.Meq.equiv_apply, LightProfinite.proj_comp_transitionMap', AlgebraicGeometry.LocallyRingedSpace.isLocalHomValStalkMap, groupHomology.eq_d₁₀_comp_inv_apply, AlgebraicGeometry.Scheme.mem_toGrothendieck_smallPretopology, AlgebraicGeometry.LocallyRingedSpace.stalkMap_congr_point, PartOrd.ext_iff, AlgebraicGeometry.Scheme.Hom.stalkMap_inv_hom, CategoryTheory.Meq.condition, MagmaCat.comp_apply, ModuleCat.RestrictionCoextensionAdj.HomEquiv.fromRestriction_hom_apply_apply, AddCommMonCat.forget_map, AlgebraicGeometry.RingedSpace.mem_basicOpen, AlgebraicGeometry.Scheme.Hom.app_injective, SheafOfModules.pushforwardPushforwardEquivalence_counit_app_val_app, groupHomology.H0π_comp_H0Iso_hom_apply, Rep.barComplex.d_single, ModuleCat.freeDesc_apply, CategoryTheory.ShortComplex.ShortExact.moduleCat_surjective_g, FinPartOrd.id_apply, Rep.mono_iff_injective, CategoryTheory.PreGaloisCategory.endMulEquivAutGalois_pi, AlgebraicGeometry.Scheme.Hom.Spec_map_residueFieldMap_fromSpecResidueField, AlgebraicGeometry.Scheme.Opens.toSpecΓ_preimage_zeroLocus, SemimoduleCat.comp_apply, AlgebraicGeometry.Scheme.Cover.fromGlued_open_map, MonCat.ofHom_apply, ModuleCat.ExtendScalars.hom_ext_iff, SemiNormedGrp.hom_inv_apply, CpltSepUniformSpace.hom_id, groupCohomology.coe_mapCocycles₂, TopCat.inv_hom_id_apply, CategoryTheory.GrothendieckTopology.Plus.toPlus_apply, CategoryTheory.Presheaf.isLocallyInjective_iff_injective_of_separated, AlgebraicGeometry.coprodMk_inr, GrpCat.comp_apply, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_bijective, CommGrpCat.forget₂_map, ModuleCat.toKernelSubobject_arrow, AlgebraicGeometry.Scheme.Hom.isProperMap, AlgebraicGeometry.StructureSheaf.algebraMap_germ_apply, AlgebraicGeometry.QuasiCompactCover.exists_isAffineOpen_of_isCompact, groupCohomology.H1π_eq_iff, Semigrp.coe_comp, AlgebraicGeometry.Scheme.Hom.asFiberHom_fiberToSpecResidueField_assoc, groupHomology.chainsMap_f_1_comp_chainsIso₁_apply, FintypeCat.homMk_apply, TopCat.pullback_fst_range, AlgebraicGeometry.Scheme.localRingHom_comp_stalkIso_apply, AlgebraicGeometry.LocallyRingedSpace.stalkMap_congr_hom, BddDistLat.hom_inv_apply, CategoryTheory.Limits.limit.w_apply, TopologicalSpace.OpenNhds.inclusionMapIso_hom_app, forget_map_eq_coe, SemiRingCat.forget_map, CategoryTheory.ShortComplex.ShortExact.moduleCat_injective_f, AlgebraicGeometry.Scheme.Hom.isOpen_smoothLocus, CategoryTheory.PreGaloisCategory.evaluation_aut_injective_of_isConnected, AddCommGrpCat.homAddEquiv_apply, UniformSpaceCat.hom_comp, groupHomology.H0π_comp_map_apply, CommMonCat.ext_iff, AlgebraicGeometry.Scheme.Opens.ι_base_apply, ModuleCat.restrictScalarsId'App_inv_apply, CategoryTheory.ShortComplex.moduleCatCyclesIso_inv_iCycles_apply, TopCat.Presheaf.stalkSpecializes_stalkPushforward_apply, PartOrd.id_apply, AlgebraicGeometry.Scheme.Hom.isDiscrete_preimage, MonCat.inv_hom_apply, bijective_of_isIso, groupHomology.π_comp_H2Iso_inv_apply, TopCat.GlueData.preimage_image_eq_image', TopCat.Presheaf.SheafConditionEqualizerProducts.res_π_apply, AlgebraicGeometry.Scheme.homOfLE_apply, ModuleCat.restrictScalarsComp'App_inv_apply, AlgebraicGeometry.IsAffineOpen.fromSpec_primeIdealOf, AddGrpCat.ofHom_apply, CategoryTheory.ShortComplex.ShortExact.ab_exact_iff_function_exact, TopCat.pullback_snd_image_fst_preimage, Stonean.effectiveEpi_tfae, AlgebraicGeometry.LocallyRingedSpace.iso_hom_base_inv_base_apply, CompHausLike.coe_comp, CompHausLike.finiteCoproduct.ι_jointly_surjective, AlgebraicGeometry.Scheme.fromSpecResidueField_apply, Rep.FiniteCyclicGroup.groupCohomologyπOdd_eq_zero_iff, CategoryTheory.PreGaloisCategory.autGaloisSystem_map_surjective, groupHomology.d₁₀_single, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.toBiprod_apply, Lat.coe_comp, AlgebraicGeometry.Scheme.Hom.coe_opensRange, AlgebraicGeometry.Scheme.stalkMap_congr, groupCohomology.cocycles₁.d₁₂_apply, AlgebraicGeometry.Scheme.Hom.stalkMap_congr_hom, ModuleCat.FilteredColimits.colimit_add_mk_eq, ModuleCat.free_hom_ext_iff, CategoryTheory.Pretriangulated.preadditiveCoyoneda_homologySequenceδ_apply, groupHomology.H2π_eq_zero_iff, AlgebraicGeometry.SheafedSpace.GlueData.ι_jointly_surjective, CommMonCat.hom_inv_apply, AlgebraicGeometry.Scheme.IdealSheafData.subschemeι_apply, AlgebraicGeometry.PresheafedSpace.stalkMap.congr_point, GrpCat.coe_id, AlgebraicGeometry.IsAffineOpen.appLE_eq_away_map, CategoryTheory.Limits.kernelSubobjectMap_arrow_apply, AlgebraicGeometry.Scheme.Modules.map_smul, AlgebraicGeometry.RingedSpace.mem_basicOpen', ModuleCat.MonoidalCategory.associator_inv_apply, MonCat.hom_inv_apply, AlgebraicGeometry.Scheme.mem_basicOpen, CategoryTheory.yonedaGrp_naturality, RingCat.forget_map, TopCat.prodIsoProd_inv_snd_apply, SemiRingCat.forget₂_addCommMonCat_map, AlgebraicGeometry.Scheme.Spec_stalkClosedPointTo_fromSpecStalk_assoc, AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv_apply, AddMagmaCat.hom_neg_apply, CategoryTheory.ShortComplex.abCyclesIso_inv_apply_iCycles, Action.FintypeCat.ofMulAction_apply, groupHomology.H1π_eq_iff, ModuleCat.biprodIsoProd_inv_comp_fst_apply, BoolAlg.ofHom_apply, groupHomology.d₃₂_comp_d₂₁_apply, BoolAlg.forget_map, ModuleCat.CoextendScalars.map_apply, SemimoduleCat.MonoidalCategory.associator_hom_apply, SheafOfModules.unitHomEquiv_apply_coe, AlgebraicGeometry.homogeneousLocalizationToStalk_stalkToFiberRingHom, AlgebraicGeometry.Scheme.Cover.fromGlued_injective, AddGrpCat.coe_id, groupHomology.chainsMap_f_2_comp_chainsIso₂_apply, AlgebraicGeometry.Spec.map_base_apply, AlgebraicGeometry.Scheme.Hom.stalkMap_surjective, AddSemigrp.ext_iff, CategoryTheory.Limits.cokernel.condition_apply, CompHausLike.finiteCoproduct.ι_desc_apply, AlgebraicGeometry.IsAffineOpen.mem_ideal_iff, AlgebraicGeometry.Scheme.toSpecΓ_base
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