| Name | Category | Theorems |
|---|
X₁ 📖 | CompOp | 398 mathmath: RightHomologyData.ofHasCokernelOfHasKernel_p, ShortExact.ab_finite_iff, FunctorEquivalence.counitIso_inv_app_app_τ₁, toCycles_comp_homologyπ, pOpcycles_comp_moduleCatOpcyclesIso_hom_apply, ShortExact.instHasSmallLocalizedShiftedHomHomologicalComplexIntUpQuasiIsoX₃CochainComplexMapSingleFunctorOfNatX₁, toCycles_comp_homologyπ_assoc, CochainComplex.triangleOfDegreewiseSplit_obj₁, Homotopy.h₀_f_assoc, LeftHomologyData.op_g', ShortExact.epi_δ, Homotopy.refl_h₀, CategoryTheory.kernelCokernelCompSequence.snakeInput_L₀_X₁, SnakeInput.naturality_φ₁, Homotopy.comm₁, Exact.epi_kernelLift, SnakeInput.op_δ, LeftHomologyData.ofHasKernelOfHasCokernel_H, cokernel_π_comp_cokernelToAbelianCoimage_assoc, pOpcycles_π_isoOpcyclesOfIsColimit_inv_assoc, SnakeInput.naturality_δ, ShortExact.hasInjectiveDimensionLT_X₁, Homotopy.sub_h₀, groupHomology.mono_δ_of_isZero, op_g, abelianImageToKernel_comp_kernel_ι_comp_cokernel_π_assoc, ShortExact.hasProjectiveDimensionLT_X₁, Homotopy.smul_h₁, zero_assoc, RightHomologyData.wp, RightHomologyData.ofAbelian_H, ShortExact.mono_δ, RightHomologyData.wp_assoc, moduleCatMk_X₁_carrier, HomologicalComplex.shortComplexFunctor'_obj_X₁, abelianImageToKernel_comp_kernel_ι, SnakeInput.exact_C₁_up, moduleCatMk_X₁_isAddCommGroup, LeftHomologyData.exact_iff_epi_f', exact_iff_kernel_ι_comp_cokernel_π_zero, CategoryTheory.Abelian.Ext.preadditiveYoneda_homologySequenceδ_singleTriangle_apply, LeftHomologyMapData.commf', Homotopy.unop_h₂, Splitting.mono_f, CochainComplex.mappingCone.homologySequenceδ_triangleh, groupHomology.comap_coinvariantsKer_pOpcycles_range_subtype_pOpcycles_eq_top, CategoryTheory.Abelian.Ext.contravariant_sequence_exact₁', ext_mk₀_f_comp_ext_mk₀_g, LeftHomologyData.f'_π_assoc, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.shortComplex_X₁, toCycles_comp_leftHomologyπ_assoc, exact_and_mono_f_iff_of_iso, exact_iff_isIso_imageToKernel', comp_τ₁, SnakeInput.w₀₂_τ₁, Hom.comm₁₂, CategoryTheory.Pretriangulated.shortComplexOfDistTriangle_X₁, Homotopy.op_h₂, LeftHomologyData.ofAbelian_H, ShortExact.comp_δ, LeftHomologyData.ofHasCokernel_π, FunctorEquivalence.unitIso_inv_app_τ₁_app, Splitting.id, op_X₁, Homotopy.add_h₁, groupHomology.H1CoresCoinfOfTrivial_X₁, CochainComplex.mappingCone.inr_f_descShortComplex_f_assoc, ShortExact.hasProjectiveDimensionLT_X₃_iff, LeftHomologyData.ofIsColimitCokernelCofork_π, LeftHomologyData.IsPreservedBy.f', LeftHomologyData.IsPreservedBy.hf', ShortExact.isIso_g_iff, Splitting.map_r, Homotopy.sub_h₁, Splitting.leftHomologyData_K, postcomp_extClass_surjective_of_projective_X₂, groupCohomology.instMonoModuleCatFH1InfRes, groupCohomology.cocyclesIso₀_inv_comp_iCocycles, abToCycles_apply_coe, ShortExact.singleTriangleIso_hom_hom₁, DerivedCategory.triangleOfSES_mor₁, CochainComplex.mappingCone.inr_descShortComplex_assoc, SnakeInput.δ_L₃_f, CategoryTheory.Functor.mapShortComplex_map_τ₁, map_f, Homotopy.unop_h₃, Exact.ab_range_eq_ker, CategoryTheory.Abelian.Ext.contravariant_sequence_exact₂', Exact.liftFromProjective_comp_assoc, moduleCat_zero_apply, Homotopy.neg_h₁, CategoryTheory.Abelian.Ext.preadditiveCoyoneda_homologySequenceδ_singleTriangle_apply, pOpcycles_comp_moduleCatOpcyclesIso_hom, exact_iff_exact_image_ι, ShortExact.extClass_hom, SnakeInput.naturality_φ₁_assoc, Homotopy.compRight_h₀, Exact.isIso_imageToKernel, Exact.liftFromProjective_comp, exact_iff_isIso_imageToKernel, RightHomologyData.wι_assoc, ShortExact.ab_injective_f, abLeftHomologyData_π, cokernel_π_comp_cokernelToAbelianCoimage, CategoryTheory.Abelian.Ext.covariant_sequence_exact₁', groupHomology.H1CoresCoinf_X₁, map_X₁, opMap_τ₃, Homotopy.compRight_h₁, HomologyData.ofEpiMonoFactorisation.g'_eq, CategoryTheory.Abelian.Ext.covariant_sequence_exact₃', Homotopy.smul_h₀, SnakeInput.mono_L₀_f, RightHomologyData.ofIsColimitCokernelCofork_Q, Exact.mono_cokernelDesc, exact_iff_surjective_abToCycles, Hom.id_τ₁, exact_iff_of_hasForget, RightHomologyData.ofIsColimitCokernelCofork_H, moduleCat_pOpcycles_eq_zero_iff, opcyclesIsoCokernel_inv, ab_exact_iff_function_exact, ChainComplex.mk_congr_succ_X₃, ModuleCat.smulShortComplex_X₁, groupHomology.epi_δ_of_isZero, HomologicalComplex.instMonoFShortComplexTruncLE, RightHomologyData.ofAbelian_Q, moduleCatCyclesIso_inv_π_assoc_apply, SnakeInput.snd_δ, RightHomologyData.IsPreservedBy.hf, SnakeInput.snd_δ_inr, Splitting.g_s, instIsIsoτ₁, Homotopy.ofEq_h₀, groupCohomology.epi_δ_of_isZero, Hom.comm₁₂_assoc, id_τ₁, precomp_extClass_surjective_of_projective_X₂, SnakeInput.mono_δ, CochainComplex.shift_f_comp_mappingConeHomOfDegreewiseSplitIso_inv, fFunctor_obj, SnakeInput.L₁_f_φ₁_assoc, Splitting.splitMono_f_retraction, instMonoAbelianImageToKernel, SnakeInput.φ₁_L₂_f_assoc, SnakeInput.L₀X₂ToP_comp_φ₁, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.mk₀_f_comp_biprodAddEquiv_symm_biprodIsoProd_hom, Exact.epi_toCycles, kernelSequence_X₁, Exact.moduleCat_range_eq_ker, Homotopy.symm_h₀, toCycles_i_assoc, SnakeInput.w₁₃_τ₁_assoc, opcyclesOpIso_hom_toCycles_op_assoc, CategoryTheory.IsPullback.mono_shortComplex'_f, CochainComplex.mappingCone.triangleRotateShortComplex_X₁, LeftHomologyData.f'_i, groupCohomology.instMonoModuleCatFShortComplexH0, LeftHomologyData.f'_π, ShortExact.singleTriangle_mor₁, SnakeInput.L₂'_X₁, ShortExact.comp_extClass_assoc, Homotopy.refl_h₁, exact_iff_epi_toCycles, exact_and_epi_g_iff_g_is_cokernel, LeftHomologyData.wπ_assoc, RightHomologyData.ofHasCokernelOfHasKernel_H, Exact.rightHomologyDataOfIsColimitCokernelCofork_Q, unop_X₁, ShortExact.mono_f, opcyclesIsoCokernel_hom, Exact.epi_f, RightHomologyData.ofIsColimitCokernelCofork_p, HomologicalComplex.opcycles_right_exact, DerivedCategory.triangleOfSES_obj₁, SnakeInput.isIso_δ, toCycles_moduleCatCyclesIso_hom, Homotopy.comp_h₁, FunctorEquivalence.inverse_obj_X₁, SnakeInput.L₀_g_δ, Splitting.leftHomologyData_π, unop_X₃, SnakeInput.snd_δ_assoc, LeftHomologyData.ofEpiOfIsIsoOfMono'_f', ShortExact.comp_extClass, CategoryTheory.CommSq.shortComplex'_X₁, kernel_ι_comp_cokernel_π_comp_cokernelToAbelianCoimage, RightHomologyData.wι, SnakeInput.w₀₂_τ₁_assoc, Exact.mono_g_iff, ab_exact_iff_ker_le_range, SnakeInput.L₁_f_φ₁, abLeftHomologyData_f', groupHomology.π_comp_H1Iso_hom_apply, CochainComplex.homOfDegreewiseSplit_f, LeftHomologyData.ofIsColimitCokernelCofork_H, sub_τ₁, HomologicalComplex.HomologySequence.δ_naturality_assoc, FunctorEquivalence.counitIso_hom_app_app_τ₁, ShortExact.moduleCat_exact_iff_function_exact, π_moduleCatCyclesIso_hom_assoc_apply, Homotopy.add_h₀, LeftHomologyData.ofHasCokernel_H, HomologicalComplex.cycles_left_exact, SnakeInput.naturality_δ_assoc, RightHomologyData.ofAbelian_p, ShortExact.isIso_δ, groupHomology.isoShortComplexH1_inv, SnakeInput.L₁'_X₁, ShortExact.injective_f, RightHomologyData.ofAbelian_ι, CategoryTheory.Abelian.Ext.covariant_sequence_exact₂', exact_iff_surjective_moduleCatToCycles, toCycles_comp_leftHomologyπ, ShortExact.singleTriangleIso_inv_hom₁, CochainComplex.mappingCone.quasiIso_descShortComplex, SnakeInput.φ₁_L₂_f, pOpcycles_comp_moduleCatOpcyclesIso_hom_assoc, exact_iff_exact_up_to_refinements, LeftHomologyData.wπ, ab_exact_iff, Exact.epi_f', abelianImageToKernel_comp_kernel_ι_assoc, Exact.exact_up_to_refinements, f'_cyclesMap', ShortExact.isIso_f_iff, SnakeInput.L₀X₂ToP_comp_φ₁_assoc, groupCohomology.H1InfRes_X₁, ShortExact.δ_comp_assoc, SnakeInput.L₁'_X₃, ShortExact.singleTriangle_obj₁, Splitting.g_s_assoc, Splitting.f_r_assoc, zero_apply, CochainComplex.mappingCone.cocycleOfDegreewiseSplit_triangleRotateShortComplexSplitting_v, LeftHomologyData.ofAbelian_π, Homotopy.op_h₃, ShortExact.homology_exact₃, LeftHomologyData.f'_i_assoc, HasRightHomology.hasKernel, RightHomologyData.IsPreservedBy.f, toCycles_naturality_assoc, CochainComplex.mappingConeHomOfDegreewiseSplitIso_inv_comp_triangle_mor₃_assoc, Homotopy.trans_h₀, RightHomologyData.ofHasCokernelOfHasKernel_ι, CategoryTheory.Functor.preservesFiniteLimits_iff_forall_exact_map_and_mono, moduleCat_exact_iff_ker_sub_range, moduleCat_exact_iff, CochainComplex.mappingCone.inl_v_descShortComplex_f_assoc, instEpiCokernelToAbelianCoimage, HomologicalComplex.shortComplexFunctor_obj_X₁, Splitting.op_s, pOpcycles_π_isoOpcyclesOfIsColimit_inv, exact_iff_mono_cokernel_desc, zero, Hom.comp_τ₁, mapNatTrans_τ₁, π_moduleCatCyclesIso_hom_apply, Exact.rightHomologyDataOfIsColimitCokernelCofork_ι, Homotopy.neg_h₀, exact_and_mono_f_iff_f_is_kernel, toComposableArrows_map, Exact.isIso_f', exact_iff_epi_imageToKernel, SnakeInput.mono_v₀₁_τ₁, exact_iff_image_eq_kernel, π₁_obj, toCycles_i, Splitting.r_f, moduleCatLeftHomologyData_f'_hom, HasLeftHomology.hasCokernel, SnakeInput.mono_L₂_f, Homotopy.comp_h₀, SnakeInput.epi_δ, ShortExact.δ_comp, unop_g, cokernelSequence_X₁, moduleCatMk_X₁_isModule, CochainComplex.shift_f_comp_mappingConeHomOfDegreewiseSplitIso_inv_assoc, ab_zero_apply, RightHomologyData.ofHasCokernelOfHasKernel_Q, LeftHomologyData.ofHasKernelOfHasCokernel_π, moduleCat_exact_iff_range_eq_ker, Homotopy.trans_h₁, FunctorEquivalence.unitIso_hom_app_τ₁_app, moduleCatMkOfKerLERange_X₁, Homotopy.comm₂, ChainComplex.mk_congr_succ_d₂, CochainComplex.mappingConeHomOfDegreewiseSplitIso_inv_comp_triangle_mor₃, Splitting.s_r, exact_iff_epi_kernel_lift, toComposableArrows_obj, abelianImageToKernel_comp_kernel_ι_comp_cokernel_π, groupCohomology.π_comp_H2Iso_hom_apply, SnakeInput.L₂'_X₂, RightHomologyData.ofIsColimitCokernelCofork_ι, Rep.coinvariantsShortComplex_X₁, HomologicalComplex.HomologySequence.δ_naturality, CochainComplex.mappingCone.inr_descShortComplex, Homotopy.ofEq_h₁, groupCohomology.π_comp_H1Iso_hom_apply, Splitting.isSplitMono_f, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.instMonoSheafAddCommGrpCatFShortComplex, op_X₃, toCycles_moduleCatCyclesIso_hom_apply, groupHomology.π_comp_H2Iso_hom_apply, CochainComplex.mappingCone.inl_v_descShortComplex_f, CategoryTheory.kernelCokernelCompSequence.snakeInput_L₃_X₁, HasRightHomology.hasCokernel, Splitting.r_f_assoc, CochainComplex.mappingCone.inr_f_descShortComplex_f, LeftHomologyMapData.commf'_assoc, moduleCatLeftHomologyData_π_hom, ModuleCat.free_shortExact_rank_add, ShortExact.extClass_comp_assoc, groupHomology.isIso_δ_of_isZero, Homotopy.compLeft_h₀, groupHomology.isoShortComplexH2_inv, HomologicalComplex.shortComplexTruncLE_X₁, exact_iff_exact_coimage_π, CategoryTheory.kernelCokernelCompSequence.snakeInput_L₁_X₁, moduleCatLeftHomologyData_H, neg_τ₁, moduleCatCyclesIso_inv_π_apply, CochainComplex.mappingConeHomOfDegreewiseSplitIso_inv_f, Splitting.ofIso_r, π_isoOpcyclesOfIsColimit_hom_assoc, unopMap_τ₃, ChainComplex.mk_d, zero_τ₁, Exact.rightHomologyDataOfIsColimitCokernelCofork_p, smul_τ₁, CategoryTheory.Abelian.Ext.contravariant_sequence_exact₃', moduleCat_pOpcycles_eq_iff, Splitting.f_r, ShortExact.hasInjectiveDimensionLT_X₃_iff, exact_iff_of_forks, HomologyData.ofEpiMonoFactorisation.f'_eq, groupHomology.π_comp_H1Iso_inv_apply, ShortExact.homology_exact₁, CategoryTheory.ObjectProperty.prop_X₁_of_shortExact, toCycles_moduleCatCyclesIso_hom_assoc_apply, opcyclesOpIso_hom_toCycles_op, LeftHomologyData.map_f', ab_exact_iff_range_eq_ker, epi_of_mono_of_epi_of_mono, Exact.isZero_X₂_iff, CategoryTheory.Abelian.Ext.covariant_sequence_exact₂, SnakeInput.w₁₃_τ₁, Exact.epi_f_iff, SnakeInput.exact_C₁_down, groupCohomology.isoShortComplexH1_inv, HomologicalComplex.shortComplexTruncLE_shortExact_δ_eq_zero, Exact.lift', CategoryTheory.CommSq.shortComplex_X₁, toCycles_moduleCatCyclesIso_hom_assoc, Exact.isIso_imageToKernel', Splitting.unop_s, f'_cyclesMap'_assoc, CategoryTheory.Abelian.Pseudoelement.pseudo_exact_of_exact, Exact.isIso_toCycles, LeftHomologyData.τ₁_ofEpiOfIsIsoOfMono_f', LeftHomologyData.unop_g', exact_iff_epi, abLeftHomologyData_H_coe, add_τ₁, Exact.lift_f, instMonoFKernelSequence, Splitting.isoBinaryBiproduct_hom, groupCohomology.isoShortComplexH2_inv, exact_iff_epi_imageToKernel', CochainComplex.mappingConeHomOfDegreewiseSplitIso_hom_f, toCycles_naturality, CategoryTheory.ObjectProperty.prop_iff_of_shortExact, Homotopy.symm_h₁, π_isoOpcyclesOfIsColimit_hom, Exact.lift_f_assoc, Homotopy.compLeft_h₁, CategoryTheory.Abelian.Ext.contravariant_sequence_exact₃, Homotopy.h₀_f, groupCohomology.mono_δ_of_isZero, fFunctor_map, ShortExact.moduleCat_injective_f, SnakeInput.instMonoFL₀'OfL₁, SnakeInput.epi_v₂₃_τ₁, groupHomology.π_comp_H2Iso_inv_apply, CategoryTheory.kernelCokernelCompSequence.snakeInput_L₂_X₁, ShortExact.homology_exact₂, ShortExact.ab_exact_iff_function_exact, isIso_iff, comp_τ₁_assoc, groupCohomology.isIso_δ_of_isZero, ShortExact.extClass_comp, f_pOpcycles, Splitting.s_r_assoc, f_pOpcycles_assoc, ShortExact.comp_δ_assoc, Splitting.isoBinaryBiproduct_inv
|
X₂ 📖 | CompOp | 541 mathmath: RightHomologyData.ofHasCokernelOfHasKernel_p, ShortExact.ab_finite_iff, pOpcycles_comp_moduleCatOpcyclesIso_hom_apply, SnakeInput.epi_L₃_g, Homotopy.h₀_f_assoc, Homotopy.comm₁, HomologyData.ofIso_right_p, Exact.epi_kernelLift, Splitting.op_r, cyclesOpIso_inv_op_iCycles_assoc, groupCohomology.isoCocycles₁_hom_comp_i_apply, LeftHomologyData.ofHasKernelOfHasCokernel_H, cokernel_π_comp_cokernelToAbelianCoimage_assoc, pOpcycles_π_isoOpcyclesOfIsColimit_inv_assoc, LeftHomologyData.unop_p, LeftHomologyData.ofAbelian_K, homology_π_ι_assoc, LeftHomologyData.cyclesIso_inv_comp_iCycles_assoc, moduleCatCyclesIso_hom_i_apply, op_g, SnakeInput.mono_v₀₁_τ₂, groupCohomology.cocyclesIso₀_hom_comp_f, abelianImageToKernel_comp_kernel_ι_comp_cokernel_π_assoc, unop_X₂, LeftHomologyData.IsPreservedBy.hg, CategoryTheory.kernelCokernelCompSequence.snakeInput_L₀_X₂, Homotopy.smul_h₁, zero_assoc, RightHomologyData.wp, RightHomologyData.ofAbelian_H, RightHomologyData.p_comp_opcyclesIso_inv, RightHomologyData.wp_assoc, ShortExact.ab_surjective_g, groupCohomology.mapCocycles₂_comp_i, abelianImageToKernel_comp_kernel_ι, Splitting.map_s, exact_iff_iCycles_pOpcycles_zero, exact_iff_kernel_ι_comp_cokernel_π_zero, ModuleCat.linearIndependent_shortExact, LeftHomologyData.ofHasKernelOfHasCokernel_K, groupCohomology.H0IsoOfIsTrivial_hom, moduleCatLeftHomologyData_i_hom, Homotopy.unop_h₂, Splitting.mono_f, CochainComplex.mappingCone.homologySequenceδ_triangleh, groupHomology.comap_coinvariantsKer_pOpcycles_range_subtype_pOpcycles_eq_top, LeftHomologyData.ofEpiOfIsIsoOfMono'_i, CategoryTheory.Abelian.Ext.contravariant_sequence_exact₁', ext_mk₀_f_comp_ext_mk₀_g, CategoryTheory.Pretriangulated.shortComplexOfDistTriangle_X₂, Splitting.s_g, moduleCatMk_X₂_isModule, mapCyclesIso_hom_iCycles_assoc, exact_and_mono_f_iff_of_iso, FunctorEquivalence.counitIso_inv_app_app_τ₂, exact_iff_isIso_imageToKernel', isoCyclesOfIsLimit_hom_iCycles_assoc, FunctorEquivalence.unitIso_inv_app_τ₂_app, ShortExact.epi_g, LeftHomologyData.instMonoI, Homotopy.compLeft_h₂, Hom.comm₁₂, groupHomology.isoCycles₁_inv_comp_iCycles_apply, RightHomologyData.unop_i, RightHomologyData.p_g'_assoc, groupCohomology.map_H0Iso_hom_f_apply, Homotopy.op_h₂, SnakeInput.L₀X₂ToP_comp_pullback_snd, LeftHomologyData.ofAbelian_H, ShortExact.comp_δ, RightHomologyData.copy_p, HomologicalComplex.pOpcycles_opcyclesIsoSc'_inv_assoc, Homotopy.g_h₃_assoc, Splitting.id, Homotopy.add_h₁, SnakeInput.w₁₃_τ₂, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.biprodAddEquiv_symm_biprodIsoProd_hom_toBiprod_apply, op_pOpcycles_opcyclesOpIso_hom, π_leftRightHomologyComparison'_ι, Exact.ab_finite, groupCohomology.mapCocycles₂_comp_i_apply, Homotopy.smul_h₂, CochainComplex.mappingCone.inr_f_descShortComplex_f_assoc, SnakeInput.exact_C₂_down, SnakeInput.instEpiGL₀', map_X₂, Splitting.s_g_assoc, groupCohomology.cocyclesMap_cocyclesIso₀_hom_f_apply, ShortExact.isIso_g_iff, isIso₂_of_shortExact_of_isIso₁₃', Splitting.map_r, Homotopy.sub_h₁, groupCohomology.instMonoModuleCatFH1InfRes, groupCohomology.cocyclesIso₀_inv_comp_iCocycles, groupCohomology.mapCocycles₂_comp_i_assoc, abToCycles_apply_coe, DerivedCategory.triangleOfSES_mor₁, p_fromOpcycles, CochainComplex.mappingCone.inr_descShortComplex_assoc, SnakeInput.δ_L₃_f, HomologicalComplex.instEpiGShortComplexTruncLE, HomologicalComplex.pOpcycles_opcyclesIsoSc'_inv, map_f, iCycles_g, Exact.ab_range_eq_ker, CategoryTheory.Abelian.Ext.contravariant_sequence_exact₂', HomologicalComplex.shortComplexFunctor_obj_X₂, moduleCat_zero_apply, Homotopy.neg_h₁, epi_of_epi_of_epi_of_epi, groupHomology.mapCycles₁_comp_i, Homotopy.refl_h₂, pOpcycles_comp_moduleCatOpcyclesIso_hom, FunctorEquivalence.inverse_obj_X₂, Hom.comp_τ₂, CochainComplex.triangleOfDegreewiseSplit_obj₂, comp_τ₂, exact_iff_exact_image_ι, gFunctor_map, LeftHomologyData.ofHasKernelOfHasCokernel_i, CochainComplex.g_shortComplexTruncLEX₃ToTruncGE, Exact.isIso_imageToKernel, leftRightHomologyComparison'_eq_descH, Splitting.epi_g, exact_iff_isIso_imageToKernel, RightHomologyData.wι_assoc, ShortExact.ab_injective_f, abLeftHomologyData_π, homology_π_ι, cokernel_π_comp_cokernelToAbelianCoimage, CategoryTheory.Abelian.Ext.covariant_sequence_exact₁', LeftHomologyData.ofAbelian_i, LeftHomologyData.map_i, moduleCatCyclesIso_hom_i_assoc, unop_f, Homotopy.compRight_h₁, instEpiPOpcycles, HomologyData.ofEpiMonoFactorisation.g'_eq, RightHomologyData.instEpiP, CategoryTheory.Functor.mapShortComplex_map_τ₂, CategoryTheory.Abelian.Ext.covariant_sequence_exact₃', SnakeInput.mono_L₀_f, HomologyData.ofIso_left_i, LeftHomologyData.IsPreservedBy.g, Exact.mono_cokernelDesc, exact_iff_surjective_abToCycles, Homotopy.ofEq_h₂, exact_iff_of_hasForget, groupHomology.mapCycles₂_comp_i, groupCohomology.map_H0Iso_hom_f, moduleCat_pOpcycles_eq_zero_iff, Homotopy.comm₃, opcyclesIsoCokernel_inv, ab_exact_iff_function_exact, op_pOpcycles_opcyclesOpIso_hom_assoc, ChainComplex.mk_congr_succ_X₃, RightHomologyData.p_comp_opcyclesIso_inv_assoc, SnakeInput.w₁₃_τ₂_assoc, HomologicalComplex.instMonoFShortComplexTruncLE, RightHomologyData.ofAbelian_Q, moduleCatCyclesIso_inv_π_assoc_apply, CochainComplex.mappingCone.triangleRotateShortComplex_X₂, SnakeInput.snd_δ, groupCohomology.H1InfRes_X₂, RightHomologyData.IsPreservedBy.hf, SnakeInput.snd_δ_inr, Splitting.g_s, HomologyData.comm_assoc, moduleCatCyclesIso_hom_i, Hom.comm₁₂_assoc, HomologicalComplex.cyclesIsoSc'_hom_iCycles_assoc, HomologicalComplex.isIso_homologyMap_shortComplexTruncLE_g, CochainComplex.shift_f_comp_mappingConeHomOfDegreewiseSplitIso_inv, fFunctor_obj, SnakeInput.L₁_f_φ₁_assoc, Splitting.splitMono_f_retraction, groupHomology.H1CoresCoinfOfTrivial_X₂, LeftHomologyData.copy_i, instMonoAbelianImageToKernel, ShortExact.singleTriangleIso_inv_hom₂, groupHomology.H1CoresCoinf_X₂, SnakeInput.φ₁_L₂_f_assoc, SnakeInput.L₀X₂ToP_comp_φ₁, Homotopy.compRight_h₂, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.mk₀_f_comp_biprodAddEquiv_symm_biprodIsoProd_hom, Exact.leftHomologyDataOfIsLimitKernelFork_K, map_g, groupHomology.mapCycles₁_comp_i_apply, cyclesIsoKernel_hom, SnakeInput.L₂'_X₃, CategoryTheory.ComposableArrows.Exact.opcyclesIsoCycles_hom_fac, groupCohomology.isoCocycles₂_hom_comp_i, p_opcyclesMap', Exact.moduleCat_range_eq_ker, toCycles_i_assoc, groupCohomology.isoCocycles₁_inv_comp_iCocycles_apply, CategoryTheory.ComposableArrows.IsComplex.opcyclesToCycles_fac, moduleCatMkOfKerLERange_X₂, abLeftHomologyData_K_coe, CategoryTheory.IsPullback.mono_shortComplex'_f, SnakeInput.epi_v₂₃_τ₂, groupCohomology.cocyclesIso₀_inv_comp_iCocycles_assoc, CategoryTheory.Functor.preservesFiniteColimits_iff_forall_exact_map_and_epi, LeftHomologyData.f'_i, groupCohomology.instMonoModuleCatFShortComplexH0, HasLeftHomology.hasKernel, ShortExact.singleTriangle_obj₂, SnakeInput.L₁'_X₂, ShortExact.singleTriangle_mor₁, HomologicalComplex.g_shortComplexTruncLEX₃ToTruncGE, ShortExact.comp_extClass_assoc, Homotopy.refl_h₁, Exact.shortExact, CategoryTheory.ObjectProperty.prop_X₂_of_shortExact, LeftHomologyData.cyclesIso_hom_comp_i_assoc, exact_and_epi_g_iff_g_is_cokernel, SnakeInput.w₀₂_τ₂, HomologicalComplex.mono_homologyMap_shortComplexTruncLE_g, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.instEpiSheafAddCommGrpCatGShortComplex, LeftHomologyData.wπ_assoc, Splitting.unop_r, RightHomologyData.ofHasCokernelOfHasKernel_H, Exact.rightHomologyDataOfIsColimitCokernelCofork_Q, groupHomology.instEpiModuleCatGH1CoresCoinf, SnakeInput.functorP_map, unopMap_τ₂, ShortExact.mono_f, opcyclesIsoCokernel_hom, HomologicalComplex.shortComplexFunctor'_obj_X₂, Splitting.ofIso_s, CategoryTheory.kernelCokernelCompSequence.snakeInput_L₂_X₂, groupHomology.isoCycles₂_hom_comp_i_apply, CategoryTheory.Abelian.Ext.covariant_sequence_exact₃, SnakeInput.exact_C₂_up, HomologicalComplex.opcycles_right_exact, groupCohomology.isoCocycles₂_inv_comp_iCocycles_apply, groupCohomology.cocyclesMap_cocyclesIso₀_hom_f_assoc, mono_τ₂_of_exact_of_mono, groupCohomology.isoCocycles₁_inv_comp_iCocycles, HomologicalComplex.quasiIsoAt_shortComplexTruncLE_g, Homotopy.comp_h₁, SnakeInput.L₀_g_δ, SnakeInput.snd_δ_assoc, ShortExact.comp_extClass, HomologicalComplex.extend.rightHomologyData_p, Homotopy.trans_h₂, kernel_ι_comp_cokernel_π_comp_cokernelToAbelianCoimage, RightHomologyData.wι, iCycles_g_assoc, Exact.mono_g_iff, exact_iff_i_p_zero, RightHomologyData.pOpcycles_comp_opcyclesIso_hom, ab_exact_iff_ker_le_range, SnakeInput.L₁_f_φ₁, π_leftRightHomologyComparison_ι_assoc, abLeftHomologyData_f', groupHomology.π_comp_H1Iso_hom_apply, LeftHomologyData.wi, ShortExact.singleTriangle_mor₂, Exact.leftHomologyDataOfIsLimitKernelFork_π, Splitting.splitEpi_g_section_, ShortExact.moduleCat_exact_iff_function_exact, π_moduleCatCyclesIso_hom_assoc_apply, opMap_τ₂, comp_homologyMap_comp_assoc, HomologicalComplex.cycles_left_exact, RightHomologyData.ofAbelian_p, CategoryTheory.ObjectProperty.prop_X₂_of_exact, groupHomology.isoShortComplexH1_inv, SnakeInput.L₁'_X₁, π_leftRightHomologyComparison'_ι_assoc, ShortExact.injective_f, groupHomology.isoCycles₁_hom_comp_i_apply, RightHomologyData.ofAbelian_ι, groupCohomology.cocyclesIso₀_hom_comp_f_assoc, CategoryTheory.Abelian.Ext.covariant_sequence_exact₂', exact_iff_surjective_moduleCatToCycles, RightHomologyData.ofEpiOfIsIsoOfMono_p, CochainComplex.mappingCone.quasiIso_descShortComplex, CategoryTheory.ObjectProperty.IsClosedUnderExtensions.prop_X₂_of_shortExact, SnakeInput.φ₁_L₂_f, CategoryTheory.IsPushout.epi_shortComplex_g, pOpcycles_comp_moduleCatOpcyclesIso_hom_assoc, exact_iff_exact_up_to_refinements, LeftHomologyData.wπ, ab_exact_iff, moduleCatCyclesIso_hom_i_assoc_apply, abelianImageToKernel_comp_kernel_ι_assoc, RightHomologyData.map_p, ShortExact.isIso_f_iff, SnakeInput.L₀X₂ToP_comp_φ₁_assoc, LeftHomologyData.op_p, ShortExact.δ_comp_assoc, epi_τ₂_of_exact_of_epi, Splitting.g_s_assoc, ShortExact.surjective_g, Splitting.f_r_assoc, zero_apply, groupCohomology.mapCocycles₁_comp_i_assoc, id_τ₂, LeftHomologyData.ofAbelian_π, ModuleCat.free_shortExact, ShortExact.homology_exact₃, abLeftHomologyData_i, groupCohomology.cocyclesMap_cocyclesIso₀_hom_f, sub_τ₂, LeftHomologyData.f'_i_assoc, HasRightHomology.hasKernel, instIsIsoτ₂, CategoryTheory.CommSq.shortComplex'_X₂, Homotopy.unop_h₁, RightHomologyData.IsPreservedBy.f, leftRightHomologyComparison'_eq_liftH, Hom.comm₂₃, CochainComplex.mappingConeHomOfDegreewiseSplitIso_inv_comp_triangle_mor₃_assoc, neg_τ₂, CategoryTheory.kernelCokernelCompSequence.snakeInput_L₁_X₂, HomologyData.exact_iff_i_p_zero, RightHomologyData.ofHasCokernelOfHasKernel_ι, CategoryTheory.Functor.preservesFiniteLimits_iff_forall_exact_map_and_mono, isIso₂_of_shortExact_of_isIso₁₃, moduleCat_exact_iff_ker_sub_range, moduleCat_exact_iff, SnakeInput.w₀₂_τ₂_assoc, groupHomology.isoCycles₂_inv_comp_iCycles_apply, groupHomology.isoCycles₂_inv_comp_iCycles_assoc, CochainComplex.mappingCone.inl_v_descShortComplex_f_assoc, instEpiCokernelToAbelianCoimage, Splitting.op_s, exact_and_epi_g_iff_of_iso, groupHomology.isoCycles₁_inv_comp_iCycles_assoc, pOpcycles_π_isoOpcyclesOfIsColimit_inv, HomologicalComplex.shortComplexTruncLE_X₂, p_opcyclesMap_assoc, exact_iff_mono_cokernel_desc, isoCyclesOfIsLimit_inv_ι_assoc, zero, π_moduleCatCyclesIso_hom_apply, Exact.rightHomologyDataOfIsColimitCokernelCofork_ι, gFunctor_obj, exact_and_mono_f_iff_f_is_kernel, toComposableArrows_map, add_τ₂, groupCohomology.isoCocycles₂_inv_comp_iCocycles, comp_τ₂_assoc, ShortExact.hasInjectiveDimensionLT_X₂, HomologicalComplex.extend.leftHomologyData_i, homologyIsoImageICyclesCompPOpcycles_ι, exact_iff_epi_imageToKernel, exact_iff_image_eq_kernel, LeftHomologyMapData.commi_assoc, π_homologyMap_ι_assoc, cyclesMap'_i_assoc, moduleCatCyclesIso_inv_iCycles, toCycles_i, Homotopy.add_h₂, ShortExact.hasProjectiveDimensionLT_X₂, Splitting.r_f, LeftHomologyData.cyclesIso_hom_comp_i, Splitting.isSplitEpi_g, HasLeftHomology.hasCokernel, groupHomology.mapCycles₂_comp_i_assoc, groupCohomology.isoCocycles₁_hom_comp_i, SnakeInput.mono_L₂_f, ShortExact.δ_comp, groupCohomology.mapCocycles₁_comp_i_apply, unop_g, Homotopy.symm_h₂, HomologicalComplex.epi_homologyMap_shortComplexTruncLE_g, mapCyclesIso_hom_iCycles, CochainComplex.shift_f_comp_mappingConeHomOfDegreewiseSplitIso_inv_assoc, ab_zero_apply, Homotopy.sub_h₂, RightHomologyData.ofHasCokernelOfHasKernel_Q, LeftHomologyData.ofHasKernelOfHasCokernel_π, moduleCat_exact_iff_range_eq_ker, ModuleCat.free_shortExact_finrank_add, Homotopy.trans_h₁, isoCyclesOfIsLimit_hom_iCycles, Homotopy.comm₂, ChainComplex.mk_congr_succ_d₂, op_f, CochainComplex.mappingConeHomOfDegreewiseSplitIso_inv_comp_triangle_mor₃, Splitting.s_r, π_leftRightHomologyComparison_ι, SnakeInput.naturality_φ₂, moduleCatLeftHomologyData_K, exact_iff_epi_kernel_lift, toComposableArrows_obj, abelianImageToKernel_comp_kernel_ι_comp_cokernel_π, groupCohomology.π_comp_H2Iso_hom_apply, SnakeInput.L₂'_X₂, HomologicalComplex.cyclesIsoSc'_hom_iCycles, groupHomology.isoCycles₁_hom_comp_i_assoc, cyclesMap_i_assoc, CochainComplex.mappingCone.inr_descShortComplex, Homotopy.ofEq_h₁, Homotopy.op_h₁, groupCohomology.π_comp_H1Iso_hom_apply, Splitting.isSplitMono_f, mapNatTrans_τ₂, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.instMonoSheafAddCommGrpCatFShortComplex, SnakeInput.L₀X₂ToP_comp_pullback_snd_assoc, groupHomology.instEpiModuleCatGShortComplexH0, toCycles_moduleCatCyclesIso_hom_apply, groupHomology.π_comp_H2Iso_hom_apply, CochainComplex.mappingCone.inl_v_descShortComplex_f, HasRightHomology.hasCokernel, Splitting.r_f_assoc, CochainComplex.mappingCone.inr_f_descShortComplex_f, moduleCatLeftHomologyData_π_hom, groupHomology.isoCycles₁_inv_comp_iCycles, groupHomology.H1CoresCoinfOfTrivial_g_epi, ShortExact.singleTriangleIso_hom_hom₂, cycles_ext_iff, ModuleCat.free_shortExact_rank_add, ShortExact.extClass_comp_assoc, CochainComplex.g_shortComplexTruncLEX₃ToTruncGE_assoc, groupHomology.isoShortComplexH2_inv, Homotopy.g_h₃, groupHomology.mapCycles₂_comp_i_apply, DerivedCategory.triangleOfSES_obj₂, Hom.comm₂₃_assoc, Homotopy.comp_h₂, groupCohomology.cocyclesIso₀_hom_comp_f_apply, exact_iff_exact_coimage_π, RightHomologyData.op_i, groupHomology.isoCycles₂_hom_comp_i, moduleCatLeftHomologyData_H, moduleCatCyclesIso_inv_iCycles_assoc_apply, RightHomologyMapData.commp_assoc, groupHomology.isoCycles₂_inv_comp_iCycles, moduleCatCyclesIso_inv_π_apply, CochainComplex.mappingConeHomOfDegreewiseSplitIso_inv_f, Splitting.ofIso_r, groupCohomology.isoCocycles₁_inv_comp_iCocycles_assoc, π_isoOpcyclesOfIsColimit_hom_assoc, ChainComplex.mk_d, Homotopy.neg_h₂, π₂_obj, FunctorEquivalence.counitIso_hom_app_app_τ₂, Exact.rightHomologyDataOfIsColimitCokernelCofork_p, CategoryTheory.Abelian.Ext.contravariant_sequence_exact₃', moduleCat_pOpcycles_eq_iff, Splitting.f_r, exact_iff_of_forks, HomologyData.ofEpiMonoFactorisation.f'_eq, smul_τ₂, groupHomology.π_comp_H1Iso_inv_apply, DerivedCategory.triangleOfSES_mor₂, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.shortComplex_X₂, RightHomologyData.pOpcycles_comp_opcyclesIso_hom_assoc, groupCohomology.isoCocycles₂_hom_comp_i_apply, ShortExact.homology_exact₁, toCycles_moduleCatCyclesIso_hom_assoc_apply, LeftHomologyData.ofEpiOfIsIsoOfMono_i, ab_exact_iff_range_eq_ker, CategoryTheory.Abelian.Ext.contravariant_sequence_exact₁, moduleCatMk_X₂_carrier, cyclesMap_i, Exact.isZero_X₂_iff, SnakeInput.naturality_φ₂_assoc, groupHomology.isoCycles₁_hom_comp_i, groupCohomology.cocyclesIso₀_inv_comp_iCocycles_apply, Rep.coinvariantsShortComplex_X₂, LeftHomologyData.wi_assoc, cyclesMap'_i, Exact.epi_f_iff, p_opcyclesMap'_assoc, mono_of_mono_of_mono_of_mono, cyclesOpIso_inv_op_iCycles, ModuleCat.span_rightExact, SnakeInput.epi_L₁_g, groupCohomology.isoShortComplexH1_inv, LeftHomologyMapData.commi, Exact.isIso_imageToKernel', comp_homologyMap_comp, cokernelSequence_X₂, Splitting.unop_s, HomologyData.comm, Exact.leftHomologyDataOfIsLimitKernelFork_i, p_fromOpcycles_assoc, FunctorEquivalence.unitIso_hom_app_τ₂_app, groupCohomology.map_H0Iso_hom_f_assoc, abLeftHomologyData_H_coe, ModuleCat.smulShortComplex_X₂, instFreeCarrierX₂ModuleCatProjectiveShortComplex, cyclesIsoKernel_inv, instMonoFKernelSequence, Splitting.isoBinaryBiproduct_hom, groupCohomology.isoShortComplexH2_inv, exact_iff_epi_imageToKernel', CochainComplex.mappingConeHomOfDegreewiseSplitIso_hom_f, groupCohomology.isoCocycles₂_hom_comp_i_assoc, CategoryTheory.ObjectProperty.prop_iff_of_shortExact, HomologicalComplex.g_shortComplexTruncLEX₃ToTruncGE_assoc, Homotopy.symm_h₁, ShortExact.moduleCat_surjective_g, π_isoOpcyclesOfIsColimit_hom, Homotopy.compLeft_h₁, ModuleCat.smulShortComplex_g_epi, Homotopy.h₀_f, groupCohomology.isoCocycles₁_hom_comp_i_assoc, RightHomologyData.ofEpiOfIsIsoOfMono'_p, Hom.id_τ₂, fFunctor_map, ShortExact.moduleCat_injective_f, groupHomology.mapCycles₁_comp_i_assoc, SnakeInput.instMonoFL₀'OfL₁, moduleCatCyclesIso_inv_iCycles_assoc, instEpiGCokernelSequence, homologyIsoImageICyclesCompPOpcycles_ι_assoc, moduleCatCyclesIso_inv_iCycles_apply, zero_τ₂, groupHomology.π_comp_H2Iso_inv_apply, ShortExact.homology_exact₂, CategoryTheory.kernelCokernelCompSequence.snakeInput_L₃_X₂, ShortExact.ab_exact_iff_function_exact, p_opcyclesMap, kernelSequence_X₂, isIso_iff, RightHomologyMapData.commp, groupCohomology.mapCocycles₁_comp_i, CategoryTheory.CommSq.shortComplex_X₂, opcycles_ext_iff, groupHomology.isoCycles₂_hom_comp_i_assoc, moduleCatMk_X₂_isAddCommGroup, op_X₂, ShortExact.extClass_comp, f_pOpcycles, groupCohomology.isoCocycles₂_inv_comp_iCocycles_assoc, isoCyclesOfIsLimit_inv_ι, LeftHomologyData.cyclesIso_inv_comp_iCycles, Splitting.s_r_assoc, abCyclesIso_inv_apply_iCycles, f_pOpcycles_assoc, ShortExact.comp_δ_assoc, π_homologyMap_ι, instMonoICycles, Splitting.isoBinaryBiproduct_inv, RightHomologyData.p_g'
|
X₃ 📖 | CompOp | 413 mathmath: ShortExact.ab_finite_iff, SnakeInput.exact_C₃_up, ShortExact.instHasSmallLocalizedShiftedHomHomologicalComplexIntUpQuasiIsoX₃CochainComplexMapSingleFunctorOfNatX₁, SnakeInput.epi_L₃_g, ShortExact.epi_δ, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.shortComplex_X₃, fromOpcycles_naturality, Exact.comp_descToInjective, Homotopy.comp_h₃, Exact.epi_kernelLift, Splitting.op_r, RightHomologyData.ι_g', SnakeInput.op_δ, groupCohomology.isoCocycles₁_hom_comp_i_apply, LeftHomologyData.ofHasKernelOfHasCokernel_H, cokernel_π_comp_cokernelToAbelianCoimage_assoc, SnakeInput.naturality_δ, LeftHomologyData.ofAbelian_K, map_X₃, groupHomology.mono_δ_of_isZero, RightHomologyData.IsPreservedBy.hg', moduleCatCyclesIso_hom_i_apply, abelianImageToKernel_comp_kernel_ι_comp_cokernel_π_assoc, LeftHomologyData.IsPreservedBy.hg, zero_assoc, RightHomologyData.ofAbelian_H, ShortExact.mono_δ, opcyclesMap'_g', ShortExact.ab_surjective_g, abelianImageToKernel_comp_kernel_ι, Splitting.map_s, exact_iff_kernel_ι_comp_cokernel_π_zero, LeftHomologyData.ofHasKernelOfHasCokernel_K, CategoryTheory.Abelian.Ext.preadditiveYoneda_homologySequenceδ_singleTriangle_apply, opMap_τ₁, moduleCatLeftHomologyData_i_hom, CochainComplex.mappingCone.homologySequenceδ_triangleh, opcyclesMap'_g'_assoc, CategoryTheory.Abelian.Ext.contravariant_sequence_exact₁', ext_mk₀_f_comp_ext_mk₀_g, Splitting.s_g, SnakeInput.mono_v₀₁_τ₃, groupHomology.H1CoresCoinf_X₃, CategoryTheory.kernelCokernelCompSequence.snakeInput_L₂_X₃, exact_iff_isIso_imageToKernel', isoCyclesOfIsLimit_hom_iCycles_assoc, ShortExact.epi_g, Homotopy.compLeft_h₂, Homotopy.op_h₀, groupHomology.isoCycles₁_inv_comp_iCycles_apply, Hom.comp_τ₃, RightHomologyData.p_g'_assoc, SnakeInput.w₀₂_τ₃_assoc, SnakeInput.w₁₃_τ₃, SnakeInput.L₀X₂ToP_comp_pullback_snd, LeftHomologyData.ofAbelian_H, ShortExact.comp_δ, RightHomologyData.map_g', Homotopy.g_h₃_assoc, Splitting.id, op_X₁, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.biprodAddEquiv_symm_biprodIsoProd_hom_toBiprod_apply, groupCohomology.mapCocycles₂_comp_i_apply, Homotopy.smul_h₂, CochainComplex.mappingCone.inr_f_descShortComplex_f_assoc, ShortExact.hasProjectiveDimensionLT_X₃_iff, SnakeInput.instEpiGL₀', Splitting.s_g_assoc, Splitting.rightHomologyData_ι, ShortExact.isIso_g_iff, CategoryTheory.kernelCokernelCompSequence.snakeInput_L₁_X₃, postcomp_extClass_surjective_of_projective_X₂, ModuleCat.smulShortComplex_X₃_isAddCommGroup, abToCycles_apply_coe, SnakeInput.w₁₃_τ₃_assoc, p_fromOpcycles, Homotopy.refl_h₃, CochainComplex.mappingCone.inr_descShortComplex_assoc, SnakeInput.δ_L₃_f, HomologicalComplex.instEpiGShortComplexTruncLE, iCycles_g, Exact.ab_range_eq_ker, CategoryTheory.Abelian.Ext.contravariant_sequence_exact₂', moduleCat_zero_apply, CategoryTheory.Abelian.Ext.preadditiveCoyoneda_homologySequenceδ_singleTriangle_apply, FunctorEquivalence.unitIso_hom_app_τ₃_app, LeftHomologyData.ofIsLimitKernelFork_H, add_τ₃, Homotopy.refl_h₂, exact_iff_exact_image_ι, gFunctor_map, LeftHomologyData.ofHasKernelOfHasCokernel_i, RightHomologyMapData.commg'_assoc, FunctorEquivalence.counitIso_inv_app_app_τ₃, ShortExact.extClass_hom, CochainComplex.g_shortComplexTruncLEX₃ToTruncGE, Exact.isIso_imageToKernel, Splitting.epi_g, exact_iff_isIso_imageToKernel, RightHomologyData.wι_assoc, abLeftHomologyData_π, cokernel_π_comp_cokernelToAbelianCoimage, CategoryTheory.Abelian.Ext.covariant_sequence_exact₁', LeftHomologyData.ofAbelian_i, moduleCatMk_X₃_isAddCommGroup, RightHomologyData.ofEpiOfIsIsoOfMono'_g'_τ₃, unop_f, HomologyData.ofEpiMonoFactorisation.g'_eq, CategoryTheory.Abelian.Ext.covariant_sequence_exact₃', Homotopy.smul_h₃, DerivedCategory.triangleOfSES_obj₃, LeftHomologyData.IsPreservedBy.g, Exact.mono_cokernelDesc, exact_iff_surjective_abToCycles, Homotopy.ofEq_h₂, moduleCatMk_X₃_carrier, unopMap_τ₁, Homotopy.comm₃, ab_exact_iff_function_exact, homologyι_comp_fromOpcycles_assoc, groupHomology.epi_δ_of_isZero, Exact.mono_g, instIsIsoτ₃, moduleCatCyclesIso_inv_π_assoc_apply, FunctorEquivalence.inverse_obj_X₃, SnakeInput.snd_δ, SnakeInput.snd_δ_inr, Splitting.g_s, LeftHomologyData.ofIsLimitKernelFork_i, groupCohomology.epi_δ_of_isZero, HomologicalComplex.isIso_homologyMap_shortComplexTruncLE_g, precomp_extClass_surjective_of_projective_X₂, SnakeInput.mono_δ, CochainComplex.shift_f_comp_mappingConeHomOfDegreewiseSplitIso_inv, HomologicalComplex.HomologySequence.quasiIso_τ₃, instMonoAbelianImageToKernel, Homotopy.compRight_h₂, CochainComplex.mappingCone.triangleRotateShortComplex_X₃, Splitting.rightHomologyData_Q, Exact.leftHomologyDataOfIsLimitKernelFork_K, map_g, groupHomology.mapCycles₁_comp_i_apply, cyclesIsoKernel_hom, Homotopy.compRight_h₃, SnakeInput.L₂'_X₃, fromOpcycles_naturality_assoc, fromOpcycles_op_cyclesOpIso_inv_assoc, Exact.moduleCat_range_eq_ker, groupCohomology.isoCocycles₁_inv_comp_iCocycles_apply, RightHomologyMapData.commg', abLeftHomologyData_K_coe, rightHomologyι_comp_fromOpcycles_assoc, RightHomologyData.unop_f', groupCohomology.H1InfRes_X₃, CategoryTheory.Functor.preservesFiniteColimits_iff_forall_exact_map_and_epi, ShortExact.singleTriangleIso_inv_hom₃, HasLeftHomology.hasKernel, SnakeInput.L₁'_X₂, HomologicalComplex.g_shortComplexTruncLEX₃ToTruncGE, SnakeInput.L₂'_X₁, ShortExact.comp_extClass_assoc, exact_and_epi_g_iff_g_is_cokernel, HomologicalComplex.mono_homologyMap_shortComplexTruncLE_g, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.instEpiSheafAddCommGrpCatGShortComplex, CategoryTheory.Functor.mapShortComplex_map_τ₃, LeftHomologyData.wπ_assoc, RightHomologyData.IsPreservedBy.g', smul_τ₃, Splitting.unop_r, RightHomologyData.ofHasCokernelOfHasKernel_H, RightHomologyData.ofEpiOfIsIsoOfMono_g', Homotopy.symm_h₃, unop_X₁, groupHomology.instEpiModuleCatGH1CoresCoinf, SnakeInput.functorP_map, Splitting.ofIso_s, HomologicalComplex.HomologySequence.epi_homologyMap_τ₃, groupHomology.isoCycles₂_hom_comp_i_apply, HomologicalComplex.opcycles_right_exact, Exact.desc', CochainComplex.triangleOfDegreewiseSplit_obj₃, groupCohomology.isoCocycles₂_inv_comp_iCocycles_apply, SnakeInput.isIso_δ, HomologicalComplex.quasiIsoAt_shortComplexTruncLE_g, ModuleCat.smulShortComplex_X₃_carrier, CategoryTheory.kernelCokernelCompSequence.snakeInput_L₃_X₃, SnakeInput.L₀_g_δ, unop_X₃, SnakeInput.snd_δ_assoc, ShortExact.comp_extClass, zero_τ₃, Homotopy.sub_h₃, HomologicalComplex.shortComplexFunctor'_obj_X₃, Homotopy.trans_h₂, kernel_ι_comp_cokernel_π_comp_cokernelToAbelianCoimage, RightHomologyData.wι, iCycles_g_assoc, Exact.mono_g_iff, ab_exact_iff_ker_le_range, abLeftHomologyData_f', groupHomology.π_comp_H1Iso_hom_apply, LeftHomologyData.wi, CochainComplex.homOfDegreewiseSplit_f, ShortExact.singleTriangle_mor₂, Exact.leftHomologyDataOfIsLimitKernelFork_π, Splitting.splitEpi_g_section_, HomologicalComplex.HomologySequence.δ_naturality_assoc, ShortExact.moduleCat_exact_iff_function_exact, π_moduleCatCyclesIso_hom_assoc_apply, Exact.comp_descToInjective_assoc, HomologicalComplex.cycles_left_exact, SnakeInput.naturality_δ_assoc, HomologicalComplex.extend.rightHomologyData_g', ShortExact.isIso_δ, groupHomology.isoShortComplexH1_inv, groupHomology.isoCycles₁_hom_comp_i_apply, RightHomologyData.ofAbelian_ι, ShortExact.singleTriangleIso_hom_hom₃, CategoryTheory.Abelian.Ext.covariant_sequence_exact₂', exact_iff_surjective_moduleCatToCycles, RightHomologyData.exact_iff_mono_g', CochainComplex.mappingCone.quasiIso_descShortComplex, exact_iff_mono, CategoryTheory.IsPushout.epi_shortComplex_g, LeftHomologyData.wπ, Exact.mono_fromOpcycles, moduleCatCyclesIso_hom_i_assoc_apply, RightHomologyData.ι_g'_assoc, abelianImageToKernel_comp_kernel_ι_assoc, ShortExact.isIso_f_iff, ShortExact.δ_comp_assoc, SnakeInput.L₁'_X₃, Splitting.g_s_assoc, Exact.isIso_g', ShortExact.surjective_g, zero_apply, CochainComplex.mappingCone.cocycleOfDegreewiseSplit_triangleRotateShortComplexSplitting_v, LeftHomologyData.ofAbelian_π, SnakeInput.epi_v₂₃_τ₃, ShortExact.homology_exact₃, abLeftHomologyData_i, Homotopy.trans_h₃, HasRightHomology.hasKernel, Homotopy.neg_h₃, Homotopy.unop_h₁, CategoryTheory.Abelian.Ext.covariant_sequence_exact₁, moduleCatMkOfKerLERange_X₃, Hom.comm₂₃, CochainComplex.mappingConeHomOfDegreewiseSplitIso_inv_comp_triangle_mor₃_assoc, RightHomologyData.ofHasCokernelOfHasKernel_ι, moduleCat_exact_iff_ker_sub_range, groupHomology.isoCycles₂_inv_comp_iCycles_apply, CochainComplex.mappingCone.inl_v_descShortComplex_f_assoc, instEpiCokernelToAbelianCoimage, Exact.g_desc_assoc, Homotopy.compLeft_h₃, exact_and_epi_g_iff_of_iso, exact_iff_mono_cokernel_desc, isoCyclesOfIsLimit_inv_ι_assoc, zero, π_moduleCatCyclesIso_hom_apply, gFunctor_obj, π₃_obj, Hom.id_τ₃, exact_and_mono_f_iff_f_is_kernel, toComposableArrows_map, RightHomologyData.op_f', FunctorEquivalence.unitIso_inv_app_τ₃_app, ShortExact.hasInjectiveDimensionLT_X₃, comp_τ₃, exact_iff_epi_imageToKernel, exact_iff_image_eq_kernel, RightHomologyData.ofIsLimitKernelFork_ι, LeftHomologyData.ofIsLimitKernelFork_K, homologyι_comp_fromOpcycles, Homotopy.add_h₂, Splitting.r_f, ShortExact.hasProjectiveDimensionLT_X₃, Splitting.isSplitEpi_g, HasLeftHomology.hasCokernel, SnakeInput.epi_δ, ShortExact.δ_comp, groupCohomology.mapCocycles₁_comp_i_apply, RightHomologyData.ofHasKernel_ι, Homotopy.symm_h₂, HomologicalComplex.epi_homologyMap_shortComplexTruncLE_g, CochainComplex.shift_f_comp_mappingConeHomOfDegreewiseSplitIso_inv_assoc, ab_zero_apply, Homotopy.sub_h₂, HomologicalComplex.HomologySequence.isIso_homologyMap_τ₃, LeftHomologyData.ofHasKernelOfHasCokernel_π, moduleCat_exact_iff_range_eq_ker, isoCyclesOfIsLimit_hom_iCycles, Homotopy.comm₂, op_f, CochainComplex.mappingConeHomOfDegreewiseSplitIso_inv_comp_triangle_mor₃, Splitting.s_r, moduleCatLeftHomologyData_K, exact_iff_epi_kernel_lift, toComposableArrows_obj, sub_τ₃, abelianImageToKernel_comp_kernel_ι_comp_cokernel_π, cokernelSequence_X₃, groupCohomology.π_comp_H2Iso_hom_apply, CategoryTheory.CommSq.shortComplex'_X₃, LeftHomologyData.ofIsLimitKernelFork_π, HomologicalComplex.HomologySequence.δ_naturality, CochainComplex.mappingCone.inr_descShortComplex, Homotopy.op_h₁, groupCohomology.π_comp_H1Iso_hom_apply, CategoryTheory.CommSq.shortComplex_X₃, moduleCatMk_X₃_isModule, CategoryTheory.Pretriangulated.shortComplexOfDistTriangle_X₃, SnakeInput.L₀X₂ToP_comp_pullback_snd_assoc, SnakeInput.exact_C₃_down, op_X₃, groupHomology.instEpiModuleCatGShortComplexH0, toCycles_moduleCatCyclesIso_hom_apply, groupHomology.π_comp_H2Iso_hom_apply, CochainComplex.mappingCone.inl_v_descShortComplex_f, CategoryTheory.Abelian.Ext.contravariant_sequence_exact₂, Splitting.r_f_assoc, CochainComplex.mappingCone.inr_f_descShortComplex_f, moduleCatLeftHomologyData_π_hom, RightHomologyData.ofHasKernel_H, ShortExact.singleTriangle_obj₃, groupHomology.H1CoresCoinfOfTrivial_g_epi, CategoryTheory.ObjectProperty.prop_X₃_of_shortExact, ModuleCat.free_shortExact_rank_add, ShortExact.extClass_comp_assoc, CochainComplex.g_shortComplexTruncLEX₃ToTruncGE_assoc, groupHomology.isIso_δ_of_isZero, groupHomology.isoShortComplexH2_inv, Homotopy.g_h₃, exact_iff_mono_fromOpcycles, Exact.g_desc, comp_τ₃_assoc, groupHomology.mapCycles₂_comp_i_apply, Hom.comm₂₃_assoc, Homotopy.comp_h₂, Homotopy.ofEq_h₃, exact_iff_exact_coimage_π, id_τ₃, moduleCatLeftHomologyData_H, moduleCatCyclesIso_inv_iCycles_assoc_apply, moduleCatCyclesIso_inv_π_apply, CochainComplex.mappingConeHomOfDegreewiseSplitIso_inv_f, Homotopy.neg_h₂, CategoryTheory.Abelian.Ext.contravariant_sequence_exact₃', ShortExact.hasInjectiveDimensionLT_X₃_iff, exact_iff_of_forks, HomologyData.ofEpiMonoFactorisation.f'_eq, groupHomology.π_comp_H1Iso_inv_apply, DerivedCategory.triangleOfSES_mor₂, groupCohomology.isoCocycles₂_hom_comp_i_apply, ShortExact.homology_exact₁, groupHomology.H1CoresCoinfOfTrivial_X₃, toCycles_moduleCatCyclesIso_hom_assoc_apply, rightHomologyι_comp_fromOpcycles, SnakeInput.w₀₂_τ₃, Homotopy.add_h₃, ab_exact_iff_range_eq_ker, Exact.isZero_X₂_iff, Exact.mono_g', Homotopy.unop_h₀, LeftHomologyData.wi_assoc, Exact.epi_f_iff, ModuleCat.smulShortComplex_X₃_isModule, HomologicalComplex.instQuasiIsoShortComplexTruncLEX₃ToTruncGE, neg_τ₃, SnakeInput.epi_L₁_g, kernelSequence_X₃, groupCohomology.isoShortComplexH1_inv, HomologicalComplex.shortComplexTruncLE_shortExact_δ_eq_zero, RightHomologyData.ofIsLimitKernelFork_H, Exact.isIso_imageToKernel', Exact.leftHomologyDataOfIsLimitKernelFork_i, CategoryTheory.kernelCokernelCompSequence.snakeInput_L₀_X₃, p_fromOpcycles_assoc, fromOpcycles_op_cyclesOpIso_inv, HomologicalComplex.HomologySequence.mono_homologyMap_τ₃, abLeftHomologyData_H_coe, FunctorEquivalence.counitIso_hom_app_app_τ₃, cyclesIsoKernel_inv, mapNatTrans_τ₃, Splitting.isoBinaryBiproduct_hom, groupCohomology.isoShortComplexH2_inv, exact_iff_epi_imageToKernel', CochainComplex.mappingConeHomOfDegreewiseSplitIso_hom_f, Rep.coinvariantsShortComplex_X₃, CategoryTheory.ObjectProperty.prop_iff_of_shortExact, HomologicalComplex.g_shortComplexTruncLEX₃ToTruncGE_assoc, ShortExact.moduleCat_surjective_g, HomologicalComplex.shortComplexTruncLE_X₃_isSupportedOutside, ModuleCat.smulShortComplex_g_epi, groupCohomology.mono_δ_of_isZero, instEpiGCokernelSequence, moduleCatCyclesIso_inv_iCycles_apply, groupHomology.π_comp_H2Iso_inv_apply, Exact.isIso_fromOpcycles, ShortExact.homology_exact₂, ShortExact.ab_exact_iff_function_exact, mono_of_epi_of_epi_of_mono, isIso_iff, groupCohomology.isIso_δ_of_isZero, ShortExact.extClass_comp, isoCyclesOfIsLimit_inv_ι, Splitting.s_r_assoc, abCyclesIso_inv_apply_iCycles, HomologicalComplex.shortComplexFunctor_obj_X₃, ShortExact.comp_δ_assoc, Splitting.isoBinaryBiproduct_inv, RightHomologyData.p_g'
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f 📖 | CompOp | 239 mathmath: RightHomologyData.ofHasCokernelOfHasKernel_p, pOpcycles_comp_moduleCatOpcyclesIso_hom_apply, Homotopy.h₀_f_assoc, Homotopy.comm₁, Exact.epi_kernelLift, LeftHomologyData.ofHasKernelOfHasCokernel_H, cokernel_π_comp_cokernelToAbelianCoimage_assoc, pOpcycles_π_isoOpcyclesOfIsColimit_inv_assoc, groupHomology.shortComplexH1_f, op_g, groupCohomology.cocyclesIso₀_hom_comp_f, abelianImageToKernel_comp_kernel_ι_comp_cokernel_π_assoc, zero_assoc, RightHomologyData.wp, cokernelSequence_f, RightHomologyData.ofAbelian_H, RightHomologyData.wp_assoc, abelianImageToKernel_comp_kernel_ι, exact_iff_kernel_ι_comp_cokernel_π_zero, ModuleCat.linearIndependent_shortExact, groupCohomology.H0IsoOfIsTrivial_hom, Splitting.mono_f, π₁Toπ₂_app, CochainComplex.mappingCone.homologySequenceδ_triangleh, groupHomology.comap_coinvariantsKer_pOpcycles_range_subtype_pOpcycles_eq_top, CategoryTheory.Abelian.Ext.contravariant_sequence_exact₁', ext_mk₀_f_comp_ext_mk₀_g, exact_and_mono_f_iff_of_iso, exact_iff_isIso_imageToKernel', Hom.comm₁₂, groupCohomology.map_H0Iso_hom_f_apply, LeftHomologyData.ofAbelian_H, LeftHomologyData.ofHasCokernel_π, Splitting.id, CochainComplex.mappingCone.inr_f_descShortComplex_f_assoc, LeftHomologyData.ofIsColimitCokernelCofork_π, groupCohomology.instMonoModuleCatFH1InfRes, groupCohomology.cocyclesIso₀_inv_comp_iCocycles, abToCycles_apply_coe, DerivedCategory.triangleOfSES_mor₁, CochainComplex.mappingCone.inr_descShortComplex_assoc, SnakeInput.δ_L₃_f, groupHomology.shortComplexH2_f, map_f, Exact.ab_range_eq_ker, CategoryTheory.Abelian.Ext.contravariant_sequence_exact₂', Exact.liftFromProjective_comp_assoc, moduleCat_zero_apply, groupCohomology.shortComplexH0_f, pOpcycles_comp_moduleCatOpcyclesIso_hom, groupCohomology.shortComplexH1_f, exact_iff_exact_image_ι, SnakeInput.L₂'_g, Exact.isIso_imageToKernel, CategoryTheory.CommSq.shortComplex_f, Exact.liftFromProjective_comp, exact_iff_isIso_imageToKernel, RightHomologyData.wι_assoc, ShortExact.ab_injective_f, cokernel_π_comp_cokernelToAbelianCoimage, CategoryTheory.Abelian.Ext.covariant_sequence_exact₁', unop_f, HomologyData.ofEpiMonoFactorisation.g'_eq, CategoryTheory.kernelCokernelCompSequence.snakeInput_L₃_f, SnakeInput.mono_L₀_f, RightHomologyData.ofIsColimitCokernelCofork_Q, SnakeInput.L₁'_f, Exact.mono_cokernelDesc, exact_iff_of_hasForget, RightHomologyData.ofIsColimitCokernelCofork_H, groupCohomology.map_H0Iso_hom_f, moduleCat_pOpcycles_eq_zero_iff, opcyclesIsoCokernel_inv, ab_exact_iff_function_exact, ChainComplex.mk_congr_succ_X₃, CategoryTheory.Pretriangulated.shortComplexOfDistTriangle_f, HomologicalComplex.instMonoFShortComplexTruncLE, RightHomologyData.ofAbelian_Q, RightHomologyData.IsPreservedBy.hf, SnakeInput.snd_δ_inr, Splitting.g_s, Hom.comm₁₂_assoc, CochainComplex.shift_f_comp_mappingConeHomOfDegreewiseSplitIso_inv, fFunctor_obj, groupCohomology.shortComplexH2_f, SnakeInput.L₁_f_φ₁_assoc, Splitting.splitMono_f_retraction, instMonoAbelianImageToKernel, SnakeInput.φ₁_L₂_f_assoc, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.mk₀_f_comp_biprodAddEquiv_symm_biprodIsoProd_hom, Exact.moduleCat_range_eq_ker, toCycles_i_assoc, CategoryTheory.IsPullback.mono_shortComplex'_f, groupCohomology.cocyclesIso₀_inv_comp_iCocycles_assoc, LeftHomologyData.f'_i, groupCohomology.instMonoModuleCatFShortComplexH0, ShortExact.singleTriangle_mor₁, exact_and_epi_g_iff_g_is_cokernel, kernelSequence_f, LeftHomologyData.wπ_assoc, RightHomologyData.ofHasCokernelOfHasKernel_H, Exact.rightHomologyDataOfIsColimitCokernelCofork_Q, ShortExact.mono_f, opcyclesIsoCokernel_hom, Exact.epi_f, RightHomologyData.ofIsColimitCokernelCofork_p, groupCohomology.π_comp_H0IsoOfIsTrivial_hom_apply, HomologicalComplex.opcycles_right_exact, Rep.coinvariantsShortComplex_f, groupCohomology.cocyclesMap_cocyclesIso₀_hom_f_assoc, HomologicalComplex.shortComplexFunctor_obj_f, HomologicalComplex.shortComplexTruncLE_f, kernel_ι_comp_cokernel_π_comp_cokernelToAbelianCoimage, RightHomologyData.wι, Exact.mono_g_iff, ab_exact_iff_ker_le_range, SnakeInput.L₁_f_φ₁, LeftHomologyData.ofIsColimitCokernelCofork_H, LeftHomologyData.ofIsColimitCokernelCofork_f', ShortExact.moduleCat_exact_iff_function_exact, LeftHomologyData.ofHasCokernel_H, HomologicalComplex.cycles_left_exact, RightHomologyData.ofAbelian_p, groupHomology.shortComplexH0_f, ShortExact.injective_f, RightHomologyData.ofAbelian_ι, groupCohomology.cocyclesIso₀_hom_comp_f_assoc, CategoryTheory.Abelian.Ext.covariant_sequence_exact₂', CochainComplex.mappingCone.quasiIso_descShortComplex, SnakeInput.φ₁_L₂_f, pOpcycles_comp_moduleCatOpcyclesIso_hom_assoc, exact_iff_exact_up_to_refinements, LeftHomologyData.wπ, ModuleCat.smulShortComplex_f, ab_exact_iff, abelianImageToKernel_comp_kernel_ι_assoc, Exact.exact_up_to_refinements, ShortExact.isIso_f_iff, ShortExact.δ_comp_assoc, Splitting.g_s_assoc, Splitting.f_r_assoc, zero_apply, LeftHomologyData.ofAbelian_π, FunctorEquivalence.inverse_obj_f, groupCohomology.cocyclesMap_cocyclesIso₀_hom_f, LeftHomologyData.f'_i_assoc, HasRightHomology.hasKernel, RightHomologyData.IsPreservedBy.f, RightHomologyData.ofHasCokernelOfHasKernel_ι, CategoryTheory.Functor.preservesFiniteLimits_iff_forall_exact_map_and_mono, moduleCat_exact_iff_ker_sub_range, moduleCat_exact_iff, CochainComplex.mappingCone.inl_v_descShortComplex_f_assoc, instEpiCokernelToAbelianCoimage, pOpcycles_π_isoOpcyclesOfIsColimit_inv, exact_iff_mono_cokernel_desc, zero, Exact.rightHomologyDataOfIsColimitCokernelCofork_ι, exact_and_mono_f_iff_f_is_kernel, moduleCatMkOfKerLERange_f, toComposableArrows_map, exact_iff_epi_imageToKernel, exact_iff_image_eq_kernel, Splitting.leftHomologyData_i, CategoryTheory.kernelCokernelCompSequence.snakeInput_L₁_f, toCycles_i, Splitting.r_f, groupHomology.H1CoresCoinfOfTrivial_f, HasLeftHomology.hasCokernel, SnakeInput.mono_L₂_f, ShortExact.δ_comp, unop_g, CochainComplex.shift_f_comp_mappingConeHomOfDegreewiseSplitIso_inv_assoc, ab_zero_apply, RightHomologyData.ofHasCokernelOfHasKernel_Q, LeftHomologyData.ofHasKernelOfHasCokernel_π, moduleCat_exact_iff_range_eq_ker, Homotopy.comm₂, ChainComplex.mk_congr_succ_d₂, op_f, exact_iff_epi_kernel_lift, abelianImageToKernel_comp_kernel_ι_comp_cokernel_π, CategoryTheory.CommSq.shortComplex'_f, moduleCatMk_f, RightHomologyData.ofIsColimitCokernelCofork_ι, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.shortComplex_f, CochainComplex.mappingCone.inr_descShortComplex, CochainComplex.triangleOfDegreewiseSplit_mor₁, Splitting.isSplitMono_f, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.instMonoSheafAddCommGrpCatFShortComplex, CochainComplex.mappingCone.inl_v_descShortComplex_f, HasRightHomology.hasCokernel, Splitting.r_f_assoc, CategoryTheory.kernelCokernelCompSequence.snakeInput_L₀_f, CochainComplex.mappingCone.inr_f_descShortComplex_f, ShortExact.extClass_comp_assoc, groupCohomology.cocyclesIso₀_hom_comp_f_apply, exact_iff_exact_coimage_π, SnakeInput.L₂'_f, π_isoOpcyclesOfIsColimit_hom_assoc, ChainComplex.mk_d, Exact.rightHomologyDataOfIsColimitCokernelCofork_p, moduleCat_pOpcycles_eq_iff, Splitting.f_r, exact_iff_of_forks, HomologyData.ofEpiMonoFactorisation.f'_eq, ShortExact.homology_exact₁, ab_exact_iff_range_eq_ker, CategoryTheory.Abelian.Ext.contravariant_sequence_exact₁, Exact.isZero_X₂_iff, CategoryTheory.Abelian.Ext.covariant_sequence_exact₂, groupCohomology.cocyclesIso₀_inv_comp_iCocycles_apply, HomologicalComplex.shortComplexFunctor'_obj_f, Exact.epi_f_iff, ModuleCat.span_rightExact, Exact.lift', groupHomology.H1CoresCoinf_f, Exact.isIso_imageToKernel', CategoryTheory.Abelian.Pseudoelement.pseudo_exact_of_exact, groupCohomology.map_H0Iso_hom_f_assoc, exact_iff_epi, CochainComplex.mappingCone.triangleRotateShortComplex_f, groupCohomology.H1InfRes_f, Exact.lift_f, instMonoFKernelSequence, exact_iff_epi_imageToKernel', π_isoOpcyclesOfIsColimit_hom, Exact.lift_f_assoc, Homotopy.h₀_f, fFunctor_map, ShortExact.moduleCat_injective_f, SnakeInput.instMonoFL₀'OfL₁, ShortExact.homology_exact₂, ShortExact.ab_exact_iff_function_exact, ShortExact.extClass_comp, f_pOpcycles, CategoryTheory.kernelCokernelCompSequence.snakeInput_L₂_f, f_pOpcycles_assoc, Splitting.isoBinaryBiproduct_inv
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fFunctor 📖 | CompOp | 2 mathmath: fFunctor_obj, fFunctor_map
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g 📖 | CompOp | 258 mathmath: SnakeInput.epi_L₃_g, CategoryTheory.CommSq.shortComplex_g, Exact.comp_descToInjective, Exact.epi_kernelLift, groupCohomology.isoCocycles₁_hom_comp_i_apply, LeftHomologyData.ofHasKernelOfHasCokernel_H, cokernel_π_comp_cokernelToAbelianCoimage_assoc, LeftHomologyData.ofAbelian_K, moduleCatCyclesIso_hom_i_apply, op_g, abelianImageToKernel_comp_kernel_ι_comp_cokernel_π_assoc, LeftHomologyData.IsPreservedBy.hg, zero_assoc, RightHomologyData.ofAbelian_H, ShortExact.ab_surjective_g, abelianImageToKernel_comp_kernel_ι, exact_iff_kernel_ι_comp_cokernel_π_zero, ModuleCat.linearIndependent_shortExact, LeftHomologyData.ofHasKernelOfHasCokernel_K, moduleCatLeftHomologyData_i_hom, CategoryTheory.kernelCokernelCompSequence.snakeInput_L₃_g, moduleCatMkOfKerLERange_g, ext_mk₀_f_comp_ext_mk₀_g, Splitting.s_g, exact_iff_isIso_imageToKernel', isoCyclesOfIsLimit_hom_iCycles_assoc, ShortExact.epi_g, moduleCatMk_g, groupHomology.isoCycles₁_inv_comp_iCycles_apply, RightHomologyData.p_g'_assoc, SnakeInput.L₀X₂ToP_comp_pullback_snd, LeftHomologyData.ofAbelian_H, ShortExact.comp_δ, Homotopy.g_h₃_assoc, Splitting.id, CategoryTheory.kernelCokernelCompSequence.snakeInput_L₁_g, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.biprodAddEquiv_symm_biprodIsoProd_hom_toBiprod_apply, groupCohomology.mapCocycles₂_comp_i_apply, CochainComplex.mappingCone.inr_f_descShortComplex_f_assoc, SnakeInput.instEpiGL₀', Splitting.s_g_assoc, ShortExact.isIso_g_iff, abToCycles_apply_coe, p_fromOpcycles, CochainComplex.mappingCone.inr_descShortComplex_assoc, HomologicalComplex.instEpiGShortComplexTruncLE, iCycles_g, Exact.ab_range_eq_ker, CategoryTheory.Abelian.Ext.contravariant_sequence_exact₂', moduleCat_zero_apply, LeftHomologyData.ofIsLimitKernelFork_H, groupCohomology.shortComplexH0_g, exact_iff_exact_image_ι, gFunctor_map, LeftHomologyData.ofHasKernelOfHasCokernel_i, SnakeInput.L₂'_g, CochainComplex.g_shortComplexTruncLEX₃ToTruncGE, Exact.isIso_imageToKernel, Splitting.epi_g, exact_iff_isIso_imageToKernel, RightHomologyData.wι_assoc, abLeftHomologyData_π, cokernel_π_comp_cokernelToAbelianCoimage, LeftHomologyData.ofAbelian_i, unop_f, cokernelSequence_g, HomologyData.ofEpiMonoFactorisation.g'_eq, CategoryTheory.Abelian.Ext.covariant_sequence_exact₃', SnakeInput.L₁'_f, LeftHomologyData.IsPreservedBy.g, Exact.mono_cokernelDesc, exact_iff_surjective_abToCycles, Splitting.rightHomologyData_p, Homotopy.comm₃, ab_exact_iff_function_exact, Exact.mono_g, moduleCatCyclesIso_inv_π_assoc_apply, SnakeInput.snd_δ, SnakeInput.snd_δ_inr, Splitting.g_s, LeftHomologyData.ofIsLimitKernelFork_i, groupHomology.H1CoresCoinf_g, ModuleCat.smulShortComplex_g, HomologicalComplex.isIso_homologyMap_shortComplexTruncLE_g, instMonoAbelianImageToKernel, Exact.leftHomologyDataOfIsLimitKernelFork_K, map_g, groupHomology.mapCycles₁_comp_i_apply, cyclesIsoKernel_hom, Exact.moduleCat_range_eq_ker, groupCohomology.isoCocycles₁_inv_comp_iCocycles_apply, abLeftHomologyData_K_coe, SnakeInput.L₁'_g, CategoryTheory.Functor.preservesFiniteColimits_iff_forall_exact_map_and_epi, HasLeftHomology.hasKernel, HomologicalComplex.g_shortComplexTruncLEX₃ToTruncGE, ShortExact.comp_extClass_assoc, CategoryTheory.kernelCokernelCompSequence.snakeInput_L₀_g, exact_and_epi_g_iff_g_is_cokernel, HomologicalComplex.mono_homologyMap_shortComplexTruncLE_g, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.instEpiSheafAddCommGrpCatGShortComplex, LeftHomologyData.wπ_assoc, RightHomologyData.ofHasCokernelOfHasKernel_H, groupHomology.instEpiModuleCatGH1CoresCoinf, SnakeInput.functorP_map, Rep.coinvariantsShortComplex_g, CochainComplex.triangleOfDegreewiseSplit_mor₂, groupHomology.isoCycles₂_hom_comp_i_apply, CategoryTheory.Abelian.Ext.covariant_sequence_exact₃, HomologicalComplex.opcycles_right_exact, Exact.desc', groupCohomology.isoCocycles₂_inv_comp_iCocycles_apply, HomologicalComplex.quasiIsoAt_shortComplexTruncLE_g, SnakeInput.L₀_g_δ, SnakeInput.snd_δ_assoc, ShortExact.comp_extClass, kernel_ι_comp_cokernel_π_comp_cokernelToAbelianCoimage, RightHomologyData.wι, iCycles_g_assoc, Exact.mono_g_iff, HomologicalComplex.shortComplexFunctor'_obj_g, ab_exact_iff_ker_le_range, abLeftHomologyData_f', groupHomology.π_comp_H1Iso_hom_apply, LeftHomologyData.wi, ShortExact.singleTriangle_mor₂, Exact.leftHomologyDataOfIsLimitKernelFork_π, Splitting.splitEpi_g_section_, ShortExact.moduleCat_exact_iff_function_exact, π_moduleCatCyclesIso_hom_assoc_apply, Exact.comp_descToInjective_assoc, HomologicalComplex.cycles_left_exact, CategoryTheory.Pretriangulated.shortComplexOfDistTriangle_g, groupHomology.isoCycles₁_hom_comp_i_apply, RightHomologyData.ofAbelian_ι, CategoryTheory.Abelian.Ext.covariant_sequence_exact₂', exact_iff_surjective_moduleCatToCycles, exact_iff_mono, CategoryTheory.IsPushout.epi_shortComplex_g, LeftHomologyData.wπ, moduleCatCyclesIso_hom_i_assoc_apply, abelianImageToKernel_comp_kernel_ι_assoc, Splitting.g_s_assoc, ShortExact.surjective_g, zero_apply, LeftHomologyData.ofAbelian_π, ShortExact.homology_exact₃, abLeftHomologyData_i, HasRightHomology.hasKernel, groupCohomology.H1InfRes_g, Hom.comm₂₃, CochainComplex.mappingConeHomOfDegreewiseSplitIso_inv_comp_triangle_mor₃_assoc, RightHomologyData.ofHasCokernelOfHasKernel_ι, groupHomology.shortComplexH2_g, moduleCat_exact_iff_ker_sub_range, CategoryTheory.GrothendieckTopology.MayerVietorisSquare.shortComplex_g, groupHomology.isoCycles₂_inv_comp_iCycles_apply, instEpiCokernelToAbelianCoimage, Exact.g_desc_assoc, exact_and_epi_g_iff_of_iso, exact_iff_mono_cokernel_desc, isoCyclesOfIsLimit_inv_ι_assoc, zero, π_moduleCatCyclesIso_hom_apply, gFunctor_obj, CategoryTheory.kernelCokernelCompSequence.snakeInput_L₂_g, exact_and_mono_f_iff_f_is_kernel, toComposableArrows_map, CategoryTheory.CommSq.shortComplex'_g, exact_iff_epi_imageToKernel, exact_iff_image_eq_kernel, π₂Toπ₃_app, RightHomologyData.ofIsLimitKernelFork_ι, LeftHomologyData.ofIsLimitKernelFork_K, Splitting.r_f, Splitting.isSplitEpi_g, HasLeftHomology.hasCokernel, groupCohomology.mapCocycles₁_comp_i_apply, RightHomologyData.ofHasKernel_ι, unop_g, HomologicalComplex.epi_homologyMap_shortComplexTruncLE_g, ab_zero_apply, LeftHomologyData.ofHasKernelOfHasCokernel_π, moduleCat_exact_iff_range_eq_ker, isoCyclesOfIsLimit_hom_iCycles, Homotopy.comm₂, op_f, CochainComplex.mappingConeHomOfDegreewiseSplitIso_inv_comp_triangle_mor₃, groupHomology.H1CoresCoinfOfTrivial_g, moduleCatLeftHomologyData_K, exact_iff_epi_kernel_lift, toComposableArrows_obj, abelianImageToKernel_comp_kernel_ι_comp_cokernel_π, CochainComplex.mappingCone.triangleRotateShortComplex_g, groupCohomology.π_comp_H2Iso_hom_apply, LeftHomologyData.ofIsLimitKernelFork_π, CochainComplex.mappingCone.inr_descShortComplex, groupCohomology.π_comp_H1Iso_hom_apply, SnakeInput.L₀X₂ToP_comp_pullback_snd_assoc, groupHomology.instEpiModuleCatGShortComplexH0, toCycles_moduleCatCyclesIso_hom_apply, groupHomology.π_comp_H2Iso_hom_apply, CategoryTheory.Abelian.Ext.contravariant_sequence_exact₂, Splitting.r_f_assoc, CochainComplex.mappingCone.inr_f_descShortComplex_f, moduleCatLeftHomologyData_π_hom, RightHomologyData.ofHasKernel_H, groupHomology.H1CoresCoinfOfTrivial_g_epi, CochainComplex.g_shortComplexTruncLEX₃ToTruncGE_assoc, HomologicalComplex.shortComplexFunctor_obj_g, Homotopy.g_h₃, Exact.g_desc, groupHomology.mapCycles₂_comp_i_apply, Hom.comm₂₃_assoc, exact_iff_exact_coimage_π, moduleCatLeftHomologyData_H, moduleCatCyclesIso_inv_iCycles_assoc_apply, moduleCatCyclesIso_inv_π_apply, CategoryTheory.Abelian.Ext.contravariant_sequence_exact₃', groupHomology.shortComplexH0_g, exact_iff_of_forks, HomologyData.ofEpiMonoFactorisation.f'_eq, groupHomology.π_comp_H1Iso_inv_apply, DerivedCategory.triangleOfSES_mor₂, groupCohomology.isoCocycles₂_hom_comp_i_apply, toCycles_moduleCatCyclesIso_hom_assoc_apply, ab_exact_iff_range_eq_ker, Exact.isZero_X₂_iff, groupCohomology.shortComplexH2_g, LeftHomologyData.wi_assoc, kernelSequence_g, Exact.epi_f_iff, FunctorEquivalence.inverse_obj_g, ModuleCat.span_rightExact, SnakeInput.epi_L₁_g, RightHomologyData.ofIsLimitKernelFork_H, Exact.isIso_imageToKernel', Exact.leftHomologyDataOfIsLimitKernelFork_i, p_fromOpcycles_assoc, groupHomology.shortComplexH1_g, abLeftHomologyData_H_coe, cyclesIsoKernel_inv, Splitting.isoBinaryBiproduct_hom, exact_iff_epi_imageToKernel', HomologicalComplex.g_shortComplexTruncLEX₃ToTruncGE_assoc, ShortExact.moduleCat_surjective_g, ModuleCat.smulShortComplex_g_epi, instEpiGCokernelSequence, moduleCatCyclesIso_inv_iCycles_apply, groupHomology.π_comp_H2Iso_inv_apply, ShortExact.homology_exact₂, ShortExact.ab_exact_iff_function_exact, RightHomologyData.ofIsLimitKernelFork_g', isoCyclesOfIsLimit_inv_ι, abCyclesIso_inv_apply_iCycles, groupCohomology.shortComplexH1_g, ShortExact.comp_δ_assoc, RightHomologyData.p_g'
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gFunctor 📖 | CompOp | 2 mathmath: gFunctor_map, gFunctor_obj
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homMk 📖 | CompOp | 8 mathmath: isoMk_inv, groupHomology.isoShortComplexH1_inv, homMk_τ₂, homMk_τ₁, homMk_τ₃, groupHomology.isoShortComplexH2_inv, groupCohomology.isoShortComplexH1_inv, groupCohomology.isoShortComplexH2_inv
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instCategory 📖 | CompOp | 540 mathmath: opcyclesMap_smul, FunctorEquivalence.counitIso_inv_app_app_τ₁, rightHomologyMap_id, SnakeInput.w₀₂_assoc, CategoryTheory.ComposableArrows.sc'MapIso_inv, homologyMap_smul, HomologyData.ofIso_right_p, isoMk_hom_τ₁, Homotopy.comp_h₃, SnakeInput.comp_f₃_assoc, HomologicalComplex.truncGE.rightHomologyMapData_φQ, leftHomologyMap_comp, groupHomology.mapShortComplexH2_id, SnakeInput.functorL₁_obj, cyclesFunctor_linear, homologyMap_add, SnakeInput.Hom.comm₂₃, toCyclesNatTrans_app, leftHomologyMap'_sub, Homotopy.sub_h₀, homologyMap_zero, π₁Toπ₂_comp_π₂Toπ₃_assoc, isIso_homologyFunctor_map_of_epi_of_isIso_of_mono, RightHomologyMapData.neg_φH, LeftHomologyMapData.id_φK, instPreservesFiniteColimitsπ₁, Homotopy.smul_h₁, leftHomologyMap'_comp, HomotopyEquiv.trans_hom, groupHomology.mapShortComplexH1_zero, mapOpcyclesIso_hom_naturality_assoc, LeftHomologyMapData.id_φH, leftHomologyMap_sub, groupHomology.mapShortComplexH2_zero, leftHomologyπNatTrans_app, HomologicalComplex.shortComplexFunctor'_obj_X₁, LeftHomologyData.map_leftHomologyMap', CochainComplex.shiftShortComplexFunctorIso_hom_app_τ₂, instIsNormalEpiCategory, CochainComplex.shiftShortComplexFunctorIso_hom_app_τ₃, hasHomology_of_preserves', opcyclesMap_sub, SnakeInput.functorL₁'_map_τ₃, opcyclesMap'_sub, RightHomologyMapData.neg_φQ, isoMk_hom_τ₂, π₁Toπ₂_app, leftHomologyMap'_neg, instPreservesColimitπ₁, SnakeInput.functorL₁'_map_τ₁, HomologicalComplex.restriction.sc'Iso_inv_τ₃, LeftHomologyMapData.map_φH, RightHomologyMapData.compatibilityOfZerosOfIsColimitCokernelCofork_φH, cyclesMap_neg, isoMk_inv, FunctorEquivalence.counitIso_inv_app_app_τ₂, HomotopyEquiv.refl_homotopyHomInvId, mapHomologyIso_hom_naturality, rightHomologyιNatTrans_app, FunctorEquivalence.unitIso_inv_app_τ₂_app, homologyMapIso_inv, Homotopy.compLeft_h₂, SnakeInput.Hom.comp_f₁, comp_τ₁, LeftHomologyMapData.compatibilityOfZerosOfIsColimitCokernelCofork_φK, opcyclesMapIso'_inv, homologyMap'_zero, functorEquivalence_counitIso, CochainComplex.shiftShortComplexFunctorIso_add'_hom_app, homologyMap'_comp, HomologicalComplex.natIsoSc'_inv_app_τ₂, HomologyMapData.add_left, unopFunctor_map, SnakeInput.id_f₂, HomologyMapData.map_left, RightHomologyMapData.compatibilityOfZerosOfIsLimitKernelFork_φH, cyclesMap_sub, FunctorEquivalence.unitIso_inv_app_τ₁_app, Homotopy.add_h₁, hasLimit_of_hasLimitπ, RightHomologyMapData.id_φH, π₃_map, groupCohomology.mapShortComplexH2_comp_assoc, SnakeInput.comp_f₀, opcyclesMap'_id, HomologyMapData.neg_left, Homotopy.smul_h₂, opMap_id, HomologicalComplex.shortComplexFunctor'_map_τ₂, cyclesMap'_comp_assoc, LeftHomologyMapData.smul_φH, cyclesMap'_sub, homologyMap'_sub, HomologicalComplex.natIsoSc'_inv_app_τ₃, Homotopy.sub_h₁, rightHomologyMap'_id, HomologyMapData.comp_left, instPreservesColimitsOfShapeπ₃, HomologicalComplex.natIsoSc'_hom_app_τ₃, HomologyData.map_homologyMap', RightHomologyMapData.quasiIso_map_iff, Homotopy.ofNullHomotopic_h₀, homologyMap'_add, CategoryTheory.Functor.mapShortComplex_map_τ₁, mapOpcyclesIso_inv_naturality, HomotopyEquiv.refl_hom, instPreservesLimitπ₁, HomologicalComplex.shortComplexFunctor_obj_X₂, HomologicalComplex.HomologySequence.mapSnakeInput_f₃, Homotopy.neg_h₁, opcyclesFunctor_linear, HomologicalComplex.HomologySequence.snakeInput_L₁, rightHomologyFunctor_linear, functorEquivalence_functor, FunctorEquivalence.unitIso_hom_app_τ₃_app, add_τ₃, instPreservesLimitsOfShapeπ₁, homologyMapIso_hom, HomologyMapData.comp_right, FunctorEquivalence.inverse_obj_X₂, comp_τ₂, π₁Toπ₂_comp_π₂Toπ₃, gFunctor_map, SnakeInput.Hom.comm₁₂, HomologicalComplex.restriction.sc'Iso_inv_τ₁, FunctorEquivalence.counitIso_inv_app_app_τ₃, instPreservesColimitsOfShapeπ₂, mapLeftHomologyIso_hom_naturality, leftHomologyMap'_comp_assoc, Homotopy.compRight_h₀, rightHomologyMapIso_inv, SnakeInput.functorL₀_map, leftHomologyFunctor_additive, rightHomologyFunctor_obj, rightHomologyFunctor_additive, opcyclesMap_add, mapHomologyIso_hom_naturality_assoc, Homotopy.compRight_h₁, SnakeInput.functorL₁'_obj, CategoryTheory.Functor.mapShortComplex_map_τ₂, Homotopy.smul_h₀, Homotopy.smul_h₃, HomologyData.ofIso_left_i, cyclesMap_smul, leftHomologyMapIso_inv, cyclesMap'_comp, leftHomologyMap_add, hasLimitsOfShape, unopFunctor_obj, LeftHomologyData.map_cyclesMap', RightHomologyData.map_rightHomologyMap', SnakeInput.functorL₁'_map_τ₂, Homotopy.ofNullHomotopic_h₁, cyclesMapIso'_inv, homologyMap'_neg, LeftHomologyMapData.neg_φH, mapRightHomologyIso_inv_naturality, SnakeInput.Hom.comm₂₃_assoc, FunctorEquivalence.inverse_obj_X₃, hasColimitsOfShape, RightHomologyMapData.zero_φH, preservesZeroMorphisms_π₃, isoMk_hom_τ₃, HomologyMapData.zero_left, CochainComplex.shiftShortComplexFunctor'_hom_app_τ₁, id_τ₁, groupCohomology.mapShortComplexH2_comp, fFunctor_obj, opcyclesFunctor_additive, hasRightHomology_of_preserves', HomologicalComplex.restriction.sc'Iso_hom_τ₂, ChainComplex.quasiIsoAt₀_iff, FunctorEquivalence.inverse_map_τ₁, Homotopy.compRight_h₂, SnakeInput.Hom.id_f₃, RightHomologyMapData.comp_φQ, groupHomology.isoShortComplexH1_hom, Homotopy.compRight_h₃, SnakeInput.functorL₀_obj, opEquiv_functor, opcyclesMap_neg, HomologyMapData.compatibilityOfZerosOfIsLimitKernelFork_left, opcyclesMapIso_inv, Homotopy.ofNullHomotopic_h₂, rightHomologyMap_comp, Homotopy.equivSubZero_symm_apply, mapHomologyIso'_hom_naturality, instIsNormalMonoCategory, SnakeInput.Hom.comm₁₂_assoc, instPreservesFiniteColimitsπ₂, quasiIso_iff_comp_right, instPreservesColimitπ₃, leftHomologyMapIso'_hom, mapNatIso_hom, quasiIso_map_of_preservesLeftHomology, SnakeInput.functorL₂_obj, rightHomologyMap'_add, homologyMap'_id, LeftHomologyMapData.quasiIso_map_iff, CategoryTheory.ComposableArrows.sc'MapIso_hom, cyclesMap_comp, rightHomologyMapIso'_inv, cyclesFunctor_obj, instPreservesLimitsOfShapeπ₂, CochainComplex.shiftShortComplexFunctor'_hom_app_τ₂, CategoryTheory.ComposableArrows.scMapIso_inv, SnakeInput.functorL₁_map, quasiIso_iff_comp_left, π₂_map, mapRightHomologyIso_hom_naturality, opcyclesMap'_comp_assoc, instPreservesColimitπ₂, groupCohomology.mapShortComplexH2_zero, quasiIso_map_iff_of_preservesRightHomology, SnakeInput.comp_f₂, CategoryTheory.Functor.mapShortComplex_map_τ₃, smul_τ₃, homologyMap_neg, HomologicalComplex.shortComplexFunctor_map_τ₁, LeftHomologyMapData.comp_φH, HomologyMapData.map_right, HomologyMapData.zero_right, SnakeInput.w₁₃_assoc, mapToComposableArrows_comp, LeftHomologyMapData.add_φH, groupCohomology.mapShortComplexH1_id, HomotopyEquiv.refl_inv, HomologyMapData.id_right, HomologicalComplex.shortComplexFunctor'_obj_X₂, groupHomology.mapShortComplexH1_id_comp, groupHomology.mapShortComplexH1_comp, Splitting.ofIso_s, opEquiv_counitIso, HomotopyEquiv.ext_iff, mapRightHomologyIso_inv_naturality_assoc, preservesMonomorphisms_π₁, Homotopy.comp_h₁, FunctorEquivalence.inverse_obj_X₁, LeftHomologyMapData.compatibilityOfZerosOfIsLimitKernelFork_φH, HomologicalComplex.shortComplexFunctor_map_τ₃, HomologicalComplex.shortComplexFunctor_obj_f, HomologicalComplex.HomologySequence.mapSnakeInput_f₁, RightHomologyMapData.id_φQ, CategoryTheory.Pretriangulated.shortComplexOfDistTriangleIsoOfIso_hom_τ₃, cyclesMap'_zero, zero_τ₃, Homotopy.sub_h₃, HomologicalComplex.shortComplexFunctor'_obj_X₃, quasiIso_map_iff_of_preservesLeftHomology, SnakeInput.w₀₂, rightHomologyMap_zero, isIso_of_isIso, HomologicalComplex.shortComplexFunctor'_obj_g, HomologicalComplex.natIsoSc'_hom_app_τ₂, SnakeInput.functorL₃_obj, HomotopyEquiv.trans_homotopyHomInvId, groupCohomology.mapShortComplexH2_id_comp_assoc, opcyclesMapIso_hom, groupHomology.mapShortComplexH2_comp, CochainComplex.shiftShortComplexFunctorIso_hom_app_τ₁, CochainComplex.shiftShortComplexFunctorIso_inv_app_τ₂, sub_τ₁, FunctorEquivalence.functor_obj_map, FunctorEquivalence.counitIso_hom_app_app_τ₁, cyclesFunctor_map, CategoryTheory.Functor.mapShortComplex_obj, instPreservesFiniteLimitsπ₁, Homotopy.add_h₀, instPreservesFiniteColimitsπ₃, SnakeInput.comp_f₀_assoc, cyclesMap_zero, leftRightHomologyComparison'_eq_leftHomologpMap'_comp_iso_hom_comp_rightHomologyMap', cyclesMap'_smul, LeftHomologyMapData.zero_φH, QuasiIsoAt.quasiIso, rightHomologyMapIso_hom, preservesMonomorphisms_π₂, SnakeInput.id_f₁, groupHomology.isoShortComplexH1_inv, instPreservesLimitsOfShapeπ₃, π₁_map, RightHomologyMapData.comp_φH, hasFiniteColimits, preservesZeroMorphisms_π₁, homologyMapIso'_hom, opcyclesMapIso'_hom, homologyMap_comp, RightHomologyMapData.smul_φH, SnakeInput.functorL₃_map, CategoryTheory.ComposableArrows.scMapIso_hom, CategoryTheory.Pretriangulated.shortComplexOfDistTriangleIsoOfIso_inv_τ₁, LeftHomologyMapData.zero_φK, HomologyMapData.add_right, rightHomologyMap_sub, cyclesFunctor_additive, SnakeInput.id_f₀, Homotopy.equivSubZero_apply, HomologicalComplex.HomologySequence.snakeInput_L₀, id_τ₂, hasLeftHomology_of_preserves', FunctorEquivalence.inverse_obj_f, mapCyclesIso_hom_naturality_assoc, homologyFunctor_obj, HomologicalComplex.shortComplexFunctor'_map_τ₁, sub_τ₂, rightHomologyMap'_comp, rightHomologyMap'_sub, Homotopy.neg_h₃, homologyMap_sub, SnakeInput.Hom.id_f₀, SnakeInput.comp_f₁_assoc, rightHomologyFunctorOpNatIso_hom_app, opcyclesMap_zero, opcyclesMap_comp, neg_τ₂, opEquiv_inverse, HomologicalComplex.HomologySequence.snakeInput_L₂, groupCohomology.mapShortComplexH1_id_comp, instPreservesLimitπ₂, CochainComplex.shiftShortComplexFunctor'_hom_app_τ₃, RightHomologyMapData.zero_φQ, homologyMap'_smul, groupHomology.isoShortComplexH2_hom, HomologicalComplex.shortComplexFunctor_obj_X₁, leftHomologyMap'_id, mapCyclesIso_inv_naturality_assoc, SnakeInput.Hom.id_f₁, Homotopy.compLeft_h₃, groupCohomology.mapShortComplexH1_comp, hasFiniteLimits, opFunctor_obj, gFunctor_obj, π₃_obj, Homotopy.neg_h₀, rightHomologyMapIso'_hom, HomologicalComplex.truncGE.rightHomologyMapData_φH, SnakeInput.comp_f₃, groupCohomology.isoShortComplexH1_hom, groupHomology.mapShortComplexH1_id, quasiIsoAt_iff', add_τ₂, rightHomologyMap'_comp_assoc, RightHomologyMapData.compatibilityOfZerosOfIsLimitKernelFork_φQ, rightHomologyMap_add, comp_τ₂_assoc, FunctorEquivalence.unitIso_inv_app_τ₃_app, comp_τ₃, HomotopyEquiv.trans_inv, CochainComplex.quasiIsoAt₀_iff, leftHomologyFunctorOpNatIso_hom_app, π₂Toπ₃_app, π₁_obj, instPreservesFiniteLimitsπ₂, HomologicalComplex.HomologySequence.snakeInput_L₃, leftHomologyMapIso'_inv, SnakeInput.Hom.comm₀₁, HomologyMapData.neg_right, SnakeInput.id_f₃, SnakeInput.functorL₂'_obj, FunctorEquivalence.functor_obj_obj, Homotopy.add_h₂, Homotopy.eq_add_nullHomotopic, Homotopy.comp_h₀, preservesEpimorphisms_π₂, SnakeInput.functorL₂'_map_τ₃, HomologyMapData.smul_right, LeftHomologyMapData.smul_φK, opcyclesFunctor_map, quasiIso_iff_evaluation, Homotopy.sub_h₂, cyclesMapIso'_hom, SnakeInput.comp_f₂_assoc, LeftHomologyMapData.neg_φK, mapCyclesIso_inv_naturality, FunctorEquivalence.unitIso_hom_app_τ₁_app, mapLeftHomologyIso_hom_naturality_assoc, cyclesMap'_id, cyclesMap_id, instPreservesLimitπ₃, unopMap_id, sub_τ₃, leftHomologyMapIso_hom, groupCohomology.mapShortComplexH1_id_comp_assoc, groupCohomology.mapShortComplexH1_zero, homologyFunctor_linear, groupCohomology.mapShortComplexH1_comp_assoc, RightHomologyMapData.add_φH, functorEquivalence_unitIso, rightHomologyMap_neg, HomologyMapData.compatibilityOfZerosOfIsLimitKernelFork_right, LeftHomologyMapData.comp_φK, RightHomologyMapData.smul_φQ, opcyclesFunctor_obj, groupCohomology.isoShortComplexH2_hom, rightHomologyFunctorOpNatIso_inv_app, homologyFunctor_map, preservesEpimorphisms_π₁, mapOpcyclesIso_inv_naturality_assoc, HomologicalComplex.restriction.sc'Iso_inv_τ₂, groupCohomology.mapShortComplexH2_id_comp, rightHomologyMap'_neg, iCyclesNatTrans_app, HomologicalComplex.shortComplexFunctor'_map_τ₃, homologyMap_id, HomologicalComplex.homologyFunctorIso_hom_app, mapRightHomologyIso_hom_naturality_assoc, HomologicalComplex.HomologySequence.mapSnakeInput_f₂, CochainComplex.shiftShortComplexFunctor'_inv_app_τ₃, HomologicalComplex.shortComplexFunctor_map_τ₂, Homotopy.compLeft_h₀, groupHomology.mapShortComplexH2_id_comp, mapHomologyIso_inv_naturality, HomologicalComplex.shortComplexFunctor_obj_g, groupHomology.isoShortComplexH2_inv, RightHomologyMapData.compatibilityOfZerosOfIsColimitCokernelCofork_φQ, comp_τ₃_assoc, instPreservesColimitsOfShapeπ₁, Homotopy.comp_h₂, cyclesMap'_add, mapHomologyIso'_hom_naturality_assoc, fromOpcyclesNatTrans_app, id_τ₃, FunctorEquivalence.functor_map_app, neg_τ₁, leftHomologyFunctor_obj, Homotopy.ofNullHomotopic_h₃, LeftHomologyMapData.add_φK, Splitting.ofIso_r, SnakeInput.Hom.comm₀₁_assoc, Homotopy.neg_h₂, π₂_obj, FunctorEquivalence.counitIso_hom_app_app_τ₂, zero_τ₁, smul_τ₁, functorEquivalence_inverse, groupCohomology.mapShortComplexH2_id, CochainComplex.shiftShortComplexFunctor'_inv_app_τ₂, HomotopyEquiv.trans_homotopyInvHomId, smul_τ₂, CategoryTheory.Pretriangulated.shortComplexOfDistTriangleIsoOfIso_inv_τ₂, quasiIso_comp, SnakeInput.functorL₂'_map_τ₂, RightHomologyMapData.map_φH, SnakeInput.Hom.comp_f₃, Homotopy.add_h₃, hasColimit_of_hasColimitπ, cyclesMap'_neg, quasiIsoAt_iff, leftHomologyFunctorOpNatIso_inv_app, opcyclesMap'_add, HomologicalComplex.shortComplexFunctor'_obj_f, rightHomologyMap'_smul, mapHomologyIso_inv_naturality_assoc, mapOpcyclesIso_hom_naturality, opEquiv_unitIso, SnakeInput.Hom.comp_f₂, CochainComplex.shiftShortComplexFunctor'_inv_app_τ₁, instPreservesFiniteLimitsπ₃, CochainComplex.shiftShortComplexFunctorIso_zero_add_hom_app, neg_τ₃, CochainComplex.shiftShortComplexFunctorIso_inv_app_τ₃, FunctorEquivalence.inverse_obj_g, SnakeInput.Hom.id_f₂, CategoryTheory.Pretriangulated.shortComplexOfDistTriangleIsoOfIso_hom_τ₁, rightHomologyMap_smul, CategoryTheory.Pretriangulated.shortComplexOfDistTriangleIsoOfIso_hom_τ₂, groupCohomology.isoShortComplexH1_inv, SnakeInput.comp_f₁, SnakeInput.functorL₂'_map_τ₁, LeftHomologyMapData.compatibilityOfZerosOfIsLimitKernelFork_φK, RightHomologyData.map_opcyclesMap', opcyclesMap'_comp, CochainComplex.ShiftSequence.shiftIso_inv_app, FunctorEquivalence.unitIso_hom_app_τ₂_app, opcyclesMap_id, leftHomologyMap_zero, leftRightHomologyComparison'_compatibility, rightHomologyFunctor_map, homologyMapIso'_inv, mapToComposableArrows_id, FunctorEquivalence.counitIso_hom_app_app_τ₃, FunctorEquivalence.inverse_map_τ₃, add_τ₁, LeftHomologyMapData.compatibilityOfZerosOfIsColimitCokernelCofork_φH, pOpcyclesNatTrans_app, leftHomologyMap'_zero, HomologicalComplex.restriction.sc'Iso_hom_τ₃, RightHomologyMapData.map_φQ, groupCohomology.isoShortComplexH2_inv, rightHomologyMap'_zero, HomologicalComplex.homologyFunctorIso_inv_app, HomologyMapData.smul_left, FunctorEquivalence.inverse_map_τ₂, opFunctor_map, SnakeInput.w₁₃, leftHomologyFunctor_linear, HomotopyEquiv.refl_homotopyInvHomId, leftHomologyMap_id, Homotopy.compLeft_h₁, leftHomologyMap'_add, opcyclesMap'_neg, CochainComplex.shiftShortComplexFunctorIso_inv_app_τ₁, SnakeInput.functorL₂_map, fFunctor_map, preservesEpimorphisms_π₃, quasiIso_map_of_preservesRightHomology, CategoryTheory.Pretriangulated.shortComplexOfDistTriangleIsoOfIso_inv_τ₃, zero_τ₂, RightHomologyMapData.add_φQ, HomologicalComplex.natIsoSc'_inv_app_τ₁, HomologicalComplex.restriction.sc'Iso_hom_τ₁, leftHomologyMap_smul, SnakeInput.Hom.comp_f₀, opcyclesMap'_smul, cyclesMap_comp_assoc, mapNatIso_inv, mapLeftHomologyIso_inv_naturality, isIso_iff, leftHomologyMap_comp_assoc, cyclesMapIso_inv, leftHomologyMap'_smul, comp_τ₁_assoc, leftHomologyFunctor_map, cyclesMap_add, HomologicalComplex.natIsoSc'_hom_app_τ₁, mapLeftHomologyIso_inv_naturality_assoc, HomologyMapData.id_left, homologyFunctor_additive, mapHomologyIso'_inv_naturality_assoc, preservesZeroMorphisms_π₂, cyclesMapIso_hom, LeftHomologyMapData.map_φK, opcyclesMap'_zero, leftHomologyMap_neg, HomologicalComplex.HomologySequence.mapSnakeInput_f₀, HomologicalComplex.shortComplexFunctor_obj_X₃, CochainComplex.ShiftSequence.shiftIso_hom_app, mapHomologyIso'_inv_naturality, mapCyclesIso_hom_naturality, preservesMonomorphisms_π₃
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instHasZeroMorphisms 📖 | CompOp | 5 mathmath: instIsNormalEpiCategory, preservesZeroMorphisms_π₃, instIsNormalMonoCategory, preservesZeroMorphisms_π₁, preservesZeroMorphisms_π₂
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instZeroHom 📖 | CompOp | 33 mathmath: SnakeInput.w₀₂_assoc, homologyMap_zero, groupHomology.mapShortComplexH1_zero, groupHomology.mapShortComplexH2_zero, homologyMap'_zero, Homotopy.ofNullHomotopic_h₀, Homotopy.ofNullHomotopic_h₁, RightHomologyMapData.zero_φH, HomologyMapData.zero_left, Homotopy.ofNullHomotopic_h₂, Homotopy.equivSubZero_symm_apply, groupCohomology.mapShortComplexH2_zero, HomologyMapData.zero_right, SnakeInput.w₁₃_assoc, cyclesMap'_zero, zero_τ₃, SnakeInput.w₀₂, rightHomologyMap_zero, cyclesMap_zero, LeftHomologyMapData.zero_φH, LeftHomologyMapData.zero_φK, Homotopy.equivSubZero_apply, opcyclesMap_zero, RightHomologyMapData.zero_φQ, groupCohomology.mapShortComplexH1_zero, Homotopy.ofNullHomotopic_h₃, zero_τ₁, leftHomologyMap_zero, leftHomologyMap'_zero, rightHomologyMap'_zero, SnakeInput.w₁₃, zero_τ₂, opcyclesMap'_zero
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isoMk 📖 | CompOp | 8 mathmath: isoMk_hom_τ₁, isoMk_hom_τ₂, isoMk_inv, isoMk_hom_τ₃, groupHomology.isoShortComplexH1_hom, groupHomology.isoShortComplexH2_hom, groupCohomology.isoShortComplexH1_hom, groupCohomology.isoShortComplexH2_hom
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map 📖 | CompOp | 113 mathmath: ShortExact.instHasSmallLocalizedShiftedHomHomologicalComplexIntUpQuasiIsoX₃CochainComplexMapSingleFunctorOfNatX₁, RightHomologyData.map_ι, ModuleCat.localizedModule_functor_map_exact, map_X₃, mapOpcyclesIso_hom_naturality_assoc, RightHomologyMapData.natTransApp_φQ, Splitting.map_s, ShortExact.map, mapCyclesIso_hom_iCycles_assoc, mapHomologyIso_hom_naturality, LeftHomologyData.map_π, RightHomologyData.map_g', hasHomology_of_preserves, HomologyMapData.natTransApp_left, RightHomologyData.map_H, map_X₂, Splitting.map_r, ShortExact.singleTriangleIso_hom_hom₁, CategoryTheory.Functor.mapShortComplex_map_τ₁, mapOpcyclesIso_inv_naturality, map_f, Module.Flat.lTensor_shortComplex_exact, mapLeftHomologyIso_hom_naturality, map_X₁, mapHomologyIso_hom_naturality_assoc, LeftHomologyData.map_i, Module.Flat.iff_rTensor_preserves_shortComplex_exact, RightHomologyData.mapOpcyclesIso_eq, CategoryTheory.Functor.mapShortComplex_map_τ₂, CategoryTheory.NatTrans.app_homology, Module.Flat.iff_lTensor_preserves_shortComplex_exact, Exact.map_of_epi_of_preservesCokernel, mapRightHomologyIso_inv_naturality, groupHomology.map_chainsFunctor_shortExact, LeftHomologyData.exact_map_iff, RightHomologyMapData.natTransApp_φH, ShortExact.singleTriangleIso_inv_hom₂, map_g, mapHomologyIso'_hom_naturality, RightHomologyData.map_Q, LeftHomologyData.map_K, mapNatIso_hom, hasRightHomology_of_preserves, CategoryTheory.Functor.preservesFiniteColimits_iff_forall_exact_map_and_epi, Exact.map, ShortExact.singleTriangleIso_inv_hom₃, homologyMap_mapNatTrans, mapRightHomologyIso_hom_naturality, CategoryTheory.Functor.mapShortComplex_map_τ₃, HomologyData.map_left, mapRightHomologyIso_inv_naturality_assoc, LeftHomologyMapData.natTransApp_φK, hasLeftHomology_of_preserves, ModuleCat.uliftFunctor_map_exact, LeftHomologyData.mapLeftHomologyIso_eq, CategoryTheory.Functor.mapShortComplex_obj, ShortExact.singleTriangleIso_hom_hom₃, ShortExact.singleTriangleIso_inv_hom₁, RightHomologyData.map_p, mapCyclesIso_hom_naturality_assoc, CategoryTheory.Functor.preservesFiniteLimits_iff_forall_exact_map_and_mono, RightHomologyData.mapRightHomologyIso_eq, CochainComplex.mappingCone.triangleRotateShortComplexSplitting_r, mapCyclesIso_inv_naturality_assoc, RightHomologyData.exact_map_iff, mapNatTrans_τ₁, CategoryTheory.Functor.map_distinguished_exact, LeftHomologyData.mapCyclesIso_eq, LeftHomologyMapData.natTransApp_φH, HomologyData.map_right, FunctorEquivalence.functor_obj_obj, mapCyclesIso_hom_iCycles, mapCyclesIso_inv_naturality, mapLeftHomologyIso_hom_naturality_assoc, LeftHomologyData.mapHomologyIso_eq, map_leftRightHomologyComparison', mapOpcyclesIso_inv_naturality_assoc, mapNatTrans_τ₂, CategoryTheory.Functor.IsHomological.exact, mapRightHomologyIso_hom_naturality_assoc, CochainComplex.mappingCone.triangleRotateShortComplexSplitting_s, ShortExact.singleTriangleIso_hom_hom₂, mapHomologyIso_inv_naturality, exact_map_iff_of_faithful, mapHomologyIso'_hom_naturality_assoc, HomologyData.map_iso, FunctorEquivalence.functor_map_app, groupCohomology.map_cochainsFunctor_shortExact, LeftHomologyData.map_f', map_comp, mapHomologyIso_inv_naturality_assoc, mapOpcyclesIso_hom_naturality, HomologicalComplex.shortExact_iff_degreewise_shortExact, CategoryTheory.Pretriangulated.preadditiveYoneda_map_distinguished, Exact.map_of_mono_of_preservesKernel, ShortExact.map_of_exact, CategoryTheory.Functor.map_distinguished_op_exact, mapNatTrans_τ₃, LeftHomologyData.map_H, HomologyMapData.natTransApp_right, Exact.map_of_preservesRightHomologyOf, exact_iff_exact_map_forget₂, Exact.map_of_preservesLeftHomologyOf, HomologicalComplex.exact_iff_degreewise_exact, mapNatIso_inv, mapLeftHomologyIso_inv_naturality, Module.Flat.rTensor_shortComplex_exact, RightHomologyData.mapHomologyIso'_eq, mapLeftHomologyIso_inv_naturality_assoc, map_id, mapHomologyIso'_inv_naturality_assoc, mapHomologyIso'_inv_naturality, mapCyclesIso_hom_naturality
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mapNatIso 📖 | CompOp | 2 mathmath: mapNatIso_hom, mapNatIso_inv
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mapNatTrans 📖 | CompOp | 18 mathmath: RightHomologyMapData.natTransApp_φQ, HomologyMapData.natTransApp_left, HomologicalComplex.HomologySequence.snakeInput_v₂₃, CategoryTheory.NatTrans.app_homology, HomologicalComplex.HomologySequence.snakeInput_v₀₁, RightHomologyMapData.natTransApp_φH, mapNatIso_hom, homologyMap_mapNatTrans, LeftHomologyMapData.natTransApp_φK, FunctorEquivalence.functor_obj_map, mapNatTrans_τ₁, LeftHomologyMapData.natTransApp_φH, HomologicalComplex.HomologySequence.snakeInput_v₁₂, mapNatTrans_τ₂, FunctorEquivalence.functor_map_app, mapNatTrans_τ₃, HomologyMapData.natTransApp_right, mapNatIso_inv
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op 📖 | CompOp | 82 mathmath: HomologyMapData.op_left, LeftHomologyData.op_g', hasRightHomology_iff_op, Splitting.op_r, cyclesOpIso_inv_op_iCycles_assoc, homologyOpIso_hom_naturality, op_g, ShortExact.op, RightHomologyMapData.op_φH, opMap_τ₁, hasLeftHomology_iff_op, shortExact_iff_op, Homotopy.op_h₀, SnakeInput.op_L₂, Homotopy.op_h₂, op_X₁, op_pOpcycles_opcyclesOpIso_hom, RightHomologyMapData.op_φK, opMap_id, LeftHomologyMapData.op_φH, HomologyData.op_right, leftHomologyMap_op, homologyOpIso_inv_naturality, exact_op_iff, cyclesOpIso_hom_naturality, LeftHomologyData.op_H, opMap_τ₃, opcyclesOpIso_hom_naturality_assoc, homologyMap'_op, LeftHomologyMapData.op_φQ, op_pOpcycles_opcyclesOpIso_hom_assoc, RightHomologyData.op_K, HomologyMapData.op_right, fromOpcycles_op_cyclesOpIso_inv_assoc, rightHomologyMap'_op, opcyclesOpIso_hom_toCycles_op_assoc, opcyclesOpIso_hom_naturality, SnakeInput.op_L₀, homologyOpIso_hom_naturality_assoc, LeftHomologyData.op_Q, instHasHomologyOppositeOp, cyclesOpIso_inv_naturality, instHasLeftHomologyOppositeOpOfHasRightHomology, RightHomologyData.op_H, opMap_τ₂, SnakeInput.op_L₃, opcyclesOpIso_inv_naturality, LeftHomologyData.op_p, HomologyData.op_left, homologyOpIso_inv_naturality_assoc, rightHomologyMap_op, Homotopy.op_h₃, quasiIso_opMap, rightHomologyFunctorOpNatIso_hom_app, Splitting.op_s, opFunctor_obj, RightHomologyData.op_f', leftHomologyFunctorOpNatIso_hom_app, Exact.op, op_f, homologyMap_op, rightHomologyFunctorOpNatIso_inv_app, Homotopy.op_h₁, SnakeInput.op_L₁, HomologyData.op_iso, op_X₃, RightHomologyData.op_i, opcyclesOpIso_hom_toCycles_op, leftHomologyFunctorOpNatIso_inv_app, cyclesOpIso_inv_op_iCycles, cyclesOpIso_inv_naturality_assoc, CategoryTheory.Pretriangulated.preadditiveYoneda_map_distinguished, fromOpcycles_op_cyclesOpIso_inv, CategoryTheory.Functor.map_distinguished_op_exact, opcyclesOpIso_inv_naturality_assoc, LeftHomologyData.op_ι, RightHomologyData.op_π, quasiIso_opMap_iff, op_X₂, leftHomologyMap'_op, instHasRightHomologyOppositeOpOfHasLeftHomology, cyclesOpIso_hom_naturality_assoc
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opEquiv 📖 | CompOp | 4 mathmath: opEquiv_functor, opEquiv_counitIso, opEquiv_inverse, opEquiv_unitIso
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opFunctor 📖 | CompOp | 8 mathmath: opEquiv_functor, opEquiv_counitIso, rightHomologyFunctorOpNatIso_hom_app, opFunctor_obj, leftHomologyFunctorOpNatIso_hom_app, rightHomologyFunctorOpNatIso_inv_app, leftHomologyFunctorOpNatIso_inv_app, opFunctor_map
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opMap 📖 | CompOp | 38 mathmath: HomologyMapData.op_left, homologyOpIso_hom_naturality, RightHomologyMapData.op_φH, opMap_τ₁, Homotopy.op_h₀, Homotopy.op_h₂, SnakeInput.op_v₁₂, RightHomologyMapData.op_φK, opMap_id, LeftHomologyMapData.op_φH, leftHomologyMap_op, homologyOpIso_inv_naturality, cyclesOpIso_hom_naturality, opMap_τ₃, opcyclesOpIso_hom_naturality_assoc, homologyMap'_op, LeftHomologyMapData.op_φQ, HomologyMapData.op_right, rightHomologyMap'_op, opcyclesOpIso_hom_naturality, homologyOpIso_hom_naturality_assoc, cyclesOpIso_inv_naturality, opMap_τ₂, opcyclesOpIso_inv_naturality, homologyOpIso_inv_naturality_assoc, rightHomologyMap_op, Homotopy.op_h₃, quasiIso_opMap, SnakeInput.op_v₂₃, SnakeInput.op_v₀₁, homologyMap_op, Homotopy.op_h₁, cyclesOpIso_inv_naturality_assoc, opcyclesOpIso_inv_naturality_assoc, opFunctor_map, quasiIso_opMap_iff, leftHomologyMap'_op, cyclesOpIso_hom_naturality_assoc
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opUnop 📖 | CompOp | — |
unop 📖 | CompOp | 44 mathmath: LeftHomologyMapData.unop_φQ, LeftHomologyData.unop_p, unop_X₂, Homotopy.unop_h₂, LeftHomologyData.unop_ι, RightHomologyData.unop_i, HomologyMapData.unop_right, unopFunctor_map, Homotopy.unop_h₃, hasLeftHomology_iff_unop, exact_unop_iff, unop_f, unopFunctor_obj, unopMap_τ₁, RightHomologyMapData.unop_φK, HomologyData.unop_left, RightHomologyData.unop_f', Splitting.unop_r, unop_X₁, unopMap_τ₂, unop_X₃, ShortExact.unop, Exact.unop, hasRightHomology_iff_unop, Homotopy.unop_h₁, RightHomologyData.unop_K, LeftHomologyData.unop_H, LeftHomologyMapData.unop_φH, shortExact_iff_unop, instHasHomologyUnopOfOpposite, unop_g, LeftHomologyData.unop_Q, unopMap_id, HomologyData.unop_iso, unopMap_τ₃, RightHomologyData.unop_π, Homotopy.unop_h₀, HomologyData.unop_right, RightHomologyData.unop_H, Splitting.unop_s, LeftHomologyData.unop_g', quasiIso_unopMap, RightHomologyMapData.unop_φH, HomologyMapData.unop_left
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unopFunctor 📖 | CompOp | 4 mathmath: unopFunctor_map, unopFunctor_obj, opEquiv_counitIso, opEquiv_inverse
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unopMap 📖 | CompOp | 16 mathmath: LeftHomologyMapData.unop_φQ, Homotopy.unop_h₂, HomologyMapData.unop_right, unopFunctor_map, Homotopy.unop_h₃, unopMap_τ₁, RightHomologyMapData.unop_φK, unopMap_τ₂, Homotopy.unop_h₁, LeftHomologyMapData.unop_φH, unopMap_id, unopMap_τ₃, Homotopy.unop_h₀, quasiIso_unopMap, RightHomologyMapData.unop_φH, HomologyMapData.unop_left
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unopOp 📖 | CompOp | — |
π₁ 📖 | CompOp | 20 mathmath: toCyclesNatTrans_app, π₁Toπ₂_comp_π₂Toπ₃_assoc, instPreservesFiniteColimitsπ₁, π₁Toπ₂_app, instPreservesColimitπ₁, instPreservesLimitπ₁, instPreservesLimitsOfShapeπ₁, π₁Toπ₂_comp_π₂Toπ₃, FunctorEquivalence.inverse_map_τ₁, preservesMonomorphisms_π₁, FunctorEquivalence.inverse_obj_X₁, instPreservesFiniteLimitsπ₁, π₁_map, preservesZeroMorphisms_π₁, FunctorEquivalence.inverse_obj_f, π₁_obj, preservesEpimorphisms_π₁, instPreservesColimitsOfShapeπ₁, FunctorEquivalence.inverse_map_τ₃, FunctorEquivalence.inverse_map_τ₂
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π₁Toπ₂ 📖 | CompOp | 7 mathmath: π₁Toπ₂_comp_π₂Toπ₃_assoc, π₁Toπ₂_app, π₁Toπ₂_comp_π₂Toπ₃, FunctorEquivalence.inverse_map_τ₁, FunctorEquivalence.inverse_obj_f, FunctorEquivalence.inverse_map_τ₃, FunctorEquivalence.inverse_map_τ₂
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π₂ 📖 | CompOp | 23 mathmath: π₁Toπ₂_comp_π₂Toπ₃_assoc, π₁Toπ₂_app, FunctorEquivalence.inverse_obj_X₂, π₁Toπ₂_comp_π₂Toπ₃, instPreservesColimitsOfShapeπ₂, FunctorEquivalence.inverse_map_τ₁, instPreservesFiniteColimitsπ₂, instPreservesLimitsOfShapeπ₂, π₂_map, instPreservesColimitπ₂, preservesMonomorphisms_π₂, FunctorEquivalence.inverse_obj_f, instPreservesLimitπ₂, π₂Toπ₃_app, instPreservesFiniteLimitsπ₂, preservesEpimorphisms_π₂, iCyclesNatTrans_app, π₂_obj, FunctorEquivalence.inverse_obj_g, FunctorEquivalence.inverse_map_τ₃, pOpcyclesNatTrans_app, FunctorEquivalence.inverse_map_τ₂, preservesZeroMorphisms_π₂
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π₂Toπ₃ 📖 | CompOp | 7 mathmath: π₁Toπ₂_comp_π₂Toπ₃_assoc, π₁Toπ₂_comp_π₂Toπ₃, FunctorEquivalence.inverse_map_τ₁, π₂Toπ₃_app, FunctorEquivalence.inverse_obj_g, FunctorEquivalence.inverse_map_τ₃, FunctorEquivalence.inverse_map_τ₂
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π₃ 📖 | CompOp | 20 mathmath: π₁Toπ₂_comp_π₂Toπ₃_assoc, π₃_map, instPreservesColimitsOfShapeπ₃, π₁Toπ₂_comp_π₂Toπ₃, FunctorEquivalence.inverse_obj_X₃, preservesZeroMorphisms_π₃, FunctorEquivalence.inverse_map_τ₁, instPreservesColimitπ₃, instPreservesFiniteColimitsπ₃, instPreservesLimitsOfShapeπ₃, π₃_obj, π₂Toπ₃_app, instPreservesLimitπ₃, fromOpcyclesNatTrans_app, instPreservesFiniteLimitsπ₃, FunctorEquivalence.inverse_obj_g, FunctorEquivalence.inverse_map_τ₃, FunctorEquivalence.inverse_map_τ₂, preservesEpimorphisms_π₃, preservesMonomorphisms_π₃
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