Documentation Verification Report

Mono

📁 Source: Mathlib/CategoryTheory/Limits/Shapes/Pullback/Mono.lean

Statistics

MetricCount
DefinitionsisLimitMkIdId, isLimitOfCompMono, isLimitOfFactors, isColimitMkIdId, isColimitOfEpiComp, isColimitOfFactors, pullbackIsPullbackOfCompMono, pushoutIsPushoutOfEpiComp, Mono
9
Theoremsfst_eq_snd_of_mono_eq, isIso_fst_of_mono_of_isLimit, isIso_snd_of_mono_of_isLimit, mono_fst_of_is_pullback_of_mono, mono_of_isLimitMkIdId, mono_snd_of_is_pullback_of_mono, epi_inl_of_is_pushout_of_epi, epi_inr_of_is_pushout_of_epi, epi_of_isColimitMkIdId, inl_eq_inr_of_epi_eq, isIso_inl_of_epi_of_isColimit, isIso_inr_of_epi_of_isColimit, epi_coprod_to_pushout, fst_eq_snd_of_mono_eq, hasPullback_of_comp_mono, hasPullback_of_left_factors_mono, hasPullback_of_right_factors_mono, hasPushout_of_epi_comp, hasPushout_of_left_factors_epi, hasPushout_of_right_factors_epi, has_cokernel_pair_of_epi, has_kernel_pair_of_mono, inl_eq_inr_of_epi_eq, isIso_fst_of_mono, isIso_inl_of_epi, isIso_inr_of_epi, isIso_snd_of_mono, mono_pullback_to_prod, fst_of_mono, snd_of_mono, pullbackSymmetry_hom_of_mono_eq, pullback_snd_iso_of_left_factors_mono, pullback_snd_iso_of_right_factors_mono, pullback_symmetry_hom_of_epi_eq, inl_of_epi, inr_of_epi, pushout_inl_iso_of_left_factors_epi, pushout_inr_iso_of_right_factors_epi
38
Total47

CategoryTheory

Definitions

NameCategoryTheorems
Mono 📖CompData
375 mathmath: Limits.MonoCoprod.instMonoInl, IsGrothendieckAbelian.instMonoIMonomorphismsRlpMonoMapFactorizationDataRlp, IsKernelPair.mono_of_eq_fst_snd, Linear.instMonoHSMulHomOfInvertible, Limits.InitialMonoClass.isInitial_mono_from, instMonoOfIsThin, TopCat.instMonoBallInclusion, Injective.ι_mono, Limits.MonoCoprod.instMonoInr, groupHomology.mono_δ_of_isZero, isCoseparator_iff_mono, Limits.limitSubobjectProduct_mono, ShortComplex.SnakeInput.mono_v₀₁_τ₂, instMonoMap'KernelCokernelCompSequenceOfNatNat, instMonoFunctorWhiskerRightOfPreservesMonomorphisms, Limits.MonoCoprod.mono_ι, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.mono, Monad.algebra_mono_of_mono, ShortComplex.ShortExact.mono_δ, Limits.instMonoι_1, op_mono_of_epi, Initial.mono_to, Under.mono_right_of_mono, SSet.modelCategoryQuillen.mono_of_cofibration, Limits.mono_of_isLimit_parallelFamily, IsPushout.mono_of_isPullback_of_mono, ShortComplex.Splitting.mono_f, IsSubterminal.mono_isTerminal_from, Abelian.mono_of_epi_of_epi_mono', HomologicalComplex.instMonoICycles, Limits.MonoCoprod.mono_of_injective_aux, ShortComplex.SnakeInput.mono_v₀₁_τ₃, Limits.MonoCoprod.binaryCofan_inl, ShortComplex.exact_and_mono_f_iff_of_iso, Endofunctor.Algebra.mono_of_mono, unop_mono_iff, Limits.Fork.IsLimit.mono, Mono.of_coproductDisjoint, Adhesive.mono_of_isPushout_of_mono_left, ShortComplex.LeftHomologyData.instMonoI, Limits.MonoFactorisation.m_mono, FinitaryExtensive.mono_ι, instMonoι, op_epi_iff, Limits.Types.mono_of_isPushout_of_isPullback, groupHomology.chainsMap_id_f_map_mono, instMonoF', ObjectProperty.isCoseparating_iff_mono, preserves_mono_of_preservesLimit, groupCohomology.instMonoModuleCatFH1InfRes, Abelian.mono_of_epi_of_mono_of_mono', GrothendieckTopology.MayerVietorisSquare.mono_f₁₃, Limits.MonoCoprod.mono_of_injective', GrpCat.mono_iff_ker_eq_bot, RegularMono.mono, DerivedCategory.HomologySequence.mono_homologyMap_mor₁_iff, Limits.instMonoLiftπ, Limits.cokernel_not_mono_of_nonzero, Adjunction.Triple.mono_leftToRight_app_iff_mono_adj₁_counit_app, Adjunction.unit_mono_of_L_faithful, Arrow.mono_left, reflects_mono_of_reflectsLimit, Limits.instMonoInr, mono_comp', mono_iff_injective, JointlyReflectMonomorphisms.mono_iff, AddCommGrpCat.instMonoι, initial_mono, AlgebraicGeometry.SurjectiveOnStalks.mono_of_injective, Limits.Types.instMonoPushoutInl, Abelian.mono_of_epi_of_mono_of_mono, FintypeCat.instMonoActionFintypeCatImageComplementIncl, HomologicalComplex.instMonoFιTruncLE, Adjunction.Quadruple.epi_leftTriple_leftToRight_iff_mono_rightTriple_rightToLeft, FinitaryExtensive.mono_inl_of_isColimit, ShortComplex.SnakeInput.mono_L₀_f, SimplicialObject.Splitting.IndexSet.eqId_iff_mono, HomologicalComplex.instMonoFTruncLE'ToRestriction, Mono.ι_of_coproductDisjoint, presheaf_mono_of_mono, ShortComplex.Exact.mono_cokernelDesc, Limits.PullbackCone.mono_of_isLimitMkIdId, Limits.wideEqualizer.ι_mono, PresheafOfModules.mono_iff_surjective, Limits.PullbackCone.mono_fst_of_is_pullback_of_mono, ComposableArrows.IsComplex.mono_cokerToKer', MorphismProperty.presheaf_mono_of_le, ShortComplex.Exact.mono_g, isCoseparating_iff_mono, HomologicalComplex.instMonoFShortComplexTruncLE, Abelian.instMonoFactorThruCoimage, Functor.ReflectsMonomorphisms.reflects, Limits.prod.mono_lift_of_mono_right, IsKernelPair.mono_of_eq_fst_snd', Limits.prod.mono_lift_of_mono_left, StrongMono.mono, Limits.image.lift_mono, Sheaf.instMonoAppArrowPLocallySurjectiveLocallyInjectiveFunctorialLocallySurjectiveInjectiveFactorization, MorphismProperty.monomorphisms.iff, CommGrpCat.mono_iff_injective, ShortComplex.SnakeInput.mono_δ, Preadditive.mono_of_isZero_kernel, Subobject.instMonoOfLE, ShortComplex.instMonoAbelianImageToKernel, Adjunction.Quadruple.epi_leftTriple_rightToLeft_iff_mono_rightTriple_leftToRight, ModuleCat.mono_iff_injective, Over.mono_homMk, Abelian.mono_cokernel_map_of_isPullback, Subobject.arrow_mono, FinitaryExtensive.mono_inr_of_isColimit, Limits.kernel.lift_mono, groupCohomology.mono_map_0_of_mono, Sheaf.Hom.mono_iff_presheaf_mono, AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.mono, instMonoAppOfFunctor, StructuredArrow.mono_iff_mono_right, Functor.homologySequence_mono_shift_map_mor₁_iff, PreGaloisCategory.exists_lift_of_mono_of_isConnected, IsPullback.mono_shortComplex'_f, Under.mono_iff_mono_right, Limits.biprod.map_mono, SplitMono.mono, HomologicalComplex.instMonoCyclesMapOfF, Limits.initial.mono_from, TopCat.Presheaf.stalk_mono_of_mono, Limits.biprod.mono_lift_of_mono_right, Pretriangulated.Triangle.mono₁, AlgebraicGeometry.PresheafedSpace.ofRestrict_mono, groupCohomology.instMonoModuleCatFShortComplexH0, Preadditive.mono_of_cancel_zero, Preadditive.mono_iff_injective, Preadditive.instMonoNegHom, ObjectProperty.IsSeparating.mono_iff, instMonoId, Limits.MonoCoprod.mono_of_injective, Adjunction.instMonoCoeEquivHomObjHomEquivOfReflectsMonomorphisms, HomologicalComplex.mono_homologyMap_shortComplexTruncLE_g, Preadditive.mono_of_kernel_iso_zero, Mono.cofanInr_of_binaryCoproductDisjoint, DerivedCategory.HomologySequence.mono_homologyMap_mor₂_iff, Abelian.mono_pushout_of_mono_g, Limits.equalizer.ι_mono, groupCohomology.cochainsMap_id_f_map_mono, Functor.map_mono, Mono.cofanInl_of_binaryCoproductDisjoint, Sheaf.mono_of_isLocallyInjective, Adjunction.Quadruple.epi_leftTriple_leftToRight_app_iff_mono_rightTriple_rightToLeft_app, TopCat.Presheaf.instMonoCommRingCatToTotalQuotientPresheafPresheaf, ShortComplex.ShortExact.mono_f, SimplexCategory.mono_iff_injective, AddGrpCat.mono_iff_injective, Subobject.instMonoOfLEMk, ShortComplex.mono_τ₂_of_exact_of_mono, Sheaf.Hom.mono_of_presheaf_mono, CompHausLike.mono_iff_injective, NatTrans.mono_iff_mono_app, AlgebraicGeometry.IsPreimmersion.instMonoScheme, MonoOver.mono, unop_mono_of_epi, ComplexShape.Embedding.mono_liftExtend_f_iff, SSet.Subcomplex.instMonoToRange, Square.IsPullback.mono_f₁₂, ShortComplex.quasiIso_iff_of_zeros, mono_of_strongMono, ShortComplex.Exact.mono_g_iff, SSet.modelCategoryQuillen.cofibration_iff, NonPreadditiveAbelian.mono_Δ, BinaryCofan.mono_inr_of_isVanKampen, Preadditive.mono_iff_isZero_kernel', ObjectProperty.IsCoseparating.mono_productTo, CostructuredArrow.mono_of_mono_left, HomologicalComplex.instMonoHomologyι, IsGrothendieckAbelian.mono_of_isColimit_monoOver, AddCommGrpCat.mono_iff_injective, AlgebraicGeometry.IsClosedImmersion.iff_isFinite_and_mono, AlgebraicGeometry.mono_pushoutSection_of_isCompact_of_flat_left_of_ringHomFlat, CochainComplex.isIso_liftCycles_iff, AlgebraicGeometry.instMonoObjWalkingSpanCompSchemeSpanForgetNoneWalkingPairSomeMapInitOfIsOpenImmersion, Limits.mono_of_source_iso_zero, NatTrans.mono_iff_mono_app', groupHomology.chainsMap_f_map_mono, ShortComplex.mono_homologyMap_of_mono_opcyclesMap', Limits.coprod.map_mono, Limits.instMonoι, Limits.MonoCoprod.mono_map'_of_injective, Limits.MonoCoprod.mono_binaryCofanSum_inl', mono_comp, ShortComplex.RightHomologyData.exact_iff_mono_g', Subobject.widePullbackι_mono, SSet.stdSimplex.objEquiv_symm_mem_nonDegenerate_iff_mono, ShortComplex.exact_iff_mono, Abelian.Ext.mono_postcomp_mk₀_of_mono, CommGrpCat.mono_iff_ker_eq_bot, PresheafOfModules.mono_of_injective, StructuredArrow.mono_right_of_mono, unop_epi_iff, ShortComplex.Exact.mono_fromOpcycles, op_mono_iff, CochainComplex.cm5b.instMonoFIntI, mono_of_mono, instMonoImageToKernel, Limits.prod.map_mono, ShortComplex.RightHomologyData.instMonoι, PreGaloisCategory.fiber_in_connected_component, GrpCat.mono_iff_injective, Limits.Types.pushoutCocone_inr_mono_of_isColimit, Abelian.mono_factorThruImage, Limits.limMap_mono', Limits.MonoCoprod.mono_inl_iff, Limits.pullback.fst_of_mono, HomologicalComplex.instMonoιTruncLE, isIso_iff_mono_and_epi, SSet.Subcomplex.instMonoι, mono_simpleSubobjectArrow, Subobject.instMonoOfMkLE, HomologicalComplex.instMonoFOfHasFiniteLimits, InjectivePresentation.mono, Limits.pullback.snd_of_mono, Preadditive.mono_iff_isZero_kernel, Limits.pullback.isIso_diagonal_iff, Functor.preservesFiniteLimits_iff_forall_exact_map_and_mono, AlgebraicTopology.DoldKan.instMonoChainComplexNatInclusionOfMooreComplexMap, Limits.MonoCoprod.mono_inj, ModuleCat.instMonoι, Subfunctor.instMonoFunctorTypeHomOfLe, ConcreteCategory.mono_iff_injective_of_preservesPullback, Limits.mono_of_isLimit_fork, CochainComplex.cm5b.instMonoIntI, Limits.image.preComp_mono, TopCat.GlueData.ι_mono, ShortComplex.exact_iff_mono_cokernel_desc, Abelian.Ext.mono_precomp_mk₀_of_epi, Mono.inr_of_binaryCoproductDisjoint, groupCohomology.cochainsMap_f_map_mono, ShortComplex.exact_and_mono_f_iff_f_is_kernel, Pretriangulated.Triangle.mono₃, GlueData.f_mono, StructuredArrow.mono_homMk, Sieve.uliftFunctorInclusion_is_mono, GlueData'.f_mono, PresheafOfModules.instMonoModuleCatCarrierObjOppositeRingCatApp, Limits.MonoCoprod.mono_binaryCofanSum_inr', AlgebraicGeometry.mono_pushoutSection_of_isCompact_of_flat_right, IsIso.mono_of_iso, ShortComplex.SnakeInput.mono_v₀₁_τ₁, Abelian.Pseudoelement.mono_of_zero_of_map_zero, Sheaf.mono_of_injective, Abelian.mono_of_kernel_ι_eq_zero, CartesianMonoidalCategory.mono_lift_of_mono_right, Limits.CoproductDisjoint.mono_inj, AlgebraicGeometry.IsClosedImmersion.iff_isProper_and_mono, AlgebraicGeometry.mono_pushoutSection_of_isCompact_of_flat_right_of_ringHomFlat, Abelian.mono_inl_of_isColimit, ShortComplex.SnakeInput.mono_L₂_f, Functor.PreservesMonomorphisms.preserves, IsKernelPair.mono_of_isIso_fst, Mono.of_binaryCoproductDisjoint_left, HomologicalComplex.HomologySequence.instMonoMap'ComposableArrows₃OfNatNat, Limits.IsZero.mono, Subfunctor.instMonoFunctorTypeι_1, Functor.mono_of_mono_map, NonPreadditiveAbelian.mono_r, StructuredArrow.mono_of_mono_right, mono_comp_iff_of_mono, Subobject.instMonoOfMkLEMk, AlgebraicGeometry.SheafedSpace.mono_of_base_injective_of_stalk_epi, mono_iff_fst_eq_snd, Limits.Types.Pushout.mono_inr, Under.mono_of_mono_right, SSet.mono_of_nonDegenerate, mono_of_nonzero_from_simple, TopCat.mono_iff_injective, Pretriangulated.Triangle.mor₁_eq_zero_iff_mono₂, MonoOver.instMonoHomDiscretePUnitObjOverForget, Limits.HasZeroObject.instMono, Mono.of_binaryCoproductDisjoint_right, GrothendieckTopology.MayerVietorisSquare.instMonoSheafAddCommGrpCatFShortComplex, Sieve.functorInclusion_is_mono, Functor.mono_map_iff_mono, CartesianMonoidalCategory.mono_lift_of_mono_left, Limits.MonoCoprod.mono_binaryCofanSum_inr, Square.IsPullback.mono_f₁₃, SSet.stdSimplex.mem_nonDegenerate_iff_mono, Preadditive.mono_iff_injective', mono_of_cofan_isVanKampen, PreGaloisCategory.exists_lift_of_mono, Abelian.mono_pushout_of_mono_f, Limits.instMonoInl, SSet.Subcomplex.mono_homOfLE, Abelian.tfae_mono, Preadditive.mono_of_isZero_kernel', ShortComplex.exact_iff_mono_fromOpcycles, Adhesive.mono_of_isPushout_of_mono_right, NonPreadditiveAbelian.instMonoFactorThruCoimage, Limits.Types.instMonoImageι, AlgebraicGeometry.mono_pushoutSection_of_isCompact_of_flat_left, TopCat.instMonoDiskBoundaryInclusion, subtype_val_mono, CostructuredArrow.mono_homMk, Adjunction.Triple.mono_leftToRight_app_iff_mono_adj₂_unit_app, Mono.inl_of_binaryCoproductDisjoint, Limits.colim.map_mono', InjectiveResolution.instMonoFNatι, Endofunctor.Coalgebra.mono_of_mono, ModuleCat.mono_iff_ker_eq_bot, Under.mono_homMk, TopModuleCat.instMonoKerι, Pretriangulated.Triangle.mor₂_eq_zero_iff_mono₃, Limits.Types.instMonoPushoutInr, mono_of_mono_fac, ShortComplex.Exact.mono_g', Abelian.mono_inl_of_factor_thru_epi_mono_factorization, Abelian.mono_inr_of_isColimit, IsSubterminal.mono_terminal_from, Limits.mono_pullback_to_prod, ConcreteCategory.mono_of_injective, Comonad.algebra_mono_of_mono, AlgebraicGeometry.instMonoObjWalkingSpanCompOverSchemeTopMorphismPropertySpanOverForgetForgetForgetNoneWalkingPairSomeMapInitOfIsOpenImmersionLeftDiscretePUnit, NormalEpiCategory.mono_of_cancel_zero, NormalEpiCategory.mono_of_zero_kernel, Limits.MonoCoprod.mono_binaryCofanSum_inl, ShortComplex.mono_of_mono_of_mono_of_mono, IsPushout.IsVanKampen.mono_of_mono_right, Pretriangulated.Triangle.mor₃_eq_zero_iff_mono₁, Limits.Multiequalizer.instMonoιPi, mono_iff_isIso_fst, ModuleCat.mono_as_hom'_subtype, Limits.biprod.mono_lift_of_mono_left, ShortComplex.instMonoRightHomologyι, HomologicalComplex.mono_homologyMap_of_mono_of_not_rel, NonemptyFinLinOrd.mono_iff_injective, Subfunctor.instMonoFunctorTypeι, mono_iff_isIso_snd, Limits.PullbackCone.mono_snd_of_is_pullback_of_mono, Pretriangulated.Triangle.mono₂, Limits.IsTerminal.mono_from, Abelian.mono_of_epi_of_epi_of_mono, ShortComplex.instMonoHomologyι, ShortComplex.instMonoFKernelSequence, AddCommGrpCat.mono_iff_ker_eq_bot, TopCat.Presheaf.mono_iff_stalk_mono, Rep.mono_iff_injective, mono_iff_isPullback, Rep.instMonoModuleCatHom, IsSplitMono.mono, Abelian.mono_of_mono_of_mono_of_mono, Limits.IsInitial.mono_from, Preadditive.mono_of_kernel_zero, AlgebraicGeometry.IsOpenImmersion.mono, groupCohomology.mono_δ_of_isZero, ShortComplex.mono_homologyMap_of_mono_opcyclesMap, Preadditive.mono_iff_cancel_zero, ShortComplex.SnakeInput.instMonoFL₀'OfL₁, mono_comp_iff_of_isIso, Functor.homologySequence_mono_shift_map_mor₂_iff, kernelCokernelCompSequence.instMonoι, ShortComplex.mono_of_epi_of_epi_of_mono, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.instMono, MonoOver.mono_obj_hom, SimplexCategory.instMonoδ, IsPushout.IsVanKampen.mono_of_mono_left, Classifier.mono_truth, Over.mono_of_mono_left, Over.mono_left_of_mono, AddGrpCat.mono_iff_ker_eq_bot, Adjunction.Triple.mono_leftToRight_app_iff, Limits.Types.mono_inl, instMonoSheafTypeImageι, Limits.MonoCoprod.binaryCofan_inr, mono_iff_forall_injective, Profinite.NobelingProof.succ_mono, Regular.instMonoDesc, ShortComplex.instMonoICycles, PreGaloisCategory.has_non_trivial_subobject_of_not_isConnected_of_not_initial, Adjunction.Quadruple.epi_leftTriple_rightToLeft_app_iff_mono_rightTriple_leftToRight_app

CategoryTheory.Limits

Definitions

NameCategoryTheorems
pullbackIsPullbackOfCompMono 📖CompOp
pushoutIsPushoutOfEpiComp 📖CompOp

Theorems

NameKindAssumesProvesValidatesDepends On
epi_coprod_to_pushout 📖mathematicalCategoryTheory.Epi
coprod
pushout
coprod.desc
pushout.inl
pushout.inr
pushout.hom_ext
coprod.desc_comp
colimit.ι_desc
fst_eq_snd_of_mono_eq 📖mathematicalpullback.fst
has_kernel_pair_of_mono
pullback.snd
PullbackCone.fst_eq_snd_of_mono_eq
has_kernel_pair_of_mono
hasPullback_of_comp_mono 📖mathematicalHasPullback
CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
hasPullback_of_left_factors_mono 📖mathematicalHasPullback
CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
CategoryTheory.Category.id_comp
hasPullback_of_comp_mono
hasPullback_of_right_iso
CategoryTheory.IsIso.id
hasPullback_of_right_factors_mono 📖mathematicalHasPullback
CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
CategoryTheory.Category.id_comp
hasPullback_of_comp_mono
hasPullback_of_left_iso
CategoryTheory.IsIso.id
hasPushout_of_epi_comp 📖mathematicalHasPushout
CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
hasPushout_of_left_factors_epi 📖mathematicalHasPushout
CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
CategoryTheory.Category.comp_id
hasPushout_of_epi_comp
hasPushout_of_right_iso
CategoryTheory.IsIso.id
hasPushout_of_right_factors_epi 📖mathematicalHasPushout
CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
CategoryTheory.Category.comp_id
hasPushout_of_epi_comp
hasPushout_of_left_iso
CategoryTheory.IsIso.id
has_cokernel_pair_of_epi 📖mathematicalHasPushout
has_kernel_pair_of_mono 📖mathematicalHasPullback
inl_eq_inr_of_epi_eq 📖mathematicalpushout.inl
has_cokernel_pair_of_epi
pushout.inr
PushoutCocone.inl_eq_inr_of_epi_eq
has_cokernel_pair_of_epi
isIso_fst_of_mono 📖mathematicalCategoryTheory.IsIso
pullback
has_kernel_pair_of_mono
pullback.fst
PullbackCone.isIso_fst_of_mono_of_isLimit
has_kernel_pair_of_mono
isIso_inl_of_epi 📖mathematicalCategoryTheory.IsIso
pushout
has_cokernel_pair_of_epi
pushout.inl
PushoutCocone.isIso_inl_of_epi_of_isColimit
has_cokernel_pair_of_epi
isIso_inr_of_epi 📖mathematicalCategoryTheory.IsIso
pushout
has_cokernel_pair_of_epi
pushout.inr
PushoutCocone.isIso_inr_of_epi_of_isColimit
has_cokernel_pair_of_epi
isIso_snd_of_mono 📖mathematicalCategoryTheory.IsIso
pullback
has_kernel_pair_of_mono
pullback.snd
PullbackCone.isIso_snd_of_mono_of_isLimit
has_kernel_pair_of_mono
mono_pullback_to_prod 📖mathematicalCategoryTheory.Mono
pullback
prod
prod.lift
pullback.fst
pullback.snd
pullback.hom_ext
prod.comp_lift
limit.lift_π
pullbackSymmetry_hom_of_mono_eq 📖mathematicalCategoryTheory.Iso.hom
pullback
has_kernel_pair_of_mono
hasPullback_symmetry
pullbackSymmetry
CategoryTheory.CategoryStruct.id
CategoryTheory.Category.toCategoryStruct
pullback.hom_ext
hasPullback_symmetry
has_kernel_pair_of_mono
fst_eq_snd_of_mono_eq
pullbackSymmetry_hom_comp_snd
CategoryTheory.Category.id_comp
pullback_snd_iso_of_left_factors_mono 📖mathematicalCategoryTheory.IsIso
pullback
CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
hasPullback_of_left_factors_mono
pullback.fst
hasPullback_of_comp_mono
hasPullback_of_right_iso
CategoryTheory.IsIso.id
limit.isoLimitCone_hom_π
hasPullback_of_left_factors_mono
CategoryTheory.Category.id_comp
CategoryTheory.IsIso.comp_isIso
CategoryTheory.Iso.isIso_hom
pullback_fst_iso_of_right_iso
pullback_snd_iso_of_right_factors_mono 📖mathematicalCategoryTheory.IsIso
pullback
CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
hasPullback_of_right_factors_mono
pullback.snd
hasPullback_of_comp_mono
hasPullback_of_left_iso
CategoryTheory.IsIso.id
limit.isoLimitCone_hom_π
hasPullback_of_right_factors_mono
CategoryTheory.Category.id_comp
CategoryTheory.IsIso.comp_isIso
CategoryTheory.Iso.isIso_hom
pullback_snd_iso_of_left_iso
pullback_symmetry_hom_of_epi_eq 📖mathematicalCategoryTheory.Iso.hom
pushout
has_cokernel_pair_of_epi
hasPushout_symmetry
pushoutSymmetry
CategoryTheory.CategoryStruct.id
CategoryTheory.Category.toCategoryStruct
pushout.hom_ext
has_cokernel_pair_of_epi
hasPushout_symmetry
inl_eq_inr_of_epi_eq
inr_comp_pushoutSymmetry_hom
CategoryTheory.Category.comp_id
pushout_inl_iso_of_left_factors_epi 📖mathematicalCategoryTheory.IsIso
pushout
CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
hasPushout_of_left_factors_epi
pushout.inl
hasPushout_of_left_factors_epi
hasPushout_of_epi_comp
hasPushout_of_right_iso
CategoryTheory.IsIso.id
CategoryTheory.Category.comp_id
colimit.isoColimitCocone_ι_inv
CategoryTheory.IsIso.comp_isIso
pushout_inl_iso_of_right_iso
CategoryTheory.Iso.isIso_inv
pushout_inr_iso_of_right_factors_epi 📖mathematicalCategoryTheory.IsIso
pushout
CategoryTheory.CategoryStruct.comp
CategoryTheory.Category.toCategoryStruct
hasPushout_of_right_factors_epi
pushout.inr
hasPushout_of_right_factors_epi
hasPushout_of_epi_comp
hasPushout_of_left_iso
CategoryTheory.IsIso.id
CategoryTheory.Category.comp_id
colimit.isoColimitCocone_ι_inv
CategoryTheory.IsIso.comp_isIso
pushout_inr_iso_of_left_iso
CategoryTheory.Iso.isIso_inv

CategoryTheory.Limits.PullbackCone

Definitions

NameCategoryTheorems
isLimitMkIdId 📖CompOp
isLimitOfCompMono 📖CompOp
isLimitOfFactors 📖CompOp

Theorems

NameKindAssumesProvesValidatesDepends On
fst_eq_snd_of_mono_eq 📖mathematicalfst
snd
CategoryTheory.cancel_mono
condition
isIso_fst_of_mono_of_isLimit 📖mathematicalCategoryTheory.IsIso
CategoryTheory.Limits.Cone.pt
CategoryTheory.Limits.WalkingCospan
CategoryTheory.Limits.WidePullbackShape.category
CategoryTheory.Limits.WalkingPair
CategoryTheory.Limits.cospan
fst
CategoryTheory.Category.id_comp
IsLimit.hom_ext
CategoryTheory.Category.assoc
IsLimit.lift_fst
CategoryTheory.Category.comp_id
fst_eq_snd_of_mono_eq
IsLimit.lift_snd
isIso_snd_of_mono_of_isLimit 📖mathematicalCategoryTheory.IsIso
CategoryTheory.Limits.Cone.pt
CategoryTheory.Limits.WalkingCospan
CategoryTheory.Limits.WidePullbackShape.category
CategoryTheory.Limits.WalkingPair
CategoryTheory.Limits.cospan
snd
isIso_fst_of_mono_of_isLimit
fst_eq_snd_of_mono_eq
mono_fst_of_is_pullback_of_mono 📖mathematicalCategoryTheory.Mono
CategoryTheory.Limits.Cone.pt
CategoryTheory.Limits.WalkingCospan
CategoryTheory.Limits.WidePullbackShape.category
CategoryTheory.Limits.WalkingPair
CategoryTheory.Limits.cospan
fst
IsLimit.hom_ext
CategoryTheory.cancel_mono
CategoryTheory.Category.assoc
condition
Mathlib.Tactic.Reassoc.eq_whisker'
mono_of_isLimitMkIdId 📖mathematicalCategoryTheory.Mono
mono_snd_of_is_pullback_of_mono 📖mathematicalCategoryTheory.Mono
CategoryTheory.Limits.Cone.pt
CategoryTheory.Limits.WalkingCospan
CategoryTheory.Limits.WidePullbackShape.category
CategoryTheory.Limits.WalkingPair
CategoryTheory.Limits.cospan
snd
IsLimit.hom_ext
CategoryTheory.cancel_mono
CategoryTheory.Category.assoc
condition
Mathlib.Tactic.Reassoc.eq_whisker'

CategoryTheory.Limits.PushoutCocone

Definitions

NameCategoryTheorems
isColimitMkIdId 📖CompOp
isColimitOfEpiComp 📖CompOp
isColimitOfFactors 📖CompOp

Theorems

NameKindAssumesProvesValidatesDepends On
epi_inl_of_is_pushout_of_epi 📖mathematicalCategoryTheory.Epi
CategoryTheory.Limits.Cocone.pt
CategoryTheory.Limits.WalkingSpan
CategoryTheory.Limits.WidePushoutShape.category
CategoryTheory.Limits.WalkingPair
CategoryTheory.Limits.span
inl
IsColimit.hom_ext
CategoryTheory.cancel_epi
condition_assoc
epi_inr_of_is_pushout_of_epi 📖mathematicalCategoryTheory.Epi
CategoryTheory.Limits.Cocone.pt
CategoryTheory.Limits.WalkingSpan
CategoryTheory.Limits.WidePushoutShape.category
CategoryTheory.Limits.WalkingPair
CategoryTheory.Limits.span
inr
IsColimit.hom_ext
CategoryTheory.cancel_epi
condition_assoc
epi_of_isColimitMkIdId 📖mathematicalCategoryTheory.Epi
inl_eq_inr_of_epi_eq 📖mathematicalinl
inr
CategoryTheory.cancel_epi
condition
isIso_inl_of_epi_of_isColimit 📖mathematicalCategoryTheory.IsIso
CategoryTheory.Limits.Cocone.pt
CategoryTheory.Limits.WalkingSpan
CategoryTheory.Limits.WidePushoutShape.category
CategoryTheory.Limits.WalkingPair
CategoryTheory.Limits.span
inl
CategoryTheory.Category.comp_id
IsColimit.inl_desc
IsColimit.hom_ext
IsColimit.inl_desc_assoc
CategoryTheory.Category.id_comp
inl_eq_inr_of_epi_eq
IsColimit.inr_desc_assoc
isIso_inr_of_epi_of_isColimit 📖mathematicalCategoryTheory.IsIso
CategoryTheory.Limits.Cocone.pt
CategoryTheory.Limits.WalkingSpan
CategoryTheory.Limits.WidePushoutShape.category
CategoryTheory.Limits.WalkingPair
CategoryTheory.Limits.span
inr
isIso_inl_of_epi_of_isColimit
inl_eq_inr_of_epi_eq

CategoryTheory.Limits.pullback

Theorems

NameKindAssumesProvesValidatesDepends On
fst_of_mono 📖mathematicalCategoryTheory.Mono
CategoryTheory.Limits.pullback
fst
CategoryTheory.Limits.PullbackCone.mono_fst_of_is_pullback_of_mono
snd_of_mono 📖mathematicalCategoryTheory.Mono
CategoryTheory.Limits.pullback
snd
CategoryTheory.Limits.PullbackCone.mono_snd_of_is_pullback_of_mono

CategoryTheory.Limits.pushout

Theorems

NameKindAssumesProvesValidatesDepends On
inl_of_epi 📖mathematicalCategoryTheory.Epi
CategoryTheory.Limits.pushout
inl
CategoryTheory.Limits.PushoutCocone.epi_inl_of_is_pushout_of_epi
inr_of_epi 📖mathematicalCategoryTheory.Epi
CategoryTheory.Limits.pushout
inr
CategoryTheory.Limits.PushoutCocone.epi_inr_of_is_pushout_of_epi

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