Abelian categories

This file contains the definition and basic properties of abelian categories.

There are many definitions of abelian category. Our definition is as follows: A category is called abelian if it is preadditive, has a finite products, kernels and cokernels, and if every monomorphism and epimorphism is normal.

It should be noted that if we also assume coproducts, then preadditivity is actually a consequence of the other properties, as we show in NonPreadditiveAbelian.lean. However, this fact is of little practical relevance, since essentially all interesting abelian categories come with a preadditive structure. In this way, by requiring preadditivity, we allow the user to pass in the "native" preadditive structure for the specific category they are working with.

Main definitions

• Abelian is the type class indicating that a category is abelian. It extends Preadditive.
• Abelian.image f is kernel (cokernel.π f), and
• Abelian.coimage f is cokernel (kernel.ι f).

Main results

• In an abelian category, mono + epi = iso.
• If f : X ⟶ Y, then the map factorThruImage f : X ⟶ image f is an epimorphism, and the map factorThruCoimage f : coimage f ⟶ Y is a monomorphism.
• Factoring through the image and coimage is a strong epi-mono factorisation. This means that
  • every abelian category has images. We provide the isomorphism imageIsoImage : abelian.image f ≅ limits.image f.
  • the canonical morphism coimageImageComparison : coimage f ⟶ image f is an isomorphism.
• We provide the alternate characterisation of an abelian category as a category with (co)kernels and finite products, and in which the canonical coimage-image comparison morphism is always an isomorphism.
• Every epimorphism is a cokernel of its kernel. Every monomorphism is a kernel of its cokernel.
• The pullback of an epimorphism is an epimorphism. The pushout of a monomorphism is a monomorphism. (This is not to be confused with the fact that the pullback of a monomorphism is a monomorphism, which is true in any category).

Implementation notes

The typeclass Abelian does not extend NonPreadditiveAbelian, to avoid having to deal with comparing the two HasZeroMorphisms instances (one from Preadditive in Abelian, and the other a field of NonPreadditiveAbelian). As a consequence, at the beginning of this file we trivially build a NonPreadditiveAbelian instance from an Abelian instance, and use this to restate a number of theorems, in each case just reusing the proof from NonPreadditiveAbelian.lean.

We don't show this yet, but abelian categories are finitely complete and finitely cocomplete. However, the limits we can construct at this level of generality will most likely be less nice than the ones that can be created in specific applications. For this reason, we adopt the following convention:

• If the statement of a theorem involves limits, the existence of these limits should be made an explicit typeclass parameter.
• If a limit only appears in a proof, but not in the statement of a theorem, the limit should not be a typeclass parameter, but instead be created using Abelian.hasPullbacks or a similar definition.

References

• [F. Borceux, Handbook of Categorical Algebra 2][borceux-vol2]
• [P. Aluffi, Algebra: Chapter 0][aluffi2016]

class CategoryTheory.Abelian (C : Type u) [CategoryTheory.Category.{v, u} C] extends CategoryTheory.Preadditive , CategoryTheory.NormalMonoCategory , CategoryTheory.NormalEpiCategory : Type (max u v)

A (preadditive) category C is called abelian if it has all finite products, all kernels and cokernels, and if every monomorphism is the kernel of some morphism and every epimorphism is the cokernel of some morphism.

(This definition implies the existence of zero objects: finite products give a terminal object, and in a preadditive category any terminal object is a zero object.)

• homGroup : (P Q : C) → AddCommGroup (P ⟶ Q)
• add_comp : ∀ (P Q R : C) (f f' : P ⟶ Q) (g : Q ⟶ R), CategoryTheory.CategoryStruct.comp (f + f') g = CategoryTheory.CategoryStruct.comp f g + CategoryTheory.CategoryStruct.comp f' g
• comp_add : ∀ (P Q R : C) (f : P ⟶ Q) (g g' : Q ⟶ R), CategoryTheory.CategoryStruct.comp f (g + g') = CategoryTheory.CategoryStruct.comp f g + CategoryTheory.CategoryStruct.comp f g'
• normalMonoOfMono : {X Y : C} → (f : X ⟶ Y) → [inst : CategoryTheory.Mono f] → CategoryTheory.NormalMono f
• normalEpiOfEpi : {X Y : C} → (f : X ⟶ Y) → [inst : CategoryTheory.Epi f] → CategoryTheory.NormalEpi f
• has_finite_products : CategoryTheory.Limits.HasFiniteProducts C
• has_kernels : CategoryTheory.Limits.HasKernels C
• has_cokernels : CategoryTheory.Limits.HasCokernels C

theorem CategoryTheory.Abelian.has_finite_products {C : Type u} [CategoryTheory.Category.{v, u} C] [self : CategoryTheory.Abelian C] : CategoryTheory.Limits.HasFiniteProducts C

theorem CategoryTheory.Abelian.has_kernels {C : Type u} [CategoryTheory.Category.{v, u} C] [self : CategoryTheory.Abelian C] : CategoryTheory.Limits.HasKernels C

theorem CategoryTheory.Abelian.has_cokernels {C : Type u} [CategoryTheory.Category.{v, u} C] [self : CategoryTheory.Abelian C] : CategoryTheory.Limits.HasCokernels C

We begin by providing an alternative constructor: a preadditive category with kernels, cokernels, and finite products, in which the coimage-image comparison morphism is always an isomorphism, is an abelian category.

@[simp]

theorem CategoryTheory.Abelian.OfCoimageImageComparisonIsIso.imageMonoFactorisation_e {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.Limits.HasCokernels C] {X : C} {Y : C} (f : X ⟶ Y) : (CategoryTheory.Abelian.OfCoimageImageComparisonIsIso.imageMonoFactorisation f).e = CategoryTheory.Limits.kernel.lift (CategoryTheory.Limits.cokernel.π f) f ⋯

@[simp]

theorem CategoryTheory.Abelian.OfCoimageImageComparisonIsIso.imageMonoFactorisation_I {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.Limits.HasCokernels C] {X : C} {Y : C} (f : X ⟶ Y) : (CategoryTheory.Abelian.OfCoimageImageComparisonIsIso.imageMonoFactorisation f).I = CategoryTheory.Abelian.image f

@[simp]

theorem CategoryTheory.Abelian.OfCoimageImageComparisonIsIso.imageMonoFactorisation_m {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.Limits.HasCokernels C] {X : C} {Y : C} (f : X ⟶ Y) : (CategoryTheory.Abelian.OfCoimageImageComparisonIsIso.imageMonoFactorisation f).m = CategoryTheory.Limits.kernel.ι (CategoryTheory.Limits.cokernel.π f)

def CategoryTheory.Abelian.OfCoimageImageComparisonIsIso.imageMonoFactorisation {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.Limits.HasCokernels C] {X : C} {Y : C} (f : X ⟶ Y) : CategoryTheory.Limits.MonoFactorisation f

The factorisation of a morphism through its abelian image.

Equations

• One or more equations did not get rendered due to their size.

theorem CategoryTheory.Abelian.OfCoimageImageComparisonIsIso.imageMonoFactorisation_e' {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.Limits.HasCokernels C] {X : C} {Y : C} (f : X ⟶ Y) : (CategoryTheory.Abelian.OfCoimageImageComparisonIsIso.imageMonoFactorisation f).e = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.cokernel.π (CategoryTheory.Limits.kernel.ι f)) (CategoryTheory.Abelian.coimageImageComparison f)

def CategoryTheory.Abelian.OfCoimageImageComparisonIsIso.imageFactorisation {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.Limits.HasCokernels C] {X : C} {Y : C} (f : X ⟶ Y) [CategoryTheory.IsIso (CategoryTheory.Abelian.coimageImageComparison f)] : CategoryTheory.Limits.ImageFactorisation f

If the coimage-image comparison morphism for a morphism f is an isomorphism, we obtain an image factorisation of f.

Equations

• One or more equations did not get rendered due to their size.

instance CategoryTheory.Abelian.OfCoimageImageComparisonIsIso.instIsIsoEImageMonoFactorisationOfHasZeroObjectOfMonoOfCoimageImageComparison {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.Limits.HasCokernels C] [CategoryTheory.Limits.HasZeroObject C] {X : C} {Y : C} (f : X ⟶ Y) [CategoryTheory.Mono f] [CategoryTheory.IsIso (CategoryTheory.Abelian.coimageImageComparison f)] : CategoryTheory.IsIso (CategoryTheory.Abelian.OfCoimageImageComparisonIsIso.imageMonoFactorisation f).e

Equations

• ⋯ = ⋯

instance CategoryTheory.Abelian.OfCoimageImageComparisonIsIso.instIsIsoMImageMonoFactorisationOfHasZeroObjectOfEpi {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.Limits.HasCokernels C] [CategoryTheory.Limits.HasZeroObject C] {X : C} {Y : C} (f : X ⟶ Y) [CategoryTheory.Epi f] : CategoryTheory.IsIso (CategoryTheory.Abelian.OfCoimageImageComparisonIsIso.imageMonoFactorisation f).m

Equations

• ⋯ = ⋯

theorem CategoryTheory.Abelian.OfCoimageImageComparisonIsIso.hasImages {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.Limits.HasCokernels C] [∀ {X Y : C} (f : X ⟶ Y), CategoryTheory.IsIso (CategoryTheory.Abelian.coimageImageComparison f)] : CategoryTheory.Limits.HasImages C

A category in which coimage-image comparisons are all isomorphisms has images.

def CategoryTheory.Abelian.OfCoimageImageComparisonIsIso.normalMonoCategory {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.Limits.HasCokernels C] [∀ {X Y : C} (f : X ⟶ Y), CategoryTheory.IsIso (CategoryTheory.Abelian.coimageImageComparison f)] [CategoryTheory.Limits.HasFiniteProducts C] : CategoryTheory.NormalMonoCategory C

A category with finite products in which coimage-image comparisons are all isomorphisms is a normal mono category.

Equations

• One or more equations did not get rendered due to their size.

def CategoryTheory.Abelian.OfCoimageImageComparisonIsIso.normalEpiCategory {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.Limits.HasCokernels C] [∀ {X Y : C} (f : X ⟶ Y), CategoryTheory.IsIso (CategoryTheory.Abelian.coimageImageComparison f)] [CategoryTheory.Limits.HasFiniteProducts C] : CategoryTheory.NormalEpiCategory C

A category with finite products in which coimage-image comparisons are all isomorphisms is a normal epi category.

Equations

• One or more equations did not get rendered due to their size.

def CategoryTheory.Abelian.ofCoimageImageComparisonIsIso {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasKernels C] [CategoryTheory.Limits.HasCokernels C] [∀ {X Y : C} (f : X ⟶ Y), CategoryTheory.IsIso (CategoryTheory.Abelian.coimageImageComparison f)] [CategoryTheory.Limits.HasFiniteProducts C] : CategoryTheory.Abelian C

A preadditive category with kernels, cokernels, and finite products, in which the coimage-image comparison morphism is always an isomorphism, is an abelian category.

The Stacks project uses this characterisation at the definition of an abelian category. See https://stacks.math.columbia.edu/tag/0109.

Equations

• CategoryTheory.Abelian.ofCoimageImageComparisonIsIso = CategoryTheory.Abelian.mk

theorem CategoryTheory.Abelian.hasFiniteBiproducts {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] : CategoryTheory.Limits.HasFiniteBiproducts C

An abelian category has finite biproducts.

@[instance 100]

instance CategoryTheory.Abelian.hasBinaryBiproducts {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] : CategoryTheory.Limits.HasBinaryBiproducts C

Equations

• ⋯ = ⋯

@[instance 100]

instance CategoryTheory.Abelian.hasZeroObject {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] : CategoryTheory.Limits.HasZeroObject C

Equations

• ⋯ = ⋯

def CategoryTheory.Abelian.nonPreadditiveAbelian {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] : CategoryTheory.NonPreadditiveAbelian C

Every abelian category is, in particular, NonPreadditiveAbelian.

Equations

• CategoryTheory.Abelian.nonPreadditiveAbelian = let __src := inst; CategoryTheory.NonPreadditiveAbelian.mk

We now promote some instances that were constructed using non_preadditive_abelian.

instance CategoryTheory.Abelian.instEpiFactorThruImage {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {P : C} {Q : C} (f : P ⟶ Q) : CategoryTheory.Epi (CategoryTheory.Abelian.factorThruImage f)

The map p : P ⟶ image f is an epimorphism

Equations

• ⋯ = ⋯

instance CategoryTheory.Abelian.isIso_factorThruImage {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {P : C} {Q : C} (f : P ⟶ Q) [CategoryTheory.Mono f] : CategoryTheory.IsIso (CategoryTheory.Abelian.factorThruImage f)

Equations

• ⋯ = ⋯

instance CategoryTheory.Abelian.instMonoFactorThruCoimage {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {P : C} {Q : C} (f : P ⟶ Q) : CategoryTheory.Mono (CategoryTheory.Abelian.factorThruCoimage f)

The canonical morphism i : coimage f ⟶ Q is a monomorphism

Equations

• ⋯ = ⋯

instance CategoryTheory.Abelian.isIso_factorThruCoimage {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {P : C} {Q : C} (f : P ⟶ Q) [CategoryTheory.Epi f] : CategoryTheory.IsIso (CategoryTheory.Abelian.factorThruCoimage f)

Equations

• ⋯ = ⋯

theorem CategoryTheory.Abelian.mono_of_kernel_ι_eq_zero {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {P : C} {Q : C} (f : P ⟶ Q) (h : CategoryTheory.Limits.kernel.ι f = 0) : CategoryTheory.Mono f

theorem CategoryTheory.Abelian.epi_of_cokernel_π_eq_zero {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {P : C} {Q : C} (f : P ⟶ Q) (h : CategoryTheory.Limits.cokernel.π f = 0) : CategoryTheory.Epi f

theorem CategoryTheory.Abelian.image_ι_comp_eq_zero {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {P : C} {Q : C} {f : P ⟶ Q} {R : C} {g : Q ⟶ R} (h : CategoryTheory.CategoryStruct.comp f g = 0) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Abelian.image.ι f) g = 0

theorem CategoryTheory.Abelian.comp_coimage_π_eq_zero {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {P : C} {Q : C} {f : P ⟶ Q} {R : C} {g : Q ⟶ R} (h : CategoryTheory.CategoryStruct.comp f g = 0) : CategoryTheory.CategoryStruct.comp f (CategoryTheory.Abelian.coimage.π g) = 0

@[simp]

theorem CategoryTheory.Abelian.imageStrongEpiMonoFactorisation_m {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {P : C} {Q : C} (f : P ⟶ Q) : (CategoryTheory.Abelian.imageStrongEpiMonoFactorisation f).m = CategoryTheory.Abelian.image.ι f

@[simp]

theorem CategoryTheory.Abelian.imageStrongEpiMonoFactorisation_e {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {P : C} {Q : C} (f : P ⟶ Q) : (CategoryTheory.Abelian.imageStrongEpiMonoFactorisation f).e = CategoryTheory.Abelian.factorThruImage f

@[simp]

theorem CategoryTheory.Abelian.imageStrongEpiMonoFactorisation_I {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {P : C} {Q : C} (f : P ⟶ Q) : (CategoryTheory.Abelian.imageStrongEpiMonoFactorisation f).I = CategoryTheory.Abelian.image f

def CategoryTheory.Abelian.imageStrongEpiMonoFactorisation {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {P : C} {Q : C} (f : P ⟶ Q) : CategoryTheory.Limits.StrongEpiMonoFactorisation f

Factoring through the image is a strong epi-mono factorisation.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem CategoryTheory.Abelian.coimageStrongEpiMonoFactorisation_e {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {P : C} {Q : C} (f : P ⟶ Q) : (CategoryTheory.Abelian.coimageStrongEpiMonoFactorisation f).e = CategoryTheory.Abelian.coimage.π f

@[simp]

theorem CategoryTheory.Abelian.coimageStrongEpiMonoFactorisation_m {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {P : C} {Q : C} (f : P ⟶ Q) : (CategoryTheory.Abelian.coimageStrongEpiMonoFactorisation f).m = CategoryTheory.Abelian.factorThruCoimage f

@[simp]

theorem CategoryTheory.Abelian.coimageStrongEpiMonoFactorisation_I {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {P : C} {Q : C} (f : P ⟶ Q) : (CategoryTheory.Abelian.coimageStrongEpiMonoFactorisation f).I = CategoryTheory.Abelian.coimage f

def CategoryTheory.Abelian.coimageStrongEpiMonoFactorisation {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {P : C} {Q : C} (f : P ⟶ Q) : CategoryTheory.Limits.StrongEpiMonoFactorisation f

Factoring through the coimage is a strong epi-mono factorisation.

Equations

• One or more equations did not get rendered due to their size.

@[instance 100]

instance CategoryTheory.Abelian.instHasStrongEpiMonoFactorisations {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] : CategoryTheory.Limits.HasStrongEpiMonoFactorisations C

An abelian category has strong epi-mono factorisations.

Equations

• ⋯ = ⋯

instance CategoryTheory.Abelian.instIsIsoCoimageImageComparison {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {X : C} {Y : C} (f : X ⟶ Y) : CategoryTheory.IsIso (CategoryTheory.Abelian.coimageImageComparison f)

The coimage-image comparison morphism is always an isomorphism in an abelian category. See CategoryTheory.Abelian.ofCoimageImageComparisonIsIso for the converse.

Equations

• ⋯ = ⋯

@[reducible, inline]

abbrev CategoryTheory.Abelian.coimageIsoImage {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {X : C} {Y : C} (f : X ⟶ Y) : CategoryTheory.Abelian.coimage f ≅ CategoryTheory.Abelian.image f

There is a canonical isomorphism between the abelian coimage and the abelian image of a morphism.

Equations

• CategoryTheory.Abelian.coimageIsoImage f = CategoryTheory.asIso (CategoryTheory.Abelian.coimageImageComparison f)

@[reducible, inline]

abbrev CategoryTheory.Abelian.coimageIsoImage' {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {X : C} {Y : C} (f : X ⟶ Y) : CategoryTheory.Abelian.coimage f ≅ CategoryTheory.Limits.image f

There is a canonical isomorphism between the abelian coimage and the categorical image of a morphism.

Equations

• CategoryTheory.Abelian.coimageIsoImage' f = (CategoryTheory.Abelian.coimageStrongEpiMonoFactorisation f).toMonoIsImage.isoExt (CategoryTheory.Limits.Image.isImage f)

theorem CategoryTheory.Abelian.coimageIsoImage'_hom {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {X : C} {Y : C} (f : X ⟶ Y) : (CategoryTheory.Abelian.coimageIsoImage' f).hom = CategoryTheory.Limits.cokernel.desc (CategoryTheory.Limits.kernel.ι f) (CategoryTheory.Limits.factorThruImage f) ⋯

theorem CategoryTheory.Abelian.factorThruImage_comp_coimageIsoImage'_inv {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {X : C} {Y : C} (f : X ⟶ Y) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.factorThruImage f) (CategoryTheory.Abelian.coimageIsoImage' f).inv = CategoryTheory.Limits.cokernel.π (CategoryTheory.Limits.kernel.ι f)

@[reducible, inline]

abbrev CategoryTheory.Abelian.imageIsoImage {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {X : C} {Y : C} (f : X ⟶ Y) : CategoryTheory.Abelian.image f ≅ CategoryTheory.Limits.image f

There is a canonical isomorphism between the abelian image and the categorical image of a morphism.

Equations

• CategoryTheory.Abelian.imageIsoImage f = (CategoryTheory.Abelian.imageStrongEpiMonoFactorisation f).toMonoIsImage.isoExt (CategoryTheory.Limits.Image.isImage f)

theorem CategoryTheory.Abelian.imageIsoImage_hom_comp_image_ι {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {X : C} {Y : C} (f : X ⟶ Y) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Abelian.imageIsoImage f).hom (CategoryTheory.Limits.image.ι f) = CategoryTheory.Limits.kernel.ι (CategoryTheory.Limits.cokernel.π f)

theorem CategoryTheory.Abelian.imageIsoImage_inv {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {X : C} {Y : C} (f : X ⟶ Y) : (CategoryTheory.Abelian.imageIsoImage f).inv = CategoryTheory.Limits.kernel.lift (CategoryTheory.Limits.cokernel.π f) (CategoryTheory.Limits.image.ι f) ⋯

def CategoryTheory.Abelian.epiIsCokernelOfKernel {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {X : C} {Y : C} {f : X ⟶ Y} [CategoryTheory.Epi f] (s : CategoryTheory.Limits.Fork f 0) (h : CategoryTheory.Limits.IsLimit s) : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.CokernelCofork.ofπ f ⋯)

In an abelian category, an epi is the cokernel of its kernel. More precisely: If f is an epimorphism and s is some limit kernel cone on f, then f is a cokernel of fork.ι s.

Equations

• CategoryTheory.Abelian.epiIsCokernelOfKernel s h = CategoryTheory.NonPreadditiveAbelian.epiIsCokernelOfKernel s h

def CategoryTheory.Abelian.monoIsKernelOfCokernel {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {X : C} {Y : C} {f : X ⟶ Y} [CategoryTheory.Mono f] (s : CategoryTheory.Limits.Cofork f 0) (h : CategoryTheory.Limits.IsColimit s) : CategoryTheory.Limits.IsLimit (CategoryTheory.Limits.KernelFork.ofι f ⋯)

In an abelian category, a mono is the kernel of its cokernel. More precisely: If f is a monomorphism and s is some colimit cokernel cocone on f, then f is a kernel of cofork.π s.

Equations

• CategoryTheory.Abelian.monoIsKernelOfCokernel s h = CategoryTheory.NonPreadditiveAbelian.monoIsKernelOfCokernel s h

def CategoryTheory.Abelian.epiDesc {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {X : C} {Y : C} (f : X ⟶ Y) [CategoryTheory.Epi f] {T : C} (g : X ⟶ T) (hg : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.kernel.ι f) g = 0) : Y ⟶ T

In an abelian category, any morphism that turns to zero when precomposed with the kernel of an epimorphism factors through that epimorphism.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem CategoryTheory.Abelian.comp_epiDesc_assoc {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {X : C} {Y : C} (f : X ⟶ Y) [CategoryTheory.Epi f] {T : C} (g : X ⟶ T) (hg : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.kernel.ι f) g = 0) {Z : C} (h : T ⟶ Z) : CategoryTheory.CategoryStruct.comp f (CategoryTheory.CategoryStruct.comp (CategoryTheory.Abelian.epiDesc f g hg) h) = CategoryTheory.CategoryStruct.comp g h

@[simp]

theorem CategoryTheory.Abelian.comp_epiDesc {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {X : C} {Y : C} (f : X ⟶ Y) [CategoryTheory.Epi f] {T : C} (g : X ⟶ T) (hg : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.kernel.ι f) g = 0) : CategoryTheory.CategoryStruct.comp f (CategoryTheory.Abelian.epiDesc f g hg) = g

def CategoryTheory.Abelian.monoLift {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {X : C} {Y : C} (f : X ⟶ Y) [CategoryTheory.Mono f] {T : C} (g : T ⟶ Y) (hg : CategoryTheory.CategoryStruct.comp g (CategoryTheory.Limits.cokernel.π f) = 0) : T ⟶ X

In an abelian category, any morphism that turns to zero when postcomposed with the cokernel of a monomorphism factors through that monomorphism.

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@[simp]

theorem CategoryTheory.Abelian.monoLift_comp_assoc {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {X : C} {Y : C} (f : X ⟶ Y) [CategoryTheory.Mono f] {T : C} (g : T ⟶ Y) (hg : CategoryTheory.CategoryStruct.comp g (CategoryTheory.Limits.cokernel.π f) = 0) {Z : C} (h : Y ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Abelian.monoLift f g hg) (CategoryTheory.CategoryStruct.comp f h) = CategoryTheory.CategoryStruct.comp g h

@[simp]

theorem CategoryTheory.Abelian.monoLift_comp {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {X : C} {Y : C} (f : X ⟶ Y) [CategoryTheory.Mono f] {T : C} (g : T ⟶ Y) (hg : CategoryTheory.CategoryStruct.comp g (CategoryTheory.Limits.cokernel.π f) = 0) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Abelian.monoLift f g hg) f = g

noncomputable def CategoryTheory.Abelian.isLimitMapConeOfKernelForkOfι {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {D : Type u_1} [CategoryTheory.Category.{u_2, u_1} D] [CategoryTheory.Limits.HasZeroMorphisms D] {X : D} {Y : D} (i : X ⟶ Y) [CategoryTheory.Limits.HasCokernel i] (F : CategoryTheory.Functor D C) [F.PreservesZeroMorphisms] [CategoryTheory.Mono (F.map i)] [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.parallelPair i 0) F] : CategoryTheory.Limits.IsLimit (F.mapCone (CategoryTheory.Limits.KernelFork.ofι i ⋯))

If F : D ⥤ C is a functor to an abelian category, i : X ⟶ Y is a morphism admitting a cokernel such that F preserves this cokernel and F.map i is a mono, then F.map X identifies to the kernel of F.map (cokernel.π i).

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noncomputable def CategoryTheory.Abelian.isColimitMapCoconeOfCokernelCoforkOfπ {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {D : Type u_1} [CategoryTheory.Category.{u_2, u_1} D] [CategoryTheory.Limits.HasZeroMorphisms D] {X : D} {Y : D} (p : X ⟶ Y) [CategoryTheory.Limits.HasKernel p] (F : CategoryTheory.Functor D C) [F.PreservesZeroMorphisms] [CategoryTheory.Epi (F.map p)] [CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.parallelPair p 0) F] : CategoryTheory.Limits.IsColimit (F.mapCocone (CategoryTheory.Limits.CokernelCofork.ofπ p ⋯))

If F : D ⥤ C is a functor to an abelian category, p : X ⟶ Y is a morphisms admitting a kernel such that F preserves this kernel and F.map p is an epi, then F.map Y identifies to the cokernel of F.map (kernel.ι p).

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@[instance 100]

instance CategoryTheory.Abelian.hasEqualizers {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] : CategoryTheory.Limits.HasEqualizers C

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@[instance 100]

instance CategoryTheory.Abelian.hasPullbacks {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] : CategoryTheory.Limits.HasPullbacks C

Any abelian category has pullbacks

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@[instance 100]

instance CategoryTheory.Abelian.hasCoequalizers {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] : CategoryTheory.Limits.HasCoequalizers C

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@[instance 100]

instance CategoryTheory.Abelian.hasPushouts {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] : CategoryTheory.Limits.HasPushouts C

Any abelian category has pushouts

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@[instance 100]

instance CategoryTheory.Abelian.hasFiniteLimits {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] : CategoryTheory.Limits.HasFiniteLimits C

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@[instance 100]

instance CategoryTheory.Abelian.hasFiniteColimits {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] : CategoryTheory.Limits.HasFiniteColimits C

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• ⋯ = ⋯

This section contains a slightly technical result about pullbacks and biproducts. We will need it in the proof that the pullback of an epimorphism is an epimorphism.

@[reducible, inline]

abbrev CategoryTheory.Abelian.PullbackToBiproductIsKernel.pullbackToBiproduct {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.Limits.HasPullbacks C] {X : C} {Y : C} {Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) : CategoryTheory.Limits.pullback f g ⟶ X ⊞ Y

The canonical map pullback f g ⟶ X ⊞ Y

Equations

• CategoryTheory.Abelian.PullbackToBiproductIsKernel.pullbackToBiproduct f g = CategoryTheory.Limits.biprod.lift (CategoryTheory.Limits.pullback.fst f g) (CategoryTheory.Limits.pullback.snd f g)

@[reducible, inline]

abbrev CategoryTheory.Abelian.PullbackToBiproductIsKernel.pullbackToBiproductFork {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.Limits.HasPullbacks C] {X : C} {Y : C} {Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) : CategoryTheory.Limits.KernelFork (CategoryTheory.Limits.biprod.desc f (-g))

The canonical map pullback f g ⟶ X ⊞ Y induces a kernel cone on the map biproduct X Y ⟶ Z induced by f and g. A slightly more intuitive way to think of this may be that it induces an equalizer fork on the maps induced by (f, 0) and (0, g).

Equations

• CategoryTheory.Abelian.PullbackToBiproductIsKernel.pullbackToBiproductFork f g = CategoryTheory.Limits.KernelFork.ofι (CategoryTheory.Abelian.PullbackToBiproductIsKernel.pullbackToBiproduct f g) ⋯

def CategoryTheory.Abelian.PullbackToBiproductIsKernel.isLimitPullbackToBiproduct {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.Limits.HasPullbacks C] {X : C} {Y : C} {Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) : CategoryTheory.Limits.IsLimit (CategoryTheory.Abelian.PullbackToBiproductIsKernel.pullbackToBiproductFork f g)

The canonical map pullback f g ⟶ X ⊞ Y is a kernel of the map induced by (f, -g).

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@[reducible, inline]

abbrev CategoryTheory.Abelian.BiproductToPushoutIsCokernel.biproductToPushout {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.Limits.HasPushouts C] {X : C} {Y : C} {Z : C} (f : X ⟶ Y) (g : X ⟶ Z) : Y ⊞ Z ⟶ CategoryTheory.Limits.pushout f g

The canonical map Y ⊞ Z ⟶ pushout f g

Equations

• CategoryTheory.Abelian.BiproductToPushoutIsCokernel.biproductToPushout f g = CategoryTheory.Limits.biprod.desc (CategoryTheory.Limits.pushout.inl f g) (CategoryTheory.Limits.pushout.inr f g)

@[reducible, inline]

abbrev CategoryTheory.Abelian.BiproductToPushoutIsCokernel.biproductToPushoutCofork {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.Limits.HasPushouts C] {X : C} {Y : C} {Z : C} (f : X ⟶ Y) (g : X ⟶ Z) : CategoryTheory.Limits.CokernelCofork (CategoryTheory.Limits.biprod.lift f (-g))

The canonical map Y ⊞ Z ⟶ pushout f g induces a cokernel cofork on the map X ⟶ Y ⊞ Z induced by f and -g.

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• One or more equations did not get rendered due to their size.

def CategoryTheory.Abelian.BiproductToPushoutIsCokernel.isColimitBiproductToPushout {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.Limits.HasPushouts C] {X : C} {Y : C} {Z : C} (f : X ⟶ Y) (g : X ⟶ Z) : CategoryTheory.Limits.IsColimit (CategoryTheory.Abelian.BiproductToPushoutIsCokernel.biproductToPushoutCofork f g)

The cofork induced by the canonical map Y ⊞ Z ⟶ pushout f g is in fact a colimit cokernel cofork.

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instance CategoryTheory.Abelian.epi_pullback_of_epi_f {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.Limits.HasPullbacks C] {X : C} {Y : C} {Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) [CategoryTheory.Epi f] : CategoryTheory.Epi (CategoryTheory.Limits.pullback.snd f g)

In an abelian category, the pullback of an epimorphism is an epimorphism. Proof from [aluffi2016, IX.2.3], cf. [borceux-vol2, 1.7.6]

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• ⋯ = ⋯

instance CategoryTheory.Abelian.epi_pullback_of_epi_g {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.Limits.HasPullbacks C] {X : C} {Y : C} {Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) [CategoryTheory.Epi g] : CategoryTheory.Epi (CategoryTheory.Limits.pullback.fst f g)

In an abelian category, the pullback of an epimorphism is an epimorphism.

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• ⋯ = ⋯

theorem CategoryTheory.Abelian.epi_snd_of_isLimit {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.Limits.HasPullbacks C] {X : C} {Y : C} {Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) [CategoryTheory.Epi f] {s : CategoryTheory.Limits.PullbackCone f g} (hs : CategoryTheory.Limits.IsLimit s) : CategoryTheory.Epi s.snd

theorem CategoryTheory.Abelian.epi_fst_of_isLimit {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.Limits.HasPullbacks C] {X : C} {Y : C} {Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) [CategoryTheory.Epi g] {s : CategoryTheory.Limits.PullbackCone f g} (hs : CategoryTheory.Limits.IsLimit s) : CategoryTheory.Epi s.fst

theorem CategoryTheory.Abelian.epi_fst_of_factor_thru_epi_mono_factorization {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.Limits.HasPullbacks C] {W : C} {X : C} {Y : C} {Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) (g₁ : Y ⟶ W) [CategoryTheory.Epi g₁] (g₂ : W ⟶ Z) [CategoryTheory.Mono g₂] (hg : CategoryTheory.CategoryStruct.comp g₁ g₂ = g) (f' : X ⟶ W) (hf : CategoryTheory.CategoryStruct.comp f' g₂ = f) (t : CategoryTheory.Limits.PullbackCone f g) (ht : CategoryTheory.Limits.IsLimit t) : CategoryTheory.Epi t.fst

Suppose f and g are two morphisms with a common codomain and suppose we have written g as an epimorphism followed by a monomorphism. If f factors through the mono part of this factorization, then any pullback of g along f is an epimorphism.

instance CategoryTheory.Abelian.mono_pushout_of_mono_f {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.Limits.HasPushouts C] {X : C} {Y : C} {Z : C} (f : X ⟶ Y) (g : X ⟶ Z) [CategoryTheory.Mono f] : CategoryTheory.Mono (CategoryTheory.Limits.pushout.inr f g)

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• ⋯ = ⋯

instance CategoryTheory.Abelian.mono_pushout_of_mono_g {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.Limits.HasPushouts C] {X : C} {Y : C} {Z : C} (f : X ⟶ Y) (g : X ⟶ Z) [CategoryTheory.Mono g] : CategoryTheory.Mono (CategoryTheory.Limits.pushout.inl f g)

Equations

• ⋯ = ⋯

theorem CategoryTheory.Abelian.mono_inr_of_isColimit {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.Limits.HasPushouts C] {X : C} {Y : C} {Z : C} (f : X ⟶ Y) (g : X ⟶ Z) [CategoryTheory.Mono f] {s : CategoryTheory.Limits.PushoutCocone f g} (hs : CategoryTheory.Limits.IsColimit s) : CategoryTheory.Mono s.inr

theorem CategoryTheory.Abelian.mono_inl_of_isColimit {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.Limits.HasPushouts C] {X : C} {Y : C} {Z : C} (f : X ⟶ Y) (g : X ⟶ Z) [CategoryTheory.Mono g] {s : CategoryTheory.Limits.PushoutCocone f g} (hs : CategoryTheory.Limits.IsColimit s) : CategoryTheory.Mono s.inl

theorem CategoryTheory.Abelian.mono_inl_of_factor_thru_epi_mono_factorization {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.Limits.HasPushouts C] {W : C} {X : C} {Y : C} {Z : C} (f : X ⟶ Y) (g : X ⟶ Z) (g₁ : X ⟶ W) [CategoryTheory.Epi g₁] (g₂ : W ⟶ Z) [CategoryTheory.Mono g₂] (hg : CategoryTheory.CategoryStruct.comp g₁ g₂ = g) (f' : W ⟶ Y) (hf : CategoryTheory.CategoryStruct.comp g₁ f' = f) (t : CategoryTheory.Limits.PushoutCocone f g) (ht : CategoryTheory.Limits.IsColimit t) : CategoryTheory.Mono t.inl

Suppose f and g are two morphisms with a common domain and suppose we have written g as an epimorphism followed by a monomorphism. If f factors through the epi part of this factorization, then any pushout of g along f is a monomorphism.

def CategoryTheory.NonPreadditiveAbelian.abelian (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.NonPreadditiveAbelian C] : CategoryTheory.Abelian C

Every NonPreadditiveAbelian category can be promoted to an abelian category.

Equations

• CategoryTheory.NonPreadditiveAbelian.abelian C = let __src := CategoryTheory.NonPreadditiveAbelian.preadditive; CategoryTheory.Abelian.mk