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 finite products, kernels, and cokernels, and if every monomorphism and epimorphism is normal.

It should be noted that if we also assume finite coproducts, then preadditivity is actually a consequence of the other properties, as we show in Mathlib/CategoryTheory/Abelian/NonPreadditive.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 Mathlib/CategoryTheory/Abelian/NonPreadditive.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) [Category.{v, u} C] extends CategoryTheory.Preadditive C, CategoryTheory.IsNormalMonoCategory C, CategoryTheory.IsNormalEpiCategory C : 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) : CategoryStruct.comp (f + f') g = CategoryStruct.comp f g + CategoryStruct.comp f' g
• comp_add (P Q R : C) (f : P ⟶ Q) (g g' : Q ⟶ R) : CategoryStruct.comp f (g + g') = CategoryStruct.comp f g + CategoryStruct.comp f g'
• normalMonoOfMono {X Y : C} (f : X ⟶ Y) [Mono f] : Nonempty (NormalMono f)
• normalEpiOfEpi {X Y : C} (f : X ⟶ Y) [Epi f] : Nonempty (NormalEpi f)
• has_finite_products : Limits.HasFiniteProducts C
• has_kernels : Limits.HasKernels C
• has_cokernels : 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.

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

The factorisation of a morphism through its abelian image.

@[simp]

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

@[simp]

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

@[simp]

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

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

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

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

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

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

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

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

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

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

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

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

@[implicit_reducible]

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

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

An abelian category has finite biproducts.

@[instance 100]

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

@[instance 100]

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

@[implicit_reducible]

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

Every abelian category is, in particular, NonPreadditiveAbelian.

We now promote some instances that were constructed using nonPreadditiveAbelian.

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

The map p : P ⟶ image f is an epimorphism

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

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

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

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

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

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

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

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

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

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

@[simp]

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

@[simp]

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

@[simp]

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

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

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

@[simp]

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

@[simp]

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

@[simp]

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

@[instance 100]

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

An abelian category has strong epi-mono factorisations.

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

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

@[reducible, inline]

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

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

@[reducible, inline]

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

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

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

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

@[simp]

theorem CategoryTheory.Abelian.image.ι_comp_eq_zero {C : Type u} [Category.{v, u} C] [Abelian C] {X Y : C} (f : X ⟶ Y) {Z : C} (g : Y ⟶ Z) : CategoryStruct.comp (image.ι f) g = 0 ↔ CategoryStruct.comp f g = 0

@[simp]

theorem CategoryTheory.Abelian.coimage.comp_π_eq_zero {C : Type u} [Category.{v, u} C] [Abelian C] {X Y : C} (f : X ⟶ Y) {Z : C} (g : Y ⟶ Z) : CategoryStruct.comp f (coimage.π g) = 0 ↔ CategoryStruct.comp f g = 0

noncomputable def CategoryTheory.Abelian.im {C : Type u} [Category.{v, u} C] [Abelian C] : Functor (Arrow C) C

Abelian.image as a functor from the arrow category.

@[simp]

theorem CategoryTheory.Abelian.im_map {C : Type u} [Category.{v, u} C] [Abelian C] {f g : Arrow C} (u : f ⟶ g) : im.map u = Limits.kernel.lift (Limits.cokernel.π g.hom) (CategoryStruct.comp (image.ι f.hom) u.right) ⋯

@[simp]

theorem CategoryTheory.Abelian.im_obj {C : Type u} [Category.{v, u} C] [Abelian C] (f : Arrow C) : im.obj f = Abelian.image f.hom

@[deprecated CategoryTheory.Abelian.im (since := "2025-10-31")]

def CategoryTheory.Abelian.imageFunctor {C : Type u} [Category.{v, u} C] [Abelian C] : Functor (Arrow C) C

Alias of CategoryTheory.Abelian.im.

---

Abelian.image as a functor from the arrow category.

noncomputable def CategoryTheory.Abelian.coim {C : Type u} [Category.{v, u} C] [Abelian C] : Functor (Arrow C) C

Abelian.coimage as a functor from the arrow category.

@[simp]

theorem CategoryTheory.Abelian.coim_obj {C : Type u} [Category.{v, u} C] [Abelian C] (f : Arrow C) : coim.obj f = Abelian.coimage f.hom

@[simp]

theorem CategoryTheory.Abelian.coim_map {C : Type u} [Category.{v, u} C] [Abelian C] {f g : Arrow C} (u : f ⟶ g) : coim.map u = Limits.cokernel.desc (Limits.kernel.ι f.hom) (CategoryStruct.comp u.left (coimage.π g.hom)) ⋯

@[deprecated CategoryTheory.Abelian.coim (since := "2025-10-31")]

def CategoryTheory.Abelian.coimageFunctor {C : Type u} [Category.{v, u} C] [Abelian C] : Functor (Arrow C) C

Alias of CategoryTheory.Abelian.coim.

---

Abelian.coimage as a functor from the arrow category.

noncomputable def CategoryTheory.Abelian.coimIsoIm {C : Type u} [Category.{v, u} C] [Abelian C] : coim ≅ im

The image and coimage of an arrow are naturally isomorphic.

@[simp]

theorem CategoryTheory.Abelian.coimIsoIm_inv_app {C : Type u} [Category.{v, u} C] [Abelian C] (X : Arrow C) : coimIsoIm.inv.app X = inv (coimageImageComparison X.hom)

@[simp]

theorem CategoryTheory.Abelian.coimIsoIm_hom_app {C : Type u} [Category.{v, u} C] [Abelian C] (X : Arrow C) : coimIsoIm.hom.app X = coimageImageComparison X.hom

@[deprecated CategoryTheory.Abelian.coimIsoIm (since := "2025-10-31")]

def CategoryTheory.Abelian.coimageFunctorIsoImageFunctor {C : Type u} [Category.{v, u} C] [Abelian C] : coim ≅ im

Alias of CategoryTheory.Abelian.coimIsoIm.

---

The image and coimage of an arrow are naturally isomorphic.

@[reducible, inline]

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

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

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

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

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

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

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

@[simp]

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

@[simp]

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

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

@[simp]

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

@[simp]

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

noncomputable def CategoryTheory.Abelian.isLimitMapConeOfKernelForkOfι {C : Type u} [Category.{v, u} C] [Abelian C] {D : Type u_1} [Category.{v_1, u_1} D] [Limits.HasZeroMorphisms D] {X Y : D} (i : X ⟶ Y) [Limits.HasCokernel i] (F : Functor D C) [F.PreservesZeroMorphisms] [Mono (F.map i)] [Limits.PreservesColimit (Limits.parallelPair i 0) F] : Limits.IsLimit (F.mapCone (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).

noncomputable def CategoryTheory.Abelian.isColimitMapCoconeOfCokernelCoforkOfπ {C : Type u} [Category.{v, u} C] [Abelian C] {D : Type u_1} [Category.{v_1, u_1} D] [Limits.HasZeroMorphisms D] {X Y : D} (p : X ⟶ Y) [Limits.HasKernel p] (F : Functor D C) [F.PreservesZeroMorphisms] [Epi (F.map p)] [Limits.PreservesLimit (Limits.parallelPair p 0) F] : Limits.IsColimit (F.mapCocone (Limits.CokernelCofork.ofπ p ⋯))

If F : D ⥤ C is a functor to an abelian category, p : X ⟶ Y is a morphism 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).

@[instance 100]

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

@[instance 100]

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

Any abelian category has pullbacks

@[instance 100]

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

@[instance 100]

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

Any abelian category has pushouts

@[instance 100]

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

@[instance 100]

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

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]

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

The canonical map pullback f g ⟶ X ⊞ Y

@[reducible, inline]

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

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

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

@[reducible, inline]

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

The canonical map Y ⊞ Z ⟶ pushout f g

@[reducible, inline]

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

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

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

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

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

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

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

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

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

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

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

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

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

@[implicit_reducible]

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

Every NonPreadditiveAbelian category can be promoted to an abelian category.

structure CategoryTheory.Abelian.AbelianStruct {C : Type u_1} [Category.{u_2, u_1} C] [Preadditive C] {X Y : C} (f : X ⟶ Y) : Type (max u_1 u_2)

A preadditive category C with finite products is abelian when this structure is nonempty for any morphism f in C, see Abelian.mk'.

• kernelFork : Limits.KernelFork f

  a limit kernel fork of f

• isLimitKernelFork : Limits.IsLimit self.kernelFork

  the kernel fork is a limit

• cokernelCofork : Limits.CokernelCofork f

  a colimit cokernel cofork of f

• isColimitCokernelCofork : Limits.IsColimit self.cokernelCofork

  the cokernel cofork is a a limit

• image : C

  the image of f

• imageπ : X ⟶ self.image

  the projection to the image

• ι_imageπ : CategoryStruct.comp (Limits.Fork.ι self.kernelFork) self.imageπ = 0
• imageIsCokernel : Limits.IsColimit (Limits.CokernelCofork.ofπ self.imageπ ⋯)

  the image is a cokernel

• imageι : self.image ⟶ Y

  the inclusion of the image

• imageι_π : CategoryStruct.comp self.imageι (Limits.Cofork.π self.cokernelCofork) = 0
• imageIsKernel : Limits.IsLimit (Limits.KernelFork.ofι self.imageι ⋯)

  the image is a kernel

• fac : CategoryStruct.comp self.imageπ self.imageι = f

@[simp]

theorem CategoryTheory.Abelian.AbelianStruct.fac_assoc {C : Type u_1} [Category.{u_2, u_1} C] [Preadditive C] {X Y : C} {f : X ⟶ Y} (self : AbelianStruct f) {Z : C} (h : Y ⟶ Z) : CategoryStruct.comp self.imageπ (CategoryStruct.comp self.imageι h) = CategoryStruct.comp f h

@[simp]

theorem CategoryTheory.Abelian.AbelianStruct.ι_imageπ_assoc {C : Type u_1} [Category.{u_2, u_1} C] [Preadditive C] {X Y : C} {f : X ⟶ Y} (self : AbelianStruct f) {Z : C} (h : self.image ⟶ Z) : CategoryStruct.comp (Limits.Fork.ι self.kernelFork) (CategoryStruct.comp self.imageπ h) = CategoryStruct.comp 0 h

@[simp]

theorem CategoryTheory.Abelian.AbelianStruct.imageι_π_assoc {C : Type u_1} [Category.{u_2, u_1} C] [Preadditive C] {X Y : C} {f : X ⟶ Y} (self : AbelianStruct f) {Z : C} (h : self.cokernelCofork.pt ⟶ Z) : CategoryStruct.comp self.imageι (CategoryStruct.comp (Limits.Cofork.π self.cokernelCofork) h) = CategoryStruct.comp 0 h

@[implicit_reducible]

noncomputable def CategoryTheory.Abelian.mk' {C : Type u_1} [Category.{u_2, u_1} C] [Preadditive C] [Limits.HasFiniteProducts C] (h : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), Nonempty (AbelianStruct f)) : Abelian C

Constructor for abelian categories. We assume that the category C is preadditive, has finite products, and that any morphism f : X ⟶ Y has a kernel i : K ⟶ X, a cokernel p : Y ⟶ Q such that f factors as f = π ≫ ι where π : X ⟶ I is a cokernel of i and ι : I ⟶ Y is a kernel of p. This assumption is packaged in a structure AbelianStruct f.