mathlib documentation

category_theory.abelian.non_preadditive

Every non_preadditive_abelian category is preadditive

In mathlib, we define an abelian category as a preadditive category with a zero object, kernels and cokernels, products and coproducts and in which every monomorphism and epimorphis is normal.

While virtually every interesting abelian category has a natural preadditive structure (which is why it is included in the definition), preadditivity is not actually needed: Every category that has all of the other properties appearing in the definition of an abelian category admits a preadditive structure. This is the construction we carry out in this file.

The proof proceeds in roughly five steps:

  1. Prove some results (for example that all equalizers exist) that would be trivial if we already had the preadditive structure but are a bit of work without it.
  2. Develop images and coimages to show that every monomorphism is the kernel of its cokernel.

The results of the first two steps are also useful for the "normal" development of abelian categories, and will be used there.

  1. For every object A, define a "subtraction" morphism σ : A ⨯ A ⟶ A and use it to define subtraction on morphisms as f - g := prod.lift f g ≫ σ.
  2. Prove a small number of identities about this subtraction from the definition of σ.
  3. From these identities, prove a large number of other identities that imply that defining f + g := f - (0 - g) indeed gives an abelian group structure on morphisms such that composition is bilinear.

The construction is non-trivial and it is quite remarkable that this abelian group structure can be constructed purely from the existence of a few limits and colimits. What's even more impressive is that all additive structures on a category are in some sense isomorphic, so for abelian categories with a natural preadditive structure, this construction manages to "almost" reconstruct this natural structure. However, we have not formalized this isomorphism.

References

@[class]
structure category_theory.non_preadditive_abelian (C : Type u) [category_theory.category C] :
Type (max u v)

We call a category non_preadditive_abelian if it has a zero object, kernels, cokernels, binary products and coproducts, and every monomorphism and every epimorphism is normal.

In a non_preadditive_abelian category, every epimorphism is strong.

theorem category_theory.non_preadditive_abelian.mono_of_cancel_zero {C : Type u} [category_theory.category C] [category_theory.non_preadditive_abelian C] {X Y : C} (f : X Y) :
(∀ (Z : C) (g : Z X), g f = 0g = 0)category_theory.mono f

If g ≫ f = 0 implies g = 0 for all g, then f is a monomorphism.

theorem category_theory.non_preadditive_abelian.epi_of_zero_cancel {C : Type u} [category_theory.category C] [category_theory.non_preadditive_abelian C] {X Y : C} (f : X Y) :
(∀ (Z : C) (g : Y Z), f g = 0g = 0)category_theory.epi f

If f ≫ g = 0 implies g = 0 for all g, then g is a monomorphism.

The kernel of the cokernel of f is called the image of f.

There is a canonical epimorphism p : P ⟶ image f for every f.

The cokernel of the kernel of f is called the coimage of f.

The composite A ⟶ A ⨯ A ⟶ cokernel (Δ A), where the first map is (𝟙 A, 0) and the second map is the canonical projection into the cokernel.

The composite A ⨯ A ⟶ cokernel (diag A) ⟶ A given by the natural projection into the cokernel followed by the inverse of r. In the category of modules, using the normal kernels and cokernels, this map is equal to the map (a, b) ↦ a - b, hence the name σ for "subtraction".

Negation of morphisms in a non_preadditive_abelian category.

Equations

Addition of morphisms in a non_preadditive_abelian category.

Equations
theorem category_theory.non_preadditive_abelian.sub_sub_sub {C : Type u} [category_theory.category C] [category_theory.non_preadditive_abelian C] {X Y : C} (a b c d : X Y) :
a - c - (b - d) = a - b - (c - d)

theorem category_theory.non_preadditive_abelian.comp_sub {C : Type u} [category_theory.category C] [category_theory.non_preadditive_abelian C] {X Y Z : C} (f : X Y) (g h : Y Z) :
f (g - h) = f g - f h

theorem category_theory.non_preadditive_abelian.sub_comp {C : Type u} [category_theory.category C] [category_theory.non_preadditive_abelian C] {X Y Z : C} (f g : X Y) (h : Y Z) :
(f - g) h = f h - g h

theorem category_theory.non_preadditive_abelian.comp_add {C : Type u} [category_theory.category C] [category_theory.non_preadditive_abelian C] (X Y Z : C) (f : X Y) (g h : Y Z) :
f (g + h) = f g + f h

theorem category_theory.non_preadditive_abelian.add_comp {C : Type u} [category_theory.category C] [category_theory.non_preadditive_abelian C] (X Y Z : C) (f g : X Y) (h : Y Z) :
(f + g) h = f h + g h