mathlib documentation

category_theory.idempotents.basic

Idempotent complete categories #

In this file, we define the notion of idempotent complete categories (also known as Karoubian categories, or pseudoabelian in the case of preadditive categories).

Main definitions #

References #

A category is idempotent complete iff for all idempotent endomorphisms, the equalizer of the identity and this idempotent exists.

theorem category_theory.idempotents.idem_of_id_sub_idem {C : Type u_1} [category_theory.category C] [category_theory.preadditive C] {X : C} (p : X X) (hp : p p = p) :
(𝟙 X - p) (𝟙 X - p) = 𝟙 X - p

In a preadditive category, when p : X ⟶ X is idempotent, then 𝟙 X - p is also idempotent.

A preadditive category is pseudoabelian iff all idempotent endomorphisms have a kernel.

@[protected, instance]

An abelian category is idempotent complete.

theorem category_theory.idempotents.split_imp_of_iso {C : Type u_1} [category_theory.category C] {X X' : C} (φ : X X') (p : X X) (p' : X' X') (hpp' : p φ.hom = φ.hom p') (h : ∃ (Y : C) (i : Y X) (e : X Y), i e = 𝟙 Y e i = p) :
∃ (Y' : C) (i' : Y' X') (e' : X' Y'), i' e' = 𝟙 Y' e' i' = p'
theorem category_theory.idempotents.split_iff_of_iso {C : Type u_1} [category_theory.category C] {X X' : C} (φ : X X') (p : X X) (p' : X' X') (hpp' : p φ.hom = φ.hom p') :
(∃ (Y : C) (i : Y X) (e : X Y), i e = 𝟙 Y e i = p) ∃ (Y' : C) (i' : Y' X') (e' : X' Y'), i' e' = 𝟙 Y' e' i' = p'

If C and D are equivalent categories, that C is idempotent complete iff D is.