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 #

is_idempotent_complete C expresses that C is idempotent complete, i.e. all idempotents in C split. Other characterisations of idempotent completeness are given by is_idempotent_complete_iff_has_equalizer_of_id_and_idempotent and is_idempotent_complete_iff_idempotents_have_kernels.
is_idempotent_complete_of_abelian expresses that abelian categories are idempotent complete.
is_idempotent_complete_iff_of_equivalence expresses that if two categories C and D are equivalent, then C is idempotent complete iff D is.
is_idempotent_complete_iff_opposite expresses that Cᵒᵖ is idempotent complete iff C is.

References #

[Stacks: Karoubian categories] https://stacks.math.columbia.edu/tag/09SF

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@[class]

structure category_theory.is_idempotent_complete (C : Type u_1) [category_theory.category C] :

Prop

idempotents_split : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → (∃ (Y : C) (i : Y ⟶ X) (e : X ⟶ Y), i ≫ e = 𝟙 Y ∧ e ≫ i = p)

A category is idempotent complete iff all idempotent endomorphisms p split as a composition p = e ≫ i with i ≫ e = 𝟙 _

Instances of this typeclass

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theorem category_theory.idempotents.is_idempotent_complete_iff_has_equalizer_of_id_and_idempotent (C : Type u_1) [category_theory.category C] :

category_theory.is_idempotent_complete C ↔ ∀ (X : C) (p : X ⟶ X), p ≫ p = p → category_theory.limits.has_equalizer (𝟙 X) p

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

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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.

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theorem category_theory.idempotents.is_idempotent_complete_iff_idempotents_have_kernels (C : Type u_1) [category_theory.category C] [category_theory.preadditive C] :

category_theory.is_idempotent_complete C ↔ ∀ (X : C) (p : X ⟶ X), p ≫ p = p → category_theory.limits.has_kernel p

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

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@[protected, instance]

def category_theory.idempotents.is_idempotent_complete_of_abelian (D : Type u_1) [category_theory.category D] [category_theory.abelian D] :

category_theory.is_idempotent_complete D

An abelian category is idempotent complete.

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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) :