open import Cat.Prelude

module Cat.Diagram.Idempotent {o h} (C : Precategory o h) where

open import Cat.Reasoning C
private variable
  A B : Ob
  f : Hom A B

Idempotents🔗

Idempotents are the categorical generalisation of projections, in the sense of linear algebra. Formally, an idempotent $e:A\to A$ is a map with $e\circ e=e\text{.}$ Keeping in line with the view that an idempotent is like a projection, we can ask what it projects onto: We would then have some subobject $F$ of fixed elements, and a decomposition of $e$ as

When this is the case, we say that $e$ is a split idempotent: We have some pair of maps $p:X\to F$ (the “projector”) and $i:F\to X\text{,}$ with $r\circ i=\mathrm{id}$ and $i\circ r=e\text{.}$

is-idempotent : Hom A A → Type _
is-idempotent e = e ∘ e ≡ e

record is-split (e : Hom A A) : Type (o ⊔ h) where
  field
    {F}     : Ob
    project : Hom A F
    inject  : Hom F A

    p∘i : project ∘ inject ≡ id
    i∘p : inject ∘ project ≡ e

is-split→is-idempotent : is-split f → is-idempotent f
is-split→is-idempotent {f = f} spl =
  f ∘ f             ≡˘⟨ ap₂ _∘_ i∘p i∘p ⟩≡˘
  (s ∘ r) ∘ (s ∘ r) ≡⟨ cancel-inner p∘i ⟩≡
  s ∘ r             ≡⟨ i∘p ⟩≡
  f                 ∎
  where open is-split spl renaming (inject to s ; project to r)

Identities are always trivially (split) idempotent:

id-is-idempotent : ∀ {A} → is-idempotent {A = A} id
id-is-idempotent = idr _

id-is-split : ∀ {A} → is-split {A = A} id
id-is-split {A} .is-split.F = A
id-is-split .is-split.project = id
id-is-split .is-split.inject = id
id-is-split .is-split.p∘i = idr _
id-is-split .is-split.i∘p = idr _

It’s not the case that idempotents are split in every category. Those where this is the case are called idempotent-complete. Every category can be embedded, by a fully faithful functor, into an idempotent-complete category; This construction is called the Karoubi envelope of $\mathcal{C}\text{.}$

is-idempotent-complete : Type _
is-idempotent-complete = ∀ {A} (f : Hom A A) → is-idempotent f → is-split f