module Cat.Instances.Karoubi {o h} (C : Precategory o h) where

Karoubi envelopes🔗

We give a construction of the Karoubi envelope C˜\~\mathcal{C} of a precategory C\mathcal{C}, a formal construction which adds a choice of splittings for every idempotent in C\mathcal{C}. Furthermore, the Karoubi envelope is the smallest idempotent-complete category which admits a map from C\mathcal{C}, in the sense that any F:CDF : \mathcal{C} \to \mathcal{D} into an idempotent-complete category D\mathcal{D} factors through C˜\~\mathcal{C}:

CC˜D \mathcal{C} \hookrightarrow \~\mathcal{C} \to \mathcal{D}

Furthermore, the embedding functor CC˜\mathcal{C} \to \~\mathcal{C} is fully faithful.

The objects in C˜\~\mathcal{C} are given by pairs of an object c:Cc : \mathcal{C} and an idempotent f:ccf : c \to c. A map between (c,f)(c,f) and (d,g)(d,g) is given by a map ϕ:cd\phi : c \to d which absorbs ff from the left and gg from the right: ϕf=ϕ=gϕ\phi \circ f = \phi = g \circ \phi.

  KOb : Type (o  h)
  KOb = Σ[ c  C.Ob ] Σ[ f  C.Hom c c ] (is-idempotent f)

  KHom : KOb  KOb  Type h
  KHom (c , f , _) (d , g , _) = Σ[ φ  C.Hom c d ] ((φ C.∘ f  φ) × (g C.∘ φ  φ))

  KH≡ :  {a b : C.Ob} {af : C.Hom a a} {bf : C.Hom b b}
          {ai : is-idempotent af} {bi : is-idempotent bf}
          {f g : KHom (a , af , ai) (b , bf , bi)}  fst f  fst g  f  g
  KH≡ = Σ-prop-path  _  hlevel 1)

We can see that these data assemble into a precategory. However, note that the identity on (c,e)(c,e) in C˜\~\mathcal{C} isn’t the identity in CC, it’s the chosen idempotent ee!

Karoubi : Precategory (o  h) h
Karoubi .Ob = KOb
Karoubi .Hom = KHom
Karoubi .Hom-set _ _ = hlevel 2

Karoubi .id {x = c , e , i} = e , i , i
Karoubi ._∘_ (f , fp , fp') (g , gp , gp') = f C.∘ g , C.pullr gp , C.pulll fp'

Karoubi .idr {x = _ , _ , i} {_ , _ , j} (f , p , q) = KH≡ {ai = i} {bi = j} p
Karoubi .idl {x = _ , _ , i} {_ , _ , j} (f , p , q) = KH≡ {ai = i} {bi = j} q
Karoubi .assoc {w = _ , _ , i} {z = _ , _ , j} _ _ _ =
  KH≡ {ai = i} {bi = j} (C.assoc _ _ _)

We can now define the embedding functor from C to its Karoubi envelope. It has object part x(x,id)x \mapsto (x, \operatorname{id}_{}); The morphism part of the functor has to send f:xyf : x \to y to some f:xyf' : x \to y which absorbs id\operatorname{id}_{} on either side; But this is just ff again.

Embed : Functor C Karoubi
Embed .F₀ x = x , , id-is-idempotent
Embed .F₁ f = f , C.idr _ , C.idl _
Embed .F-id = KH≡ {ai = id-is-idempotent} {bi = id-is-idempotent} refl
Embed .F-∘ f g = KH≡ {ai = id-is-idempotent} {bi = id-is-idempotent} refl

An elementary argument shows that the morphism part of Embed has an inverse given by projecting the first component of the pair; Hence, Embed is fully faithful.

Embed-is-fully-faithful : is-fully-faithful Embed
Embed-is-fully-faithful = is-iso→is-equiv $
  iso fst  _  Σ-prop-path  _  hlevel 1) refl) λ _  refl


We now show that any idempotent f:(A,e)(A,e)f : (A, e) \to (A, e) admits a splitting in C˜\~\mathcal{C}. First, note that since ff is (by assumption) idempotent, we have an object given by (A,f)(A, f); We’ll split ff as a map

(A,e)(A,f)(A,e) (A, e) \to (A, f) \to (A, e)

The first map is given by the underlying map of f:AAf : A \to A. We must show that fe=ff \circ e = f, but we have this by the definition of maps in C˜\~\mathcal{C}. In the other direction, we can again take f:AAf : A \to A, which also satisfies ef=fe \circ f = f.

is-idempotent-complete-Karoubi : Karoubi
is-idempotent-complete-Karoubi {A = A , e , i} (f , p , q) idem = spl where

  f-idem : f C.∘ f  f
  f-idem i = idem i .fst

  spl : Karoubi (f , p , q)
  spl .F = A , f , f-idem
  spl .project = f , p , f-idem
  spl .inject = f , f-idem , q

For this to be a splitting of ff, we must show that ff=ff \circ f = f as a map (A,e)(A,e)(A, e) \to (A, e), which we have by assumption; And we must show that ff=idf \circ f = \operatorname{id}_{} as a map (A,f)(A,f)(A, f) \to (A, f). But recall that the identity on (A,f)(A, f) is ff, so we also have this by assumption!

  spl .p∘i = KH≡ {ai = f-idem} {bi = f-idem} f-idem
  spl .i∘p = KH≡ {ai = i} {bi = i} f-idem

Hence C˜\~\mathcal{C} is an idempotent-complete category which admits CC as a full subcategory.

If C\mathcal{C} was already idempotent-complete, then Embed is an equivalence of categories between C\mathcal{C} and C˜\~\mathcal{C}:

Karoubi-is-completion : is-idempotent-complete  is-equivalence Embed
Karoubi-is-completion complete = ff+split-eso→is-equivalence Embed-is-fully-faithful eso where
  eso : is-split-eso Embed
  eso (c , e , ie) = c' , same where
    open is-split (complete e ie) renaming (F to c'; inject to i; project to p)
    module Karoubi = Cat.Morphism Karoubi
    open Inverses
    same : (c' , , _) Karoubi.≅ (c , e , ie)
    same .to = i , C.idr _ , C.rswizzle (sym i∘p) p∘i
    same .from = p , C.lswizzle (sym i∘p) p∘i , C.idl _
    same .inverses .invl = KH≡ {ai = ie} {bi = ie} i∘p
    same .inverses .invr = KH≡ {ai = id-is-idempotent} {bi = id-is-idempotent} p∘i

This makes the Karoubi envelope an “idempotent completion” in two different technical senses1!

  1. it is a completion that adds splittings of idempotents, and a completion that is idempotent↩︎