open import Cat.Diagram.Coproduct.Indexed
open import Cat.Diagram.Colimit.Base
open import Cat.Instances.Discrete
open import Cat.Instances.Product
open import Cat.Prelude

import Cat.Reasoning

module Cat.Diagram.Coproduct.Copower where

Copowers🔗

Let $\mathcal{C}$ be a category admitting $\kappa$ -small indexed coproducts, for example a $\kappa$ -cocomplete category. In the same way that (in ordinary arithmetic) the iterated product of a bunch of copies of the same factor

$\underbrace{a \times a \times \dots \times a}_{n}$

is called a “power”, we refer to the coproduct $\coprod_{X} A$ of many copies of an object $X$ indexed by a $\kappa$ -small set $X$ as the copower of $A$ by $X$ , and alternatively denote it $X \otimes A$ . If $\mathcal{C}$ does indeed admit coproducts indexed by any $\kappa$ -small set, then the copowering construction extends to a functor ${{\mathbf{Sets}}}_\kappa \times \mathcal{C} \to \mathcal{C}$ .

The notion of copowering gives us slick terminology for a category which admits all $\kappa$ -small coproducts, but not necessarily all $\kappa$ -small colimits: Such a category is precisely one copowered over ${{\mathbf{Sets}}}_\kappa$ .

  _⊗_ : Set ℓ → Ob → Ob
  X ⊗ A = coprods X (λ _ → A) .ΣF

The action of the copowering functor is given by simultaneously changing the indexing along a function of sets $f : X \to Y$ and changing the underlying object by a morphism $g : A \to B$ . This is functorial by the uniqueness properties of colimiting maps.

  Copowering : Functor (Sets ℓ ×ᶜ C) C
  Copowering .F₀ (X , A) = X ⊗ A
  Copowering .F₁ {X , A} {Y , B} (idx , obj) =
    coprods X (λ _ → A) .match λ i → coprods Y (λ _ → B) .ι (idx i) ∘ obj
  Copowering .F-id {X , A} = sym $
    coprods X (λ _ → A) .unique _ λ i → sym id-comm
  Copowering .F-∘ {X , A} f g = sym $
    coprods X (λ _ → A) .unique _ λ i →
      pullr (coprods _ _ .commute) ∙ extendl (coprods _ _ .commute)

cocomplete→copowering
  : ∀ {o ℓ} {C : Precategory o ℓ}
  → is-cocomplete ℓ ℓ C → Functor (Sets ℓ ×ᶜ C) C
cocomplete→copowering colim = Copowering λ S F →
  Colimit→IC _ (is-hlevel-suc 2 (S .is-tr)) (Disc-adjunct F) (colim _)