open import 1Lab.Reflection.Record

open import Cat.Instances.Functor
open import Cat.Instances.Product
open import Cat.Diagram.Product
open import Cat.Prelude
open import Cat.Strict

module Cat.Instances.StrictCat where

open Product
open is-product
open Precategory
open Functor

private variable
  o h : Level

The category of strict categories🔗

Recall that a precategory is said strict if its space of objects is a Set. While general precategories are too homotopically interesting to fit into a Precategory (because functor spaces will not, in general, be h-sets), the strict categories do form a precategory, which we denote $\mathfrak{Cat}_\mathrm{strict}$ .

private unquoteDecl eqv = declare-record-iso eqv (quote Functor)

Functor-is-set : ∀ {o h} {C D : Precategory o h} → is-set (Ob D)
               → is-set (Functor C D)
Functor-is-set {o = o} {h} {C} {D} dobset =
  Iso→is-hlevel 2 eqv (hlevel 2)
  where
    open Precategory.HLevel-instance D
    instance
      Dob : H-Level (Ob D) 2
      Dob = basic-instance 2 dobset

Strict-cats : ∀ o h → Precategory _ _
Strict-cats o h .Ob = Σ[ C ∈ Precategory o h ] (is-strict C)
Strict-cats o h .Hom (C , _) (D , _) = Functor C D
Strict-cats o h .id  = Id
Strict-cats o h ._∘_ = _F∘_
Strict-cats o h .idr _       = Functor-path (λ _ → refl) λ _ → refl
Strict-cats o h .idl _       = Functor-path (λ _ → refl) λ _ → refl
Strict-cats o h .assoc _ _ _ = Functor-path (λ _ → refl) λ _ → refl

This assembles into a Precategory because the only bit of a Functor that doesn’t have a fixed h-level is the object mapping; By asking that D be a strict category, this fixes the functors to be sets.

Strict-cats o h .Hom-set _ (D , dset) = Functor-is-set dset

Products🔗

We prove that Strict-cats has products. This is because $(\mathcal{C} \times \mathcal{D})_0$ is $\mathcal{C}_0 \times \mathcal{D}_0$ , and h-levels are closed under products.

Strict-cats-products
  : {C D : Precategory o h}
  → (cob : is-set (Ob C)) (dob : is-set (Ob D))
  → Product (Strict-cats o h) (C , cob) (D , dob)
Strict-cats-products {C = C} {D = D} cob dob = prod where
  prod : Product (Strict-cats _ _) (C , cob) (D , dob)
  prod .apex = C ×ᶜ D , ×-is-hlevel 2 cob dob
  prod .π₁ = Fst {C = C} {D = D}
  prod .π₂ = Snd {C = C} {D = D}
  prod .has-is-product .⟨_,_⟩ p q = Cat⟨ p , q ⟩Cat
  prod .has-is-product .π₁∘factor = Functor-path (λ _ → refl) λ _ → refl
  prod .has-is-product .π₂∘factor = Functor-path (λ _ → refl) λ _ → refl
  prod .has-is-product .unique other p q =
    Functor-path (λ x i → F₀ (p i) x , F₀ (q i) x) λ f i → F₁ (p i) f , F₁ (q i) f