open import Cat.Instances.Functor
open import Cat.Prelude

open import Data.Bool

import Cat.Reasoning

module Cat.Instances.Shape.Isomorphism where

The isomorphism category🔗

The isomorphism category is the category with two points, along with a unique isomorphism between them.

0≅1 : Precategory lzero lzero
0≅1 .Precategory.Ob = Bool
0≅1 .Precategory.Hom _ _ = ⊤
0≅1 .Precategory.Hom-set _ _ = hlevel 
0≅1 .Precategory.id = tt
0≅1 .Precategory._∘_ tt tt = tt
0≅1 .Precategory.idr tt i = tt
0≅1 .Precategory.idl tt i = tt
0≅1 .Precategory.assoc tt tt tt i = tt

private
  module 0≅1 = Cat.Reasoning 0≅1

open Cat.Reasoning using (Isomorphism)

Note that the space of isomorphisms between any 2 objects is contractible.

0≅1-iso-contr : ∀ X Y → is-contr (Isomorphism 0≅1 X Y)
0≅1-iso-contr _ _ .centre =
  0≅1.make-iso tt tt (hlevel  _ _) (hlevel  _ _)
0≅1-iso-contr _ _ .paths p = trivial!

The isomorphism category is strict, as its objects form a set.

0≅1-is-strict : is-set 0≅1.Ob
0≅1-is-strict = hlevel 

The isomorphism category is not univalent🔗

The isomorphism category is the canonical example of a non-univalent category. If it were univalent, then we’d get a path between true and false!

0≅1-not-univalent : ¬ is-category 0≅1
0≅1-not-univalent is-cat =
  true≠false $ is-cat .to-path $
  0≅1-iso-contr true false .centre

Functors out of the isomorphism category🔗

One important fact about the isomorphism category is that it classifies isomorphisms in categories, in the sense that functors out of 0≅1 into some category $\mathcal{C}$ are equivalent to isomorphisms in $\mathcal{C}\text{.}$

Isos : ∀ {o ℓ} → Precategory o ℓ → Type (o ⊔ ℓ)
Isos 𝒞 = Σ[ A ∈ 𝒞 ] Σ[ B ∈ 𝒞 ] (A 𝒞.≅ B)
  where module 𝒞 = Cat.Reasoning 𝒞

To prove this, we fix some category $\mathcal{C}\text{,}$ and construct an isomorphism between functors out of 0≅1 and isomorphisms in $\mathcal{C}\text{.}$

module _ {o ℓ} {𝒞 : Precategory o ℓ} where
  private
    module 𝒞 = Cat.Reasoning 𝒞
    open Functor
    open 𝒞._≅_

For the forward direction, we use the fact that all objects in 0≅1 are isomorphic to construct an iso between true and false, and then use the fact that functors preserve isomorphisms to obtain an isomorphism in $\mathcal{C}\text{.}$

  functor→iso : (F : Functor 0≅1 𝒞) → Isos 𝒞
  functor→iso F = _ , _ , F-map-iso F (0≅1-iso-contr true false .centre)

For the backwards direction, we are given an isomorphism $X\cong Y$ in $\mathcal{C}\text{.}$ Our functor will map true to $X\text{,}$ and false to $Y\text{:}$ this is somewhat arbitrary, but lines up with our choices for the forward direction. We then perform a big case bash to construct the mapping of morphisms, and unpack the components of the provided isomorphism into place. Functoriality follows by the fact that the provided isomorphism is indeed an isomorphism.

  iso→functor : Isos 𝒞 → Functor 0≅1 𝒞
  iso→functor (X , Y , isom) = fun where
    fun : Functor _ _
    fun .F₀ true = X
    fun .F₀ false = Y
    fun .F₁ {true}  {true}  _ = 𝒞.id
    fun .F₁ {true}  {false} _ = isom .to
    fun .F₁ {false} {true}  _ = isom .from
    fun .F₁ {false} {false} _ = 𝒞.id
    fun .F-id {true}  = refl
    fun .F-id {false} = refl
    fun .F-∘ {true}  {true}  {z}     f g = sym $ 𝒞.idr _
    fun .F-∘ {true}  {false} {true}  f g = sym $ isom .invr
    fun .F-∘ {true}  {false} {false} f g = sym $ 𝒞.idl _
    fun .F-∘ {false} {true}  {true}  f g = sym $ 𝒞.idl _
    fun .F-∘ {false} {true}  {false} f g = sym $ isom .invl
    fun .F-∘ {false} {false} {z}     f g = sym $ 𝒞.idr _

Showing that this function is an equivalence is relatively simple: the only tricky part is figuring out which lemmas to use to characterise path spaces!

  iso≃functor : is-equiv iso→functor
  iso≃functor = is-iso→is-equiv (iso functor→iso rinv linv) where
    rinv : is-right-inverse functor→iso iso→functor
    rinv F = Functor-path
      (λ { true → refl ; false → refl })
      (λ { {true}  {true}  _  → sym (F .F-id)
         ; {true}  {false} tt → refl
         ; {false} {true}  tt → refl
         ; {false} {false} tt → sym (F .F-id) })

    linv : is-left-inverse functor→iso iso→functor
    linv F = Σ-pathp refl $ Σ-pathp refl $ trivial!