open import Cat.Prelude

module Cat.Instances.Sets where

The category of SetsπŸ”—

We prove that the category of Sets is univalent. Recall that this means that, fixing a set AA, the type βˆ‘(B:Set)(Aβ‰…B)\sum_{(B : \set)} (A \cong B) is contractible. We first exhibit a contraction directly, using ua, and then provide an alternative proof in terms of univalence for nn-types.

Direct proofπŸ”—

The direct proof is surprisingly straightforward, in particular because the heavy lifting is done by a plethora of existing lemmas: Iso→Equiv to turn an isomorphism into an equivalence, path→ua-pathp to reduce dependent paths over ua to non-dependent paths, ≅-pathp to characterise dependent paths in _≅_, etc.

module _ {β„“} where
  import Cat.Reasoning (Sets β„“) as Sets

We must first rearrange _β‰…_ to _≃_, for which we can use Isoβ†’Equiv. We must then show that an isomorphism in the category of Sets is the same thing as an isomorphism of types; But the only difference between these types can be patched by happly/funext.

  isoβ†’equiv : {A B : Set β„“} β†’ A Sets.β‰… B β†’ ∣ A ∣ ≃ ∣ B ∣
  iso→equiv x = Iso→Equiv (x.to , iso x.from (happly x.invl) (happly x.invr))
    where module x = Sets._β‰…_ x

Using univalence for nn-types, function extensionality and the computation rule for univalence, it is almost trivial to show that categorical isomorphisms of sets are an identity system.

  Sets-is-category : is-category (Sets β„“)
  Sets-is-category .to-path i = n-ua (iso→equiv i)
  Sets-is-category .to-path-over p = Sets.β‰…-pathp refl _ $
    funextP λ a → path→ua-pathp _ refl

Indirect proofπŸ”—

While the proof above is fairly simple, we can give a different formulation, which might be more intuitive. Let’s start by showing that the rearrangement isoβ†’equiv is an equivalence:

  equivβ†’iso : {A B : Set β„“} β†’ ∣ A ∣ ≃ ∣ B ∣ β†’ A Sets.β‰… B
  equiv→iso (f , f-eqv) =
    Sets.make-iso f (equiv→inverse f-eqv)
      (funext (equiv→counit f-eqv))
      (funext (equiv→unit f-eqv))

  equiv≃iso : {A B : Set β„“} β†’ (A Sets.β‰… B) ≃ (∣ A ∣ ≃ ∣ B ∣)
  equiv≃iso {A} {B} = Isoβ†’Equiv (isoβ†’equiv , iso equivβ†’iso p q)
    where
      p : is-right-inverse (equiv→iso {A} {B}) iso→equiv
      p x = Ξ£-prop-path is-equiv-is-prop refl

      q : is-left-inverse (equiv→iso {A} {B}) iso→equiv
      q x = Sets.β‰…-pathp refl refl refl

We then use univalence for nn-types to directly establish that (A≑B)≃(Aβ‰…B)(A \equiv B) \simeq (A \cong B):

  is-categoryβ€²-Sets : βˆ€ {A B : Set β„“} β†’ (A ≑ B) ≃ (A Sets.β‰… B)
  is-categoryβ€²-Sets {A} {B} =
    (A ≑ B)         β‰ƒβŸ¨ n-univalence e⁻¹ βŸ©β‰ƒ
    (∣ A ∣ ≃ ∣ B ∣) β‰ƒβŸ¨ equiv≃iso e⁻¹ βŸ©β‰ƒ
    (A Sets.β‰… B)    β‰ƒβˆŽ