module Cat.Instances.Shape.Involution where

The walking involution🔗

The walking involution is the category with a single non-trivial morphism satisfying We can encode this by defining morphisms as booleans, and using xor as our composition operation.

∙⤮∙ : Precategory lzero lzero
∙⤮∙ .Precategory.Ob = 
∙⤮∙ .Precategory.Hom _ _ = Bool
∙⤮∙ .Precategory.Hom-set _ _ = Bool-is-set
∙⤮∙ .Precategory.id = false
∙⤮∙ .Precategory._∘_ = xor
∙⤮∙ .Precategory.idr f = xor-falser f
∙⤮∙ .Precategory.idl f = refl
∙⤮∙ .Precategory.assoc f g h = xor-associative f g h

open Cat.Reasoning ∙⤮∙

This is the smallest precategory with a non-trivial isomorphism.

Bool≃∙⤮∙-isos : Bool  (tt  tt)
Bool≃∙⤮∙-isos =
  Iso→Equiv (Bool→Iso , iso Iso→Bool right-inv left-inv)
  where
    Bool→Iso : Bool  tt  tt
    Bool→Iso true = make-iso true true refl refl
    Bool→Iso false = id-iso

    Iso→Bool : tt  tt  Bool
    Iso→Bool i = i .to

    right-inv : is-right-inverse Iso→Bool Bool→Iso
    right-inv f = Bool-elim  b  b  f .to  Bool→Iso b  f) ext ext (f .to) refl

    left-inv : is-left-inverse Iso→Bool Bool→Iso
    left-inv true = refl
    left-inv false = refl

Skeletality and (non)-univalence🔗

As the walking involution only has one object, it is trivially skeletal.

∙⤮∙-skeletal : is-skeletal ∙⤮∙
∙⤮∙-skeletal .to-path _ = refl
∙⤮∙-skeletal .to-path-over _ =
  is-prop→pathp  _  squash) (inc id-iso) _

However, it is not univalent, since its only object has more internal automorphisms than it has loops. By definition, univalence for the walking involution would mean that there are two paths in the unit type: but this contradicts the unit type being a set.

∙⤮∙-not-univalent : ¬ is-category ∙⤮∙
∙⤮∙-not-univalent is-cat = true≠false (Bool-is-prop true false) where

  Bool≃Ob-path : Bool  (tt  tt)
  Bool≃Ob-path = Bool≃∙⤮∙-isos ∙e identity-system-gives-path is-cat

  Bool-is-prop : is-prop Bool
  Bool-is-prop = Equiv→is-hlevel 1 Bool≃Ob-path hlevel!