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

open import Data.Bool

import Cat.Reasoning

module Cat.Instances.Shape.Isomorphism where

The isomorphism categoryπŸ”—

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

0β‰…1 : Precategory lzero lzero
0β‰…1 .Precategory.Ob = Bool
0β‰…1 .Precategory.Hom _ _ = ⊀
0β‰…1 .Precategory.Hom-set _ _ = hlevel!
0β‰…1 .Precategory.id = tt
0β‰…1 .Precategory._∘_ tt tt = tt
0β‰…1 .Precategory.idr tt i = tt
0β‰…1 .Precategory.idl tt i = tt
0β‰…1 .Precategory.assoc tt tt tt i = tt

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

0β‰…1-iso-contr : βˆ€ X Y β†’ is-contr (Isomorphism 0β‰…1 X Y)
0β‰…1-iso-contr _ _ .centre =
  0β‰…1.make-iso tt tt (hlevel 1 _ _) (hlevel 1 _ _)
0β‰…1-iso-contr _ _ .paths p =
  0β‰…1.β‰…-pathp refl refl refl

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

–
-- 0β‰…1-is-strict : is-set 0β‰…1.Ob
-- 0β‰…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`{.Agda} and `false`{.Agda}!

-- ```agda
-- 0β‰…1-not-univalent : Β¬ is-category 0β‰…1
-- 0β‰…1-not-univalent is-cat =
--   true≠false $ is-cat .to-path $
--   0β‰…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 `0β‰…1`{.Agda}
-- into some category $\cC$ are equivalent to isomorphisms in $\cC$.

-- ```agda
-- Isos : βˆ€ {o β„“} β†’ Precategory o β„“ β†’ Type (o βŠ” β„“)
-- Isos π’ž = Ξ£[ A ∈ π’ž.Ob ] Ξ£[ B ∈ π’ž.Ob ] (A π’ž.β‰… B)
--   where module π’ž = Cat.Reasoning π’ž
-- ```

-- To prove this, we fix some category $\cC$, and construct an
-- isomorphism between functors out of `0β‰…1`{.Agda} and isomorphisms
-- in $\cC$.

-- ```agda
-- module _ {o β„“} {π’ž : Precategory o β„“} where
--   private
--     module π’ž = Cat.Reasoning π’ž
--     open Functor
--     open π’ž._β‰…_
-- ```

-- For the forward direction, we use the fact that all objects in
-- `0β‰…1`{.Agda} are isomorphic to construct an iso between `true`{.Agda}
-- and `false`{.Agda}, and then use the fact that functors preserve
-- isomorphisms to obtain an isomorphism in $\cC$.

-- ```agda
--   functorβ†’iso : (F : Functor 0β‰…1 π’ž) β†’ Isos π’ž
--   functor→iso F =
--     _ , _ , F-map-iso F (0β‰…1-iso-contr true false .centre)
-- ```

-- For the backwards direction, we are given an isomorphism $X \cong Y$
-- in $\cC$. Our functor will map `true`{.Agda} to $X$, and `false`
-- to $Y$: 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.

-- ```agda
--   isoβ†’functor : Isos π’ž β†’ Functor 0β‰…1 π’ž
--   iso→functor (X , Y , isom) = fun
--     where
--       fun : Functor _ _
--       fun .Fβ‚€ true = X
--       fun .Fβ‚€ false = Y
--       fun .F₁ {true} {true} _ = π’ž.id
--       fun .F₁ {true} {false} _ = to isom
--       fun .F₁ {false} {true} _ = from isom
--       fun .F₁ {false} {false} _ = π’ž.id
--       fun .F-id {true} = refl
--       fun .F-id {false} = refl
--       fun .F-∘ {true} {true} {z} f g = sym $ π’ž.idr _
--       fun .F-∘ {true} {false} {true} f g = sym $ invr isom
--       fun .F-∘ {true} {false} {false} f g = sym $ π’ž.idl _
--       fun .F-∘ {false} {true} {true} f g = sym $ π’ž.idl _
--       fun .F-∘ {false} {true} {false} f g = sym $ invl isom
--       fun .F-∘ {false} {false} {z} f g = sym $ π’ž.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!

-- ```agda
--   iso≃functor : is-equiv isoβ†’functor
--   iso≃functor = is-isoβ†’is-equiv (iso functorβ†’iso rinv linv) where
--     rinv : is-right-inverse functor→iso iso→functor
--     rinv F =
--       Functor-path
--         (Ξ» { true β†’ refl ; false β†’ refl })
--         (Ξ» { {true} {true} _ β†’ sym (F-id F)
--            ; {true} {false} tt β†’ refl
--            ; {false} {true} tt β†’ refl
--            ; {false} {false} tt β†’ sym (F-id F) })

--     linv : is-left-inverse functor→iso iso→functor
--     linv F = Ξ£-pathp refl $ Ξ£-pathp refl $ π’ž.β‰…-pathp refl refl refl
-- ```