open import Cat.Allegory.Base
open import Cat.Prelude

import Cat.Allegory.Reasoning as Ar

module Cat.Allegory.Maps where

Maps in allegories🔗

Suppose we have an allegory $\mathcal{A}$ — for definiteness you could consider the archetypal allegory $\mathbf{Rel}$ — but what we really wanted was to get our hands on an ordinary category. That is: we want, given some category of “sets and relations”, to recover the associated category of “sets and functions”. So, let’s start by thinking about $\mathbf{Rel}$!

If you have a relation $R \hookrightarrow A \times B$, when does it correspond to a function $A \to B$? Well, it must definitely be functional: if we want $R(x, y)$ to represent “$f(x) = y$”, then surely if $R(x, y) \land R(x, z)$, we must have $y = z$. In allegorical language, we would say $RR^o \le \mathrm{id}_{}$, which we can calculate to mean (in $\mathbf{Rel}$) that, for all $x, y$,

$$(\exists z, R(z, x) \land R(z, y)) \to (x = y)\text{.}$$

It must also be an entire relation: Every $x$ must at least one $y$ it is related to, and by functionality, exactly one $y$ it is related to. This is the “value of” the function at $y$.

module _ {o ℓ h} (A : Allegory o ℓ h) where
  open Allegory A
  record is-map x y (f : Hom x y) : Type h where
    constructor mapping
    field
      functional : f ∘ f † ≤ id
      entire     : id ≤ f † ∘ f

module _ {o ℓ h} {A : Allegory o ℓ h} where
  open Allegory A
  open is-map
  instance
    H-Level-is-map : ∀ {x y} {f : Hom x y} {n} → H-Level (is-map A x y f) (suc n)
    H-Level-is-map = prop-instance {T = is-map A _ _ _} λ where
      x y i .functional → ≤-thin (x .functional) (y .functional) i
      x y i .entire     → ≤-thin (x .entire) (y .entire) i

We can calculate that a functional entire relation is precisely a function between sets: Functionality ensures that (fixing a point $a$ in the domain) the space of “points in the codomain related to $a$” is a subsingleton, and entireness ensures that it is inhabited; By unique choice, we can project the value of the function.

Rel-map→function
  : ∀ {ℓ} {x y : Set ℓ} {f : Allegory.Hom (Rel ℓ) x y}
  → is-map (Rel ℓ) x y f
  → ∣ x ∣ → ∣ y ∣
Rel-map→function {x = x} {y} {rel} map elt =
  ∥-∥-rec {P = Σ ∣ y ∣ λ b → ∣ rel elt b ∣}
    (λ { (x , p) (y , q) → Σ-prop-path (λ _ → hlevel!) (functional′ p q) })
    (λ x → x)
    (entire′ elt) .fst
  where
    module map = is-map map
    functional′ : ∀ {a b c} → ∣ rel a b ∣ → ∣ rel a c ∣ → b ≡ c
    functional′ r1 r2 = out! (map.functional _ _ (inc (_ , r1 , r2)))

    entire′ : ∀ a → ∃ ∣ y ∣ λ b → ∣ rel a b ∣
    entire′ a =
      □-rec! (λ { (x , y , R) → inc (x , R) }) (map.entire a a (inc refl))

The category of maps🔗

Given an allegory $\mathcal{A}$, we can form a category $\mathrm{Map}(\mathcal{A})$ with the same objects as $\mathcal{A}$, but considering only the maps (rather than arbitrary relations).

module _ {o ℓ h} (A : Allegory o ℓ h) where
  private module A = Ar A
  open is-map
  open Precategory hiding (_∘_ ; id)
  open A using (_† ; _∘_ ; id ; _◀_ ; _▶_)

  Maps[_] : Precategory _ _
  Maps[_] .Ob  = A.Ob
  Maps[_] .Hom x y = Σ (A.Hom x y) (is-map A x y)
  Maps[_] .Hom-set x y = hlevel 

Proving that maps compose is a bit annoying, but it follows from the properties of the duality involution. By the way: This is essentially the proof that adjunctions compose! This is because, sneakily, the property “being a map” is defined to be $f \dashv f^o$, but without using those words.

  Maps[_] .Precategory.id = A.id , mapping
    (subst (A._≤ id) (sym (A.idl _ ∙ dual-id A)) A.≤-refl)
    (subst (id A.≤_) (sym (A.idr _ ∙ dual-id A)) A.≤-refl)

  Maps[_] .Precategory._∘_ (f , m) (g , m′) = f A.∘ g , mapping
    ( (f ∘ g) ∘ (f ∘ g) †     A.=⟨ A.†-inner refl ⟩A.=
      f ∘ (g ∘ g †) ∘ f †     A.≤⟨ f ▶ m′ .functional ◀ f † ⟩A.≤
      f ∘ id ∘ f †            A.=⟨ A.refl⟩∘⟨ A.idl _ ⟩A.=
      f ∘ f †                 A.≤⟨ m .functional ⟩A.≤
      id                      A.≤∎ )
    ( id                   A.≤⟨ m′ .entire ⟩A.≤
      g † ∘ g              A.=⟨ ap (g † ∘_) (sym (A.idl _)) ⟩A.=
      g † ∘ id ∘ g         A.≤⟨ g † ▶ m .entire ◀ g ⟩A.≤
      g † ∘ (f † ∘ f) ∘ g  A.=⟨ A.pulll (A.pulll (sym A.dual-∘)) ∙ A.pullr refl ⟩A.=
      (f ∘ g) † ∘ (f ∘ g)  A.≤∎ )

  Maps[_] .idr f = Σ-prop-path (λ _ → hlevel ) (A.idr _)
  Maps[_] .idl f = Σ-prop-path (λ _ → hlevel ) (A.idl _)
  Maps[_] .assoc f g h = Σ-prop-path (λ _ → hlevel ) (A.assoc _ _ _)