module Cat.Allegory.Maps where

Maps in allegoriesπŸ”—

Suppose we have an allegory β€” for definiteness you could consider the archetypal allegory β€” but what we really wanted was to get our hands on an ordinary category. That is: we want, given some category of β€œsets and relations”, to recover the associated category of β€œsets and functions”. So, let’s start by thinking about

If you have a relation when does it correspond to a function Well, it must definitely be functional: if we want to represent β€œβ€, then surely if we must have In allegorical language, we would say which we can calculate to mean (in that, for all

It must also be an entire relation: Every must at least one it is related to, and by functionality, exactly one it is related to. This is the β€œvalue of” the function at

module _ {o β„“ h} (A : Allegory o β„“ h) where
  open Cat.Allegory.Morphism A
  record is-map x y (f : Hom x y) : Type h where
    constructor mapping
      functional : is-functional f
      entire     : is-entire f

module _ {o β„“ h} {A : Allegory o β„“ h} where
  open Allegory A
  open is-map
    H-Level-is-map : βˆ€ {x y} {f : Hom x y} {n} β†’ H-Level (is-map A x y f) (suc n)
    H-Level-is-map = prop-instance {T = is-map A _ _ _} Ξ» where
      x y i .functional β†’ ≀-thin (x .functional) (y .functional) i
      x y i .entire     β†’ ≀-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 in the domain) the space of β€œpoints in the codomain related to ” is a subsingleton, and entireness ensures that it is inhabited; By unique choice, we can project the value of the function.

  : βˆ€ {β„“} {x y : Set β„“} {f : Allegory.Hom (Rel β„“) x y}
  β†’ is-map (Rel β„“) x y f
  β†’ ∣ x ∣ β†’ ∣ y ∣
Rel-map→function {x = x} {y} {rel} map elt =
  βˆ₯-βˆ₯-rec {P = Ξ£ ∣ y ∣ Ξ» b β†’ ∣ rel elt b ∣}
    (Ξ» { (x , p) (y , q) β†’ Ξ£-prop-path! (functional' p q) })
    (Ξ» x β†’ x)
    (entire' elt) .fst
    module map = is-map map
    functional' : βˆ€ {a b c} β†’ ∣ rel a b ∣ β†’ ∣ rel a c ∣ β†’ b ≑ c
    functional' r1 r2 = β–‘-out! (map.functional _ _ (inc (_ , r1 , r2)))

    entire' : βˆ€ a β†’ βˆƒ ∣ y ∣ Ξ» b β†’ ∣ rel a b ∣
    entire' a = case map.entire a a (inc refl) of Ξ» x _ a~x β†’ inc (x , a~x)

The category of mapsπŸ”—

Given an allegory we can form a category with the same objects as but considering only the maps (rather than arbitrary relations).

module _ {o β„“ h} (A : Allegory o β„“ h) where
  private module A = Cat.Allegory.Morphism A
  open is-map
  open Precategory hiding (_∘_ ; id)
  open A using (_† ; _∘_ ; id ; _β—€_ ; _β–Ά_)

  Maps[_] : Precategory _ _
  Maps[_] .Ob  = A.Ob
  Maps[_] .Hom x y = Ξ£ (A.Hom x y) (is-map A x y)
  Maps[_] .Hom-set x y = hlevel 2

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 β€œbeing a map” is defined to be but without using those words.

  Maps[_] = , mapping
  Maps[_] .Precategory._∘_ (f , m) (g , m') =
    f A.∘ g , mapping
      (A.functional-∘ (m .functional) (m' .functional))
      (A.entire-∘ (m .entire) (m' .entire))
  Maps[_] .idr f = Ξ£-prop-path! (A.idr _)
  Maps[_] .idl f = Ξ£-prop-path! (A.idl _)
  Maps[_] .assoc f g h = Ξ£-prop-path! (A.assoc _ _ _)