open import 1Lab.Prelude

module Data.Set.Coequaliser where

Set CoequalisersπŸ”—

In their most general form, colimits can be pictured as taking disjoint unions and then β€œgluing together” some parts. The β€œgluing together” part of that definition is where coequalisers come in: If you have parallel maps f,g:Aβ†’Bf, g : A \to B, then the coequaliser coeq(f,g)\id{coeq}(f,g) can be thought of as β€œBB, with the images of ff and gg identified”.

data Coeq (f g : A β†’ B) : Type (level-of A βŠ” level-of B) where
  inc    : B β†’ Coeq f g
  glue   : βˆ€ x β†’ inc (f x) ≑ inc (g x)
  squash : is-set (Coeq f g)

The universal property of coequalisers, being a type of colimit, is a mapping-out property: Maps from coeq(f,g)\id{coeq}(f,g) are maps out of BB, satisfying a certain property. Specifically, for a map h:Bβ†’Ch : B \to C, if we have h∘f=h∘gh \circ f = h \circ g, then the map ff factors (uniquely) through inc. The situation can be summarised with the diagram below.

We refer to this unique factoring as Coeq-rec.

Coeq-rec : βˆ€ {β„“} {C : Type β„“} {f g : A β†’ B}
      β†’ is-set C β†’ (h : B β†’ C)
      β†’ (βˆ€ x β†’ h (f x) ≑ h (g x)) β†’ Coeq f g β†’ C
Coeq-rec cset h h-coeqs (inc x) = h x
Coeq-rec cset h h-coeqs (glue x i) = h-coeqs x i
Coeq-rec cset h h-coeqs (squash x y p q i j) =
  cset (g x) (g y) (Ξ» i β†’ g (p i)) (Ξ» i β†’ g (q i)) i j
  where g = Coeq-rec cset h h-coeqs

An alternative phrasing of the desired universal property is precomposition with inc induces an equivalence between the β€œspace of maps Bβ†’CB \to C which coequalise ff and gg” and the maps coeq(f,g)β†’C\id{coeq}(f,g) \to C. In this sense, inc is the universal map which coequalises ff and gg.

To construct the map above, we used Coeq-elim-prop, which has not yet been defined. It says that, to define a dependent function from Coeq to a family of propositions, it suffices to define how it acts on inc: The path constructions don’t matter.

Coeq-elim-prop : βˆ€ {β„“} {f g : A β†’ B} {C : Coeq f g β†’ Type β„“}
              β†’ (βˆ€ x β†’ is-prop (C x))
              β†’ (βˆ€ x β†’ C (inc x))
              β†’ βˆ€ x β†’ C x
Coeq-elim-prop cprop cinc (inc x) = cinc x

Since C was assumed to be a family of propositions, we automatically get the necessary coherences for glue and squash.

Coeq-elim-prop {f = f} {g = g} cprop cinc (glue x i) =
  is-prop→pathp (λ i → cprop (glue x i)) (cinc (f x)) (cinc (g x)) i
Coeq-elim-prop cprop cinc (squash x y p q i j) =
  is-prop→squarep (λ i j → cprop (squash x y p q i j))
    (Ξ» i β†’ g x) (Ξ» i β†’ g (p i)) (Ξ» i β†’ g (q i)) (Ξ» i β†’ g y) i j
  where g = Coeq-elim-prop cprop cinc

Since β€œthe space of maps h:Bβ†’Ch : B \to C which coequalise ff and gg” is a bit of a mouthful, we introduce an abbreviation: Since a colimit is defined to be a certain universal (co)cone, we call these Coeq-cones.

private
  coeq-cone : βˆ€ {β„“} (f g : A β†’ B) β†’ Type β„“ β†’ Type _
  coeq-cone {B = B} f g C = Ξ£[ h ∈ (B β†’ C) ] (h ∘ f ≑ h ∘ g)

The universal property of Coeq then says that Coeq-cone is equivalent to the maps coeq(f,g)β†’C\id{coeq}(f,g) \to C, and this equivalence is given by inc, the β€œuniversal Coequalising map”.

Coeq-univ : βˆ€ {β„“} {C : Type β„“} {f g : A β†’ B}
          β†’ is-set C
          β†’ is-equiv {A = Coeq f g β†’ C} {B = coeq-cone f g C}
            (Ξ» h β†’ h ∘ inc , Ξ» i z β†’ h (glue z i))
Coeq-univ {C = C} {f = f} {g = g} cset =
  is-iso→is-equiv (iso cr' (λ x → refl) islinv)
  where
    open is-iso
    cr' : coeq-cone f g C β†’ Coeq f g β†’ C
    cr' (f , f-coeqs) = Coeq-rec cset f (happly f-coeqs)

    islinv : is-left-inverse cr' (Ξ» h β†’ h ∘ inc , Ξ» i z β†’ h (glue z i))
    islinv f = funext (Coeq-elim-prop (Ξ» x β†’ cset _ _) Ξ» x β†’ refl)

EliminationπŸ”—

Above, we defined what it means to define a dependent function (x:coeq(f,g))β†’CΒ x(x : \id{coeq}(f,g)) \to C\ x when CC is a family of propositions, and what it means to define a non-dependent function coeq(f,g)β†’C\id{coeq}(f,g) \to C. Now, we combine the two notions, and allow dependent elimination into families of sets:

Coeq-elim : βˆ€ {β„“} {f g : A β†’ B} {C : Coeq f g β†’ Type β„“}
          β†’ (βˆ€ x β†’ is-set (C x))
          β†’ (ci : βˆ€ x β†’ C (inc x))
          β†’ (βˆ€ x β†’ PathP (Ξ» i β†’ C (glue x i)) (ci (f x)) (ci (g x)))
          β†’ βˆ€ x β†’ C x
Coeq-elim cset ci cg (inc x) = ci x
Coeq-elim cset ci cg (glue x i) = cg x i
Coeq-elim cset ci cg (squash x y p q i j) =
  is-set→squarep (λ i j → cset (squash x y p q i j))
    (Ξ» i β†’ g x) (Ξ» i β†’ g (p i)) (Ξ» i β†’ g (q i)) (Ξ» i β†’ g y) i j
  where g = Coeq-elim cset ci cg

There is a barrage of combined eliminators, whose definitions are not very enlightening β€” you can mouse over these links to see their types: Coeq-elim-propβ‚‚ Coeq-elim-prop₃ Coeq-recβ‚‚.

QuotientsπŸ”—

With dependent sums, we can recover quotients as a special case of coequalisers. Observe that, by taking the total space of a relation R:A→A→TypeR : A \to A \to \ty, we obtain two projection maps which have as image all of the possible related elements in AA. By coequalising these projections, we obtain a space where any related objects are identified: The quotient A/RA/R.

private
  tot : βˆ€ {β„“} β†’ (A β†’ A β†’ Type β„“) β†’ Type (level-of A βŠ” β„“)
  tot {A = A} R = Σ[ x ∈ A ] Σ[ y ∈ A ] R x y

  /-left : βˆ€ {β„“} {R : A β†’ A β†’ Type β„“} β†’ tot R β†’ A
  /-left (x , _ , _) = x

  /-right : βˆ€ {β„“} {R : A β†’ A β†’ Type β„“} β†’ tot R β†’ A
  /-right (_ , x , _) = x

We form the quotient as the aforementioned coequaliser of the two projections from the total space of RR:

_/_ : βˆ€ {β„“ β„“'} (A : Type β„“) (R : A β†’ A β†’ Type β„“') β†’ Type (β„“ βŠ” β„“')
A / R = Coeq (/-left {R = R}) /-right

quot : βˆ€ {β„“ β„“'} {A : Type β„“} {R : A β†’ A β†’ Type β„“'} {x y : A} β†’ R x y
    β†’ Path (A / R) (inc x) (inc y)
quot r = glue (_ , _ , r)

Using Coeq-elim, we can recover the elimination principle for quotients:

Quot-elim : βˆ€ {β„“} {B : A / R β†’ Type β„“}
          β†’ (βˆ€ x β†’ is-set (B x))
          β†’ (f : βˆ€ x β†’ B (inc x))
          β†’ (βˆ€ x y (r : R x y) β†’ PathP (Ξ» i β†’ B (quot r i)) (f x) (f y))
          β†’ βˆ€ x β†’ B x
Quot-elim bset f r = Coeq-elim bset f Ξ» { (x , y , w) β†’ r x y w }

EffectivityπŸ”—

The most well-behaved case of quotients is when R:A→A→TypeR : A \to A \to \ty takes values in propositions, is reflexive, transitive and symmetric (an equivalence relation). In this case, we have that the quotient A/RA / R is effective: The map quot is an equivalence.

record Congruence {β„“} (A : Type β„“) β„“β€² : Type (β„“ βŠ” lsuc β„“β€²) where
  field
    _∼_         : A β†’ A β†’ Type β„“β€²
    has-is-prop : βˆ€ x y β†’ is-prop (x ∼ y)
    reflᢜ : βˆ€ {x} β†’ x ∼ x
    _βˆ™αΆœ_  : βˆ€ {x y z} β†’ x ∼ y β†’ y ∼ z β†’ x ∼ z
    symᢜ  : βˆ€ {x y}   β†’ x ∼ y β†’ y ∼ x

  relation = _∼_

  quotient : Type _
  quotient = A / _∼_

module _ {A : Type β„“} (R : Congruence A β„“') where
  private module R = Congruence R

We will show this using an encode-decode method. For each x:Ax : A, we define a type family Codex(p)\id{Code}_x(p), which represents an equality inc(x)=y\id{inc}(x) = y. Importantly, the fibre over inc(y)\id{inc}(y) will be R(x,y)R(x, y), so that the existence of functions converting between Codex(y)\id{Code}_x(y) and paths inc(x)=y\id{inc}(x) = y is enough to establish effectivity of the quotient.

  private
    Code : A β†’ R.quotient β†’ Prop β„“'
    Code x = Quot-elim
      (Ξ» x β†’ n-Type-is-hlevel 1)
      (Ξ» y β†’ el (x R.∼ y) (R.has-is-prop x y) {- 1 -})
      Ξ» y z r β†’
        n-ua (prop-ext (R.has-is-prop _ _) (R.has-is-prop _ _)
          (Ξ» z β†’ z R.βˆ™αΆœ r)
          Ξ» z β†’ z R.βˆ™αΆœ (R.symᢜ r))

We do quotient induction into the type of propositions, which by univalence is a set. Since the fibre over inc(y)\id{inc}(y) must be R(x,y)R(x, y), that’s what we give for the inc constructor ({- 1 -}, above). For this to respect the quotient, it suffices to show that, given R(y,z)R(y,z), we have R(x,y)⇔R(x,z)R(x,y) \Leftrightarrow R(x,z), which follows from the assumption that RR is an equivalence relation ({- 2 -}).

    encode : βˆ€ x y (p : inc x ≑ y) β†’ ∣ Code x y ∣
    encode x y p = subst (Ξ» y β†’ ∣ Code x y ∣) p R.reflᢜ

    decode : βˆ€ x y (p : ∣ Code x y ∣) β†’ inc x ≑ y
    decode x y p =
      Coeq-elim-prop {C = Ξ» y β†’ (p : ∣ Code x y ∣) β†’ inc x ≑ y}
        (Ξ» _ β†’ Ξ -is-hlevel 1 Ξ» _ β†’ squash _ _) (Ξ» y r β†’ quot r) y p

For encode, it suffices to transport the proof that RR is reflexive along the given proof, and for decoding, we eliminate from the quotient to a proposition. It boils down to establishing that R(x,y)β†’inc(x)≑inc(y)R(x,y) \to \id{inc}(x) \equiv \id{inc}(y), which is what the constructor quot says. Putting this all together, we get a proof that equivalence relations are effective.

  effective : βˆ€ {x y : A} β†’ is-equiv (quot {R = R.relation})
  effective {x = x} {y} =
    prop-ext (R.has-is-prop x y) (squash _ _) (decode x (inc y)) (encode x (inc y)) .snd