open import 1Lab.Prelude

module Data.Set.Coequaliser where


# Set Coequalisers🔗

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

data Coeq (f g : A → B) : Type (level-of A ⊔ level-of B) where
inc    : B → Coeq f g
glue   : ∀ x → 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 $\id{coeq}(f,g)$ are maps out of $B$, satisfying a certain property. Specifically, for a map $h : B \to C$, if we have $h \circ f = h \circ g$, then the map $f$ factors (uniquely) through inc. The situation can be summarised with the diagram below. We refer to this unique factoring as Coeq-rec.

Coeq-rec : ∀ {ℓ} {C : Type ℓ} {f g : A → B}
→ is-set C → (h : B → C)
→ (∀ x → h (f x) ≡ h (g x)) → Coeq f g → 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) (λ i → g (p i)) (λ i → 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 “space of maps $B \to C$ which coequalise $f$ and $g$” and the maps $\id{coeq}(f,g) \to C$. In this sense, inc is the universal map which coequalises $f$ and $g$.

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 : ∀ {ℓ} {f g : A → B} {C : Coeq f g → Type ℓ}
→ (∀ x → is-prop (C x))
→ (∀ x → C (inc x))
→ ∀ x → 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))
(λ i → g x) (λ i → g (p i)) (λ i → g (q i)) (λ i → g y) i j
where g = Coeq-elim-prop cprop cinc


Since “the space of maps $h : B \to C$ which coequalise $f$ and $g$” 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 : ∀ {ℓ} (f g : A → B) → Type ℓ → Type _
coeq-cone {B = B} f g C = Σ[ h ∈ (B → C) ] (h ∘ f ≡ h ∘ g)


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

Coeq-univ : ∀ {ℓ} {C : Type ℓ} {f g : A → B}
→ is-set C
→ is-equiv {A = Coeq f g → C} {B = coeq-cone f g C}
(λ h → h ∘ inc , λ i z → 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 → Coeq f g → C
cr' (f , f-coeqs) = Coeq-rec cset f (happly f-coeqs)

islinv : is-left-inverse cr' (λ h → h ∘ inc , λ i z → h (glue z i))
islinv f = funext (Coeq-elim-prop (λ x → cset _ _) λ x → refl)


# Elimination🔗

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

Coeq-elim : ∀ {ℓ} {f g : A → B} {C : Coeq f g → Type ℓ}
→ (∀ x → is-set (C x))
→ (ci : ∀ x → C (inc x))
→ (∀ x → PathP (λ i → C (glue x i)) (ci (f x)) (ci (g x)))
→ ∀ x → 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))
(λ i → g x) (λ i → g (p i)) (λ i → g (q i)) (λ i → g y) i j
where g = Coeq-elim cset ci cg


There is a barrage of combined eliminators, whose definitions are not very enlightening — you can mouse over these links to see their types: Coeq-elim-prop₂ Coeq-elim-prop₃ Coeq-elim₂ Coeq-rec₂.

# Quotients🔗

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

private
tot : ∀ {ℓ} → (A → A → Type ℓ) → Type (level-of A ⊔ ℓ)
tot {A = A} R = Σ[ x ∈ A ] Σ[ y ∈ A ] R x y

/-left : ∀ {ℓ} {R : A → A → Type ℓ} → tot R → A
/-left (x , _ , _) = x

/-right : ∀ {ℓ} {R : A → A → Type ℓ} → tot R → A
/-right (_ , x , _) = x


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

_/_ : ∀ {ℓ ℓ'} (A : Type ℓ) (R : A → A → Type ℓ') → Type (ℓ ⊔ ℓ')
A / R = Coeq (/-left {R = R}) /-right

quot : ∀ {ℓ ℓ'} {A : Type ℓ} {R : A → A → Type ℓ'} {x y : A} → R x y
→ Path (A / R) (inc x) (inc y)
quot r = glue (_ , _ , r)


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

Quot-elim : ∀ {ℓ} {B : A / R → Type ℓ}
→ (∀ x → is-set (B x))
→ (f : ∀ x → B (inc x))
→ (∀ x y (r : R x y) → PathP (λ i → B (quot r i)) (f x) (f y))
→ ∀ x → B x
Quot-elim bset f r = Coeq-elim bset f λ { (x , y , w) → r x y w }


## Effectivity🔗

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

module _ {A : Type ℓ} {R : A → A → Type ℓ'}
(Rp : ∀ x y → is-prop (R x y))
(rr : ∀ {x} → R x x)
(rt : ∀ {x y z} → R x y → R y z → R x z)
(rs : ∀ {x y} → R x y → R y x)
where


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

  private
Code : A → A / R → Prop ℓ'
Code x = Quot-elim
(λ x → n-Type-is-hlevel 1)
(λ y → {- 1 -} R x y , Rp x y)
λ y z r →
n-ua (prop-ext (Rp _ _) (Rp _ _) (λ z → rt z r) λ z → rt z (rs r))


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

    encode : ∀ x y (p : inc x ≡ y) → ∣ Code x y ∣
encode x y p = subst (λ y → ∣ Code x y ∣) p rr

decode : ∀ x y (p : ∣ Code x y ∣) → inc x ≡ y
decode x y p =
Coeq-elim-prop {C = λ y → (p : ∣ Code x y ∣) → inc x ≡ y}
(λ _ → Π-is-hlevel 1 λ _ → squash _ _) (λ y r → quot r) y p


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

  effective : ∀ {x y : A} → is-equiv (quot {R = R})
effective {x = x} {y} =
prop-ext (Rp x y) (squash _ _) (decode x (inc y)) (encode x (inc y)) .snd