open import Cat.Instances.Shape.Parallel
open import Cat.Instances.Shape.Terminal
open import Cat.Diagram.Limit.Base
open import Cat.Diagram.Equaliser
open import Cat.Functor.Kan.Base
open import Cat.Prelude

module Cat.Diagram.Limit.Equaliser {o h} (C : Precategory o h) where

open import Cat.Reasoning C

open is-equaliser
open Equaliser
open Functor
open _=>_


We establish the correspondence between Equaliser and the Limit of a parallel arrows diagram.

is-equaliser→is-limit
: ∀ {e} (F : Functor ·⇉· C) {equ : Hom e (F .F₀ false)}
→ (eq : is-equaliser C (forkl F) (forkr F) equ)
→ is-limit {C = C} F e (Fork→Cone F (is-equaliser.equal eq))
is-equaliser→is-limit {e} F {equ} is-eq =
to-is-limitp ml λ where
{true} → refl
{false} → refl
where
module is-eq = is-equaliser is-eq
open make-is-limit

ml : make-is-limit F e
ml .ψ true = forkl F ∘ equ
ml .ψ false = equ
ml .commutes {true} {true} tt = eliml (F .F-id)
ml .commutes {false} {true} true = sym is-eq.equal
ml .commutes {false} {true} false = refl
ml .commutes {false} {false} tt = eliml (F .F-id)
ml .universal eps p =
is-eq.universal (p {false} {true} false ∙ sym (p {false} {true} true))
ml .factors {true} eps p =
pullr is-eq.factors ∙ p {false} {true} false
ml .factors {false} eps p =
is-eq.factors
ml .unique eps p other q =
is-eq.unique (q false)

is-limit→is-equaliser
: ∀ (F : Functor ·⇉· C) {K : Functor ⊤Cat C}
→ {eps : K F∘ !F => F}
→ is-ran !F F K eps
→ is-equaliser C (forkl F) (forkr F) (eps .η false)
is-limit→is-equaliser F {K} {eps} lim = eq where
module lim = is-limit lim

parallel
: ∀ {x} → Hom x (F .F₀ false)
→ (j : Bool) → Hom x (F .F₀ j)
parallel e' true = forkl F ∘ e'
parallel e' false = e'

parallel-commutes
: ∀ {x} {e' : Hom x (F .F₀ false)}
→ forkl F ∘ e' ≡ forkr F ∘ e'
→ ∀ i j → (h : ·⇉· .Precategory.Hom i j)
→ F .F₁ {i} {j} h ∘ parallel e' i ≡ parallel e' j
parallel-commutes p true true tt = eliml (F .F-id)
parallel-commutes p false true true = sym p
parallel-commutes p false true false = refl
parallel-commutes p false false tt = eliml (F .F-id)

eq : is-equaliser C (forkl F) (forkr F) (eps .η false)
eq .equal =
sym (eps .is-natural false true false) ∙ eps .is-natural false true true
eq .universal {e' = e'} p =
lim.universal (parallel e') (λ {i} {j} h → parallel-commutes p i j h)
eq .factors = lim.factors {j = false} _ _
eq .unique {p = p} {other = other} q =
lim.unique _ _ _ λ where
true →
ap (_∘ other) (intror (K .F-id) ∙ eps .is-natural false true true)
·· pullr q
·· sym p
false → q

Equaliser→Limit : ∀ {F : Functor ·⇉· C} → Equaliser C (forkl F) (forkr F) → Limit F
Equaliser→Limit {F = F} eq = to-limit (is-equaliser→is-limit F (has-is-eq eq))

Limit→Equaliser : ∀ {a b} {f g : Hom a b} → Limit {C = C} (Fork f g) → Equaliser C f g
Limit→Equaliser lim .apex = _
Limit→Equaliser lim .equ = _
Limit→Equaliser {f = f} {g} lim .has-is-eq =
is-limit→is-equaliser (Fork f g) (Limit.has-limit lim)