open import Cat.Prelude

module Cat.Diagram.Equaliser {ℓ ℓ′} (C : Precategory ℓ ℓ′) where

Equalisers🔗

The equaliser of two maps $f, g : A \to B$, when it exists, represents the largest subobject of $A$ where $f$ and $g$ agree. In this sense, the equaliser is the categorical generalisation of a solution set: The solution set of an equation in one variable is largest subset of the domain (i.e. what the variable ranges over) where the left- and right-hand-sides agree.

record is-equaliser {E} (f g : Hom A B) (equ : Hom E A) : Type (ℓ ⊔ ℓ′) where
  field
    equal     : f ∘ equ ≡ g ∘ equ
    limiting  : ∀ {F} {e′ : Hom F A} (p : f ∘ e′ ≡ g ∘ e′) → Hom F E
    universal : ∀ {F} {e′ : Hom F A} {p : f ∘ e′ ≡ g ∘ e′} → equ ∘ limiting p ≡ e′
    unique
      : ∀ {F} {e′ : Hom F A} {p : f ∘ e′ ≡ g ∘ e′} {lim' : Hom F E}
      → e′ ≡ equ ∘ lim'
      → lim' ≡ limiting p

  equal-∘ : f ∘ equ ∘ h ≡ g ∘ equ ∘ h
  equal-∘ {h = h} =
    f ∘ equ ∘ h ≡⟨ extendl equal ⟩≡
    g ∘ equ ∘ h ∎

  unique₂
    : ∀ {F} {e′ : Hom F A} {p : f ∘ e′ ≡ g ∘ e′} {lim' lim'' : Hom F E}
    → e′ ≡ equ ∘ lim'
    → e′ ≡ equ ∘ lim''
    → lim' ≡ lim''
  unique₂ {p = p} q r = unique {p = p} q ∙ sym (unique r)

We can visualise the situation using the commutative diagram below:

There is also a convenient bundling of an equalising arrow together with its domain:

record Equaliser (f g : Hom A B) : Type (ℓ ⊔ ℓ′) where
  field
    {apex}  : Ob
    equ     : Hom apex A
    has-is-eq : is-equaliser f g equ

  open is-equaliser has-is-eq public

Equalisers are monic🔗

As a small initial application, we prove that equaliser arrows are always monic:

is-equaliser→is-monic
  : ∀ {E} (equ : Hom E A)
  → is-equaliser f g equ
  → is-monic equ
is-equaliser→is-monic equ equalises g h p =
  g                ≡⟨ unique (sym p) ⟩≡
  limiting equal-∘ ≡˘⟨ unique refl ⟩≡˘
  h ∎
  where open is-equaliser equalises