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Equations Inline Footnotes

open import Cat.Prelude

import Cat.Reasoning

module Cat.Diagram.Coequaliser where

module _ {o ℓ} (C : Precategory o ℓ) where
  open Cat.Reasoning C
  private variable
    A B : Ob
    f g h : Hom A B

Coequalisers🔗

The coequaliser of two maps $f,g:A\to B$ (if it exists), represents the largest quotient object of $B$ that identifies $f$ and $g\text{.}$

  record is-coequaliser {E} (f g : Hom A B) (coeq : Hom B E) : Type (o ⊔ ℓ) where
    field
      coequal   : coeq ∘ f ≡ coeq ∘ g
      universal : ∀ {F} {e' : Hom B F} (p : e' ∘ f ≡ e' ∘ g) → Hom E F
      factors
        : ∀ {F} {e' : Hom B F} {p : e' ∘ f ≡ e' ∘ g}
        → universal p ∘ coeq ≡ e'
      unique
        : ∀ {F} {e' : Hom B F} {p : e' ∘ f ≡ e' ∘ g} {colim : Hom E F}
        → colim ∘ coeq ≡ e' → universal p ≡ colim

    unique₂
      : ∀ {F} {e' : Hom B F}  {o1 o2 : Hom E F}
      → (e' ∘ f ≡ e' ∘ g)
      → o1 ∘ coeq ≡ e'
      → o2 ∘ coeq ≡ e'
      → o1 ≡ o2
    unique₂ p q r = sym (unique {p = p} q) ∙ unique r

    id-coequalise : universal coequal ≡ id
    id-coequalise = unique (idl _)

There is also a convenient bundling of an coequalising arrow together with its codomain:

  record Coequaliser (f g : Hom A B) : Type (o ⊔ ℓ) where
    field
      {coapex}  : Ob
      coeq      : Hom B coapex
      has-is-coeq : is-coequaliser f g coeq

    open is-coequaliser has-is-coeq public

Coequalisers are epic🔗

Dually to the situation with equalisers, coequaliser arrows are always epic.

module _ {o ℓ} {C : Precategory o ℓ} where
  open Cat.Reasoning C
  private variable
    A B : Ob
    f g h : Hom A B

  is-coequaliser→is-epic
    : ∀ {E} (coequ : Hom A E)
    → is-coequaliser C f g coequ
    → is-epic coequ
  is-coequaliser→is-epic {f = f} {g = g} equ equalises h i p =
    h                           ≡˘⟨ unique p ⟩≡˘
    universal (extendr coequal) ≡⟨ unique refl ⟩≡
    i                           ∎
    where open is-coequaliser equalises

  coequaliser-unique
    : ∀ {E E'} {c1 : Hom A E} {c2 : Hom A E'}
    → is-coequaliser C f g c1
    → is-coequaliser C f g c2
    → E ≅ E'
  coequaliser-unique {c1 = c1} {c2} co1 co2 = make-iso
    (co1 .universal {e' = c2} (co2 .coequal))
    (co2 .universal {e' = c1} (co1 .coequal))
    (unique₂ co2 (co2 .coequal) (pullr (co2 .factors) ∙ co1 .factors) (idl _))
    (unique₂ co1 (co1 .coequal) (pullr (co1 .factors) ∙ co2 .factors) (idl _))
    where open is-coequaliser

Categories with all coequalisers🔗

We also define a helper module for working with categories that have coequalisers of all parallel pairs of morphisms.

has-coequalisers : ∀ {o ℓ} → Precategory o ℓ → Type _
has-coequalisers C = ∀ {a b} (f g : Hom a b) → Coequaliser C f g
  where open Precategory C

module Coequalisers
  {o ℓ}
  (C : Precategory o ℓ)
  (all-coequalisers : has-coequalisers C)
  where
  open Cat.Reasoning C
  module coequaliser {a b} (f g : Hom a b) = Coequaliser (all-coequalisers f g)

  Coequ : ∀ {a b} (f g : Hom a b) → Ob
  Coequ = coequaliser.coapex