open import Cat.Functor.FullSubcategory
open import Cat.Diagram.Equaliser
open import Cat.Diagram.Initial
open import Cat.Diagram.Pushout
open import Cat.Instances.Comma
open import Cat.Instances.Slice
open import Cat.Diagram.Image
open import Cat.Prelude

import Cat.Reasoning

module Cat.Diagram.Equaliser.RegularMono where

module _ {o ℓ} (C : Precategory o ℓ) where
  open Cat.Reasoning C
  private variable a b : Ob

Regular monomorphisms🔗

A regular monomorphism is a morphism that behaves like an embedding, i.e. it is an isomorphism onto its image. Since images of arbitrary morphisms do not exist in every category, we must find a definition which implies this property but only speaks diagrammatically about objects directly involved in the definition.

The definition is as follows: A regular monomorphism $f : a ↪ b$ is an equaliser of some pair of arrows $g, h : b \to c .$

  record is-regular-mono (f : Hom a b) : Type (o ⊔ ℓ) where
    no-eta-equality
    field
      {c}       : Ob
      arr₁ arr₂ : Hom b c
      has-is-eq : is-equaliser C arr₁ arr₂ f

    open is-equaliser has-is-eq public

From the definition we can directly conclude that regular monomorphisms are in fact monomorphisms:

    is-regular-mono→is-mono : is-monic f
    is-regular-mono→is-mono = is-equaliser→is-monic _ has-is-eq

  open is-regular-mono using (is-regular-mono→is-mono) public

Effective monomorphisms🔗

Proving that a map $f$ is a regular monomorphism involves finding two maps which it equalises, but if $C$ is a category with pushouts, there is often a canonical choice: The cokernel pair of $f,$ that is, the pushout of $f$ along with itself. Morphisms which a) have a cokernel pair and b) equalise their cokernel pair are called effective monomorphisms.

  record is-effective-mono (f : Hom a b) : Type (o ⊔ ℓ) where
    no-eta-equality
    field
      {cokernel}       : Ob
      i₁ i₂            : Hom b cokernel
      is-cokernel-pair : is-pushout C f i₁ f i₂
      has-is-equaliser : is-equaliser C i₁ i₂ f

Every effective monomorphism is a regular monomorphism, since it equalises the inclusions of its cokernel pair.

    is-effective-mono→is-regular-mono : is-regular-mono f
    is-effective-mono→is-regular-mono = rm where
      open is-regular-mono
      rm : is-regular-mono f
      rm .c = _
      rm .arr₁ = _
      rm .arr₂ = _
      rm .has-is-eq = has-is-equaliser

If $f$ has a cokernel pair, and it is a regular monomorphism, then it is also effective — it is the equaliser of its cokernel pair.

module _ {o ℓ} (C : Precategory o ℓ) where
  open Cat.Reasoning C
  open Initial
  open ↓Obj
  open ↓Hom
  open /-Obj
  open /-Hom

  is-regular-mono→is-effective-mono
    : ∀ {a b} {f : Hom a b}
    → Pushout C f f
    → is-regular-mono C f
    → is-effective-mono C f
  is-regular-mono→is-effective-mono {f = f} cokern reg = mon where
    module f⊔f = Pushout cokern
    module reg = is-regular-mono reg

Let $f : a ↪ b$ be the equaliser of $ar r_{1}, ar r_{2} : b \to c .$ By the universal property of the cokernel pair of $f,$ we have a map $φ : B ⊔_{A} B \to C$ such that $φ \circ i_{1} = ar r_{1}$ and $φ \circ i_{2} = ar r_{2} .$

    phi : Hom f⊔f.coapex reg.c
    phi = f⊔f.universal reg.equal

    open is-effective-mono
    mon : is-effective-mono C f
    mon .cokernel = f⊔f.coapex
    mon .i₁ = f⊔f.i₁
    mon .i₂ = f⊔f.i₂
    mon .is-cokernel-pair = f⊔f.has-is-po
    mon .has-is-equaliser = eq where

To show that $f$ also has the universal property of the equaliser of $i_{1}, i_{2},$ suppose that $e' : E \to b$ also equalises the injections. Then we can calculate:

ar r_{1} e' = (φ i_{1}) e' = (φ i_{2}) e' = ar r_{2} e'

So $e'$ equalises the same arrows that $f$ does, hence there is a universal map $E \to a$ which commutes with “everything in sight”:

      open is-equaliser
      eq : is-equaliser _ _ _ _
      eq .equal     = f⊔f.square
      eq .universal {F = F} {e' = e'} p = reg.universal p' where
        p' : reg.arr₁ ∘ e' ≡ reg.arr₂ ∘ e'
        p' =
          reg.arr₁ ∘ e'       ≡˘⟨ ap (_∘ e') f⊔f.universal∘i₁ ⟩
          (phi ∘ f⊔f.i₁) ∘ e' ≡⟨ extendr p ⟩
          (phi ∘ f⊔f.i₂) ∘ e' ≡⟨ ap (_∘ e') f⊔f.universal∘i₂ ⟩
          reg.arr₂ ∘ e'       ∎
      eq .factors = reg.factors
      eq .unique = reg.unique

If $f : A \to B$ has a canonical choice of pushout along itself, then it suffices to check that it equalises those injections to show it is an effective mono.

  equalises-cokernel-pair→is-effective-mono
    : ∀ {a b} {f : Hom a b}
    → (P : Pushout C f f)
    → is-equaliser C (P .Pushout.i₁) (P .Pushout.i₂) f
    → is-effective-mono C f
  equalises-cokernel-pair→is-effective-mono P eq = em where
    open is-effective-mono
    em : is-effective-mono _ _
    em .cokernel = _
    em .i₁ = _
    em .i₂ = _
    em .is-cokernel-pair = P .Pushout.has-is-po
    em .has-is-equaliser = eq

Images of regular monos🔗

Let $f : A ↪ B$ be an effective mono, or, in a category with pushouts, a regular mono. We show that $f$ admits an image relative to the class of regular monomorphisms. The construction of the image is as follows: We let $im f = A$ and factor $f$ as

A id A f B

This factorisation is very straightforwardly shown to be universal, as the code below demonstrates.

  is-effective-mono→image
    : ∀ {a b} {f : Hom a b}
    → is-effective-mono C f
    → M-image C (is-regular-mono C , is-regular-mono→is-mono) f
  is-effective-mono→image {f = f} mon = im where
    module eff = is-effective-mono mon

    itself : ↓Obj _ _
    itself .dom = tt
    itself .cod = cut f , eff.is-effective-mono→is-regular-mono
    itself .map = record { map = id ; com = idr _ }

    im : Initial _
    im .bot = itself
    im .has⊥ other = contr hom unique where
      hom : ↓Hom _ _ itself other
      hom .top = tt
      hom .bot = other .map
      hom .com = ext refl

      unique : ∀ x → hom ≡ x
      unique x = ↓Hom-path _ _ refl
        (ext (intror refl ∙ ap map (x .com) ∙ elimr refl))

Hence the characterisation of regular monomorphisms given in the introductory paragraph: In a category with pushouts, every regular monomorphism “is an isomorphism” onto its image. In reality, it gives its own image!