module Cat.Diagram.Equaliser.RegularMono {o β„“} (C : Precategory o β„“) where

Regular monomorphismsπŸ”—

A regular monomorphism is a morphism that behaves like an embedding, i.e.Β 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 is an equaliser of some pair of arrows

record is-regular-mono (f : Hom a b) : Type (o βŠ” β„“) where
    {c}       : Ob
    arr₁ arrβ‚‚ : Hom b c
    has-is-eq : is-equaliser C arr₁ arrβ‚‚ 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 C _ has-is-eq

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

Effective monomorphismsπŸ”—

Proving that a map is a regular monomorphism involves finding two maps which it equalises, but if is a category with pushouts, there is often a canonical choice: The cokernel pair of that is, the pushout of 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 βŠ” β„“) where
    {cokernel}       : Ob
    i₁ iβ‚‚            : Hom b cokernel
    is-cokernel-pair : is-pushout C f i₁ f iβ‚‚
    has-is-equaliser : is-equaliser C i₁ iβ‚‚ 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 .arrβ‚‚ = _
    rm .has-is-eq = has-is-equaliser

If has a cokernel pair, and it is a regular monomorphism, then it is also effective β€” it is the equaliser of its cokernel pair.

  : βˆ€ {a b} {f : Hom a b}
  β†’ Pushout C f f
  β†’ is-regular-mono f
  β†’ is-effective-mono f
is-regular-mono→is-effective-mono {f = f} cokern reg = mon where
  module fβŠ”f = Pushout cokern
  module reg = is-regular-mono reg

Let be the equaliser of By the universal property of the cokernel pair of we have a map such that and

  phi : Hom fβŠ”f.coapex reg.c
  phi = fβŠ”f.universal reg.equal

  open is-effective-mono
  mon : is-effective-mono f
  mon .cokernel = fβŠ”f.coapex
  mon .i₁ = fβŠ”f.i₁
  mon .iβ‚‚ = fβŠ”f.iβ‚‚
  mon .is-cokernel-pair = fβŠ”f.has-is-po
  mon .has-is-equaliser = eq where

To show that also has the universal property of the equaliser of suppose that also equalises the injections. Then we can calculate:

So equalises the same arrows that does, hence there is a universal map which commutes with β€œeverything in sight”:

    open is-equaliser
    eq : is-equaliser _ _ _ _
    eq .equal     = fβŠ”f.square
    eq .universal {F = F} {e' = e'} p = reg.universal p' where
      p' : reg.arr₁ ∘ e' ≑ reg.arrβ‚‚ ∘ e'
      p' =
        reg.arr₁ ∘ e'       β‰‘Λ˜βŸ¨ ap (_∘ e') fβŠ”f.iβ‚βˆ˜universal βŸ©β‰‘Λ˜
        (phi ∘ fβŠ”f.i₁) ∘ e' β‰‘βŸ¨ extendr p βŸ©β‰‘
        (phi ∘ fβŠ”f.iβ‚‚) ∘ e' β‰‘βŸ¨ ap (_∘ e') fβŠ”f.iβ‚‚βˆ˜universal βŸ©β‰‘
        reg.arrβ‚‚ ∘ e'       ∎
    eq .factors = reg.factors
    eq .unique = reg.unique

If 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.

  : βˆ€ {a b} {f : Hom a b}
  β†’ (P : Pushout C f f)
  β†’ is-equaliser C (P .Pushout.i₁) (P .Pushout.iβ‚‚) f
  β†’ is-effective-mono f
equalises-cokernel-pair→is-effective-mono P eq = em where
  open is-effective-mono
  em : is-effective-mono _
  em .cokernel = _
  em .i₁ = _
  em .iβ‚‚ = _
  em .is-cokernel-pair = P .Pushout.has-is-po
  em .has-is-equaliser = eq

Images of regular monosπŸ”—

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

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

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

  itself : ↓Obj _ _
  itself .x = tt
  itself .y = restrict (cut f)→is-regular-mono
  itself .map = record { map = id ; commutes = idr _ }

  im : Initial _
  im .bot = itself
  im .hasβŠ₯ other = contr hom unique where
    hom : ↓Hom _ _ itself other
    hom .Ξ± = tt
    hom .Ξ² = other .map
    hom .sq = trivial!

    unique : βˆ€ x β†’ hom ≑ x
    unique x = ↓Hom-path _ _ refl
      (ext (intror refl βˆ™ ap map (x .sq) βˆ™ elimr refl))

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