This is an archived version pinned as of the submission of my master's thesis. An up-to-date version may be found online.

Monomorphisms

module foundation.monomorphisms where

open import foundation.action-on-identifications-functions
open import foundation.dependent-pair-types
open import foundation.embeddings
open import foundation.function-extensionality
open import foundation.functoriality-function-types
open import foundation.postcomposition-functions
open import foundation.universe-levels
open import foundation.whiskering-homotopies-composition

open import foundation-core.equivalences
open import foundation-core.function-types
open import foundation-core.homotopies
open import foundation-core.identity-types
open import foundation-core.propositional-maps
open import foundation-core.propositions
open import foundation-core.truncation-levels

Idea

A function f : A → B is a monomorphism if whenever we have two functions g h : X → A such that f ∘ g = f ∘ h, then in fact g = h. The way to state this in Homotopy Type Theory is to say that postcomposition by f is an embedding.

Definition

module _
  {l1 l2 : Level} (l3 : Level)
  {A : UU l1} {B : UU l2} (f : A → B)
  where

  is-mono-Prop : Prop (l1 ⊔ l2 ⊔ lsuc l3)
  is-mono-Prop = Π-Prop (UU l3) λ X → is-emb-Prop (postcomp X f)

  is-mono : UU (l1 ⊔ l2 ⊔ lsuc l3)
  is-mono = type-Prop is-mono-Prop

  is-prop-is-mono : is-prop is-mono
  is-prop-is-mono = is-prop-type-Prop is-mono-Prop

Properties

If f : A → B is a monomorphism then for any g h : X → A we have an equivalence (f ∘ g = f ∘ h) ≃ (g = h). In particular, if f ∘ g = f ∘ h then g = h.

module _
  {l1 l2 : Level} (l3 : Level)
  {A : UU l1} {B : UU l2} (f : A → B)
  (p : is-mono l3 f) {X : UU l3} (g h : X → A)
  where

  equiv-postcomp-is-mono : (g ＝ h) ≃ ((f ∘ g) ＝ (f ∘ h))
  pr1 equiv-postcomp-is-mono = ap (f ∘_)
  pr2 equiv-postcomp-is-mono = p X g h

  is-injective-postcomp-is-mono : (f ∘ g) ＝ (f ∘ h) → g ＝ h
  is-injective-postcomp-is-mono = map-inv-equiv equiv-postcomp-is-mono

A function is a monomorphism if and only if it is an embedding.

module _
  {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B)
  where

  is-mono-is-emb : is-emb f → {l3 : Level} → is-mono l3 f
  is-mono-is-emb is-emb-f X =
    is-emb-is-prop-map
      ( is-trunc-map-postcomp-is-trunc-map neg-one-𝕋 f
        ( is-prop-map-is-emb is-emb-f)
        ( X))

  is-emb-is-mono : ({l3 : Level} → is-mono l3 f) → is-emb f
  is-emb-is-mono is-mono-f =
    is-emb-is-prop-map
      ( is-trunc-map-is-trunc-map-postcomp neg-one-𝕋 f
        ( λ X → is-prop-map-is-emb (is-mono-f X)))

We construct an explicit equivalence for postcomposition for homotopies between functions (rather than equality) when the map is an embedding.

module _
  {l1 l2 l3 : Level}
  {A : UU l1} {B : UU l2} (f : A ↪ B)
  {X : UU l3} (g h : X → A)
  where

  map-inv-equiv-htpy-postcomp-is-emb :
    (pr1 f ∘ g) ~ (pr1 f ∘ h) → g ~ h
  map-inv-equiv-htpy-postcomp-is-emb H x =
    map-inv-is-equiv (pr2 f (g x) (h x)) (H x)

  is-section-map-inv-equiv-htpy-postcomp-is-emb :
    (pr1 f ·l_) ∘ map-inv-equiv-htpy-postcomp-is-emb ~ id
  is-section-map-inv-equiv-htpy-postcomp-is-emb H =
    eq-htpy (λ x →
      is-section-map-inv-is-equiv (pr2 f (g x) (h x)) (H x))

  is-retraction-map-inv-equiv-htpy-postcomp-is-emb :
    map-inv-equiv-htpy-postcomp-is-emb ∘ (pr1 f ·l_) ~ id
  is-retraction-map-inv-equiv-htpy-postcomp-is-emb H =
    eq-htpy (λ x →
      is-retraction-map-inv-is-equiv (pr2 f (g x) (h x)) (H x))

  equiv-htpy-postcomp-is-emb :
    (g ~ h) ≃ ((pr1 f ∘ g) ~ (pr1 f ∘ h))
  pr1 equiv-htpy-postcomp-is-emb = pr1 f ·l_
  pr2 equiv-htpy-postcomp-is-emb =
    is-equiv-is-invertible
      map-inv-equiv-htpy-postcomp-is-emb
      is-section-map-inv-equiv-htpy-postcomp-is-emb
      is-retraction-map-inv-equiv-htpy-postcomp-is-emb