Monomorphisms in the category of groups

module group-theory.monomorphisms-groups where

Imports

open import category-theory.monomorphisms-large-precategories

open import foundation.propositions
open import foundation.universe-levels

open import group-theory.groups
open import group-theory.homomorphisms-groups
open import group-theory.isomorphisms-groups
open import group-theory.precategory-of-groups

Idea

A group homomorphism f : x → y is a monomorphism if whenever we have two group homomorphisms g h : w → x 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) (G : Group l1)
  (H : Group l2) (f : type-hom-Group G H)
  where

  is-mono-Group-Prop : Prop (l1 ⊔ l2 ⊔ lsuc l3)
  is-mono-Group-Prop =
    is-mono-Large-Precategory-Prop Group-Large-Precategory l3 G H f

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

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

Properties

Isomorphisms are monomorphisms

module _
  {l1 l2 : Level} (l3 : Level) (G : Group l1)
  (H : Group l2) (f : type-iso-Group G H)
  where

  is-mono-iso-Group :
    is-mono-Group l3 G H (hom-iso-Group G H f)
  is-mono-iso-Group =
    is-mono-iso-Large-Precategory Group-Large-Precategory l3 G H f