Monomorphisms of concrete groups

Content created by Fredrik Bakke, Jonathan Prieto-Cubides and Egbert Rijke.

Created on 2022-07-09.
Last modified on 2023-05-03.

module group-theory.monomorphisms-concrete-groups where
open import foundation.dependent-pair-types
open import foundation.embeddings
open import foundation.propositions
open import foundation.universe-levels

open import group-theory.concrete-groups
open import group-theory.homomorphisms-concrete-groups


A monomorphism of concrete groups from G to H is a faithful pointed map BH →∗ BG. Recall that a map is said to be faithful if it induces embeddings on identity types. In particular, any faithful pointed map BH →∗ BG induces an embedding ΩBH → ΩBG, i.e., an inclusion of the underlying groups of a concrete group.

module _
  {l1 l2 : Level} (l3 : Level)
  (G : Concrete-Group l1) (H : Concrete-Group l2)
  (f : hom-Concrete-Group G H)

  is-mono-hom-Concrete-Group-Prop : Prop (l1  l2  lsuc l3)
  is-mono-hom-Concrete-Group-Prop =
      ( Concrete-Group l3)
      ( λ F  is-emb-Prop (comp-hom-Concrete-Group F G H f))

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

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

module _
  {l1 : Level} (l2 : Level) (G : Concrete-Group l1)

  mono-Concrete-Group : UU (l1  lsuc l2)
  mono-Concrete-Group =
    Σ ( Concrete-Group l2)
      ( λ H 
        Σ (hom-Concrete-Group H G) λ f  is-mono-hom-Concrete-Group l2 H G f)

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