Totally faithful morphisms of undirected graphs

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

Created on 2022-06-15.
Last modified on 2023-10-13.

module graph-theory.totally-faithful-morphisms-undirected-graphs where
open import foundation.dependent-pair-types
open import foundation.embeddings
open import foundation.functoriality-dependent-pair-types
open import foundation.universe-levels

open import graph-theory.morphisms-undirected-graphs
open import graph-theory.undirected-graphs


A totally faithful morphism of undirected graphs is a morphism f : G → H of undirected graphs such that for edge e in H there is at most one edge in G that f maps to e.


module _
  {l1 l2 l3 l4 : Level}
  (G : Undirected-Graph l1 l2) (H : Undirected-Graph l3 l4)

  is-totally-faithful-hom-Undirected-Graph :
    hom-Undirected-Graph G H  UU (lsuc lzero  l1  l2  l4)
  is-totally-faithful-hom-Undirected-Graph f =
    is-emb (tot (edge-hom-Undirected-Graph G H f))

  totally-faithful-hom-Undirected-Graph : UU (lsuc lzero  l1  l2  l3  l4)
  totally-faithful-hom-Undirected-Graph =
    Σ (hom-Undirected-Graph G H) is-totally-faithful-hom-Undirected-Graph

module _
  {l1 l2 l3 l4 : Level}
  (G : Undirected-Graph l1 l2) (H : Undirected-Graph l3 l4)
  (f : totally-faithful-hom-Undirected-Graph G H)

  hom-totally-faithful-hom-Undirected-Graph : hom-Undirected-Graph G H
  hom-totally-faithful-hom-Undirected-Graph = pr1 f

  vertex-totally-faithful-hom-Undirected-Graph :
    vertex-Undirected-Graph G  vertex-Undirected-Graph H
  vertex-totally-faithful-hom-Undirected-Graph =
    vertex-hom-Undirected-Graph G H hom-totally-faithful-hom-Undirected-Graph

  unordered-pair-vertices-totally-faithful-hom-Undirected-Graph :
    unordered-pair-vertices-Undirected-Graph G 
    unordered-pair-vertices-Undirected-Graph H
  unordered-pair-vertices-totally-faithful-hom-Undirected-Graph =
    unordered-pair-vertices-hom-Undirected-Graph G H

  edge-totally-faithful-hom-Undirected-Graph :
    (p : unordered-pair-vertices-Undirected-Graph G) 
    edge-Undirected-Graph G p 
    edge-Undirected-Graph H
      ( unordered-pair-vertices-totally-faithful-hom-Undirected-Graph p)
  edge-totally-faithful-hom-Undirected-Graph =
    edge-hom-Undirected-Graph G H hom-totally-faithful-hom-Undirected-Graph

  is-totally-faithful-totally-faithful-hom-Undirected-Graph :
    is-totally-faithful-hom-Undirected-Graph G H
  is-totally-faithful-totally-faithful-hom-Undirected-Graph = pr2 f

See also

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