Geometric realizations of undirected graphs

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

Created on 2023-02-09.
Last modified on 2023-06-10.

module graph-theory.geometric-realizations-undirected-graphs where
open import foundation.dependent-pair-types
open import foundation.equivalences
open import foundation.function-types
open import foundation.symmetric-identity-types
open import foundation.universe-levels

open import graph-theory.reflecting-maps-undirected-graphs
open import graph-theory.undirected-graphs


The geometric realization of an undirected graph G is the initial type X equipped with a reflecting map from G into X.


precomp-reflecting-map-Undirected-Graph :
  {l1 l2 l3 l4 : Level} (G : Undirected-Graph l1 l2) {X : UU l3} (Y : UU l4) 
  reflecting-map-Undirected-Graph G X 
  (X  Y)  reflecting-map-Undirected-Graph G Y
pr1 (precomp-reflecting-map-Undirected-Graph G Y f h) =
  h  vertex-reflecting-map-Undirected-Graph G f
pr2 (precomp-reflecting-map-Undirected-Graph G Y f h) v e =
  map-symmetric-Id h
    ( unordered-pair-vertices-reflecting-map-Undirected-Graph G f v)
    ( edge-reflecting-map-Undirected-Graph G f v e)

is-geometric-realization-reflecting-map-Undirected-Graph :
  {l1 l2 l3 : Level} (l : Level) (G : Undirected-Graph l1 l2) {X : UU l3}
  (f : reflecting-map-Undirected-Graph G X)  UU (l1  l2  l3  lsuc l)
is-geometric-realization-reflecting-map-Undirected-Graph l G f =
  (Y : UU l)  is-equiv (precomp-reflecting-map-Undirected-Graph G Y f)

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