Library UniMath.CategoryTheory.categories.Graph
Bicategory of graphs
Benedikt Ahrens, Marco Maggesi May 2018 Revised June 2019 *********************************************************************************Require Import UniMath.Foundations.PartB.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.Combinatorics.Graph.
NB: pregraph is the same as precategory_ob_mor.
Should be moved in Combinatorics/Graph.v,
but it depends on code from CategoryTheory for now.
Definition graph_mor_eq {G H : pregraph} (p q : graph_mor G H)
(e₀ : ∏ x : vertex G, onvertex p x = onvertex q x)
(e₁ : ∏ x y (f : edge G x y),
UniMath.CategoryTheory.Core.Univalence.double_transport
(e₀ x) (e₀ y) (onedge p f) =
onedge q f)
: p = q
:= UniMath.CategoryTheory.Core.Functors.functor_data_eq G H p q e₀ e₁.
Lemma isaprop_has_edgesets (G : pregraph)
: isaprop (has_edgesets G).
Show proof.
(e₀ : ∏ x : vertex G, onvertex p x = onvertex q x)
(e₁ : ∏ x y (f : edge G x y),
UniMath.CategoryTheory.Core.Univalence.double_transport
(e₀ x) (e₀ y) (onedge p f) =
onedge q f)
: p = q
:= UniMath.CategoryTheory.Core.Functors.functor_data_eq G H p q e₀ e₁.
Lemma isaprop_has_edgesets (G : pregraph)
: isaprop (has_edgesets G).
Show proof.
Definition pregraph_precategory_ob_mor : precategory_ob_mor
:= make_precategory_ob_mor pregraph graph_mor.
Definition pregraph_precategory_data : precategory_data
:= make_precategory_data
pregraph_precategory_ob_mor
graph_mor_id
(@graph_mor_comp).
Lemma is_precategory_pregraph : is_precategory pregraph_precategory_data.
Show proof.
apply is_precategory_one_assoc_to_two.
repeat apply make_dirprod; cbn.
- exact @graph_mor_id_left.
- exact @graph_mor_id_right.
- apply @graph_mor_comp_assoc.
repeat apply make_dirprod; cbn.
- exact @graph_mor_id_left.
- exact @graph_mor_id_right.
- apply @graph_mor_comp_assoc.
Definition pregraph_category : precategory
:= make_precategory
pregraph_precategory_data
is_precategory_pregraph.
Definition graph_precategory_ob_mor : precategory_ob_mor
:= make_precategory_ob_mor graph graph_mor.
Definition graph_precategory_data : precategory_data
:= make_precategory_data
graph_precategory_ob_mor
(λ G : graph, graph_mor_id G)
(λ G H K : graph, graph_mor_comp).
Lemma is_precategory_graph : is_precategory graph_precategory_data.
Show proof.
apply is_precategory_one_assoc_to_two.
repeat apply make_dirprod; cbn.
- exact @graph_mor_id_left.
- exact @graph_mor_id_right.
- exact @graph_mor_comp_assoc.
repeat apply make_dirprod; cbn.
- exact @graph_mor_id_left.
- exact @graph_mor_id_right.
- exact @graph_mor_comp_assoc.
Definition graph_precategory : precategory
:= make_precategory
graph_precategory_data
is_precategory_graph.
Lemma has_homsets_graph : has_homsets graph_precategory_ob_mor.
Show proof.