nLab
loop graph object

Contents

Contents

Idea

A loop graph object internal to a category with finite products is an object that behaves in that category like loop graphs do in Set.

Definition

In a category with finite products

A loop graph object in a category 𝒞\mathcal{C} with finite products is a loop digraph object (V,E,R:EV×V)(V, E, R:E \to V \times V) with a morphism i:EEi:E \to E such that

  • p 1R=p 2Rsymp_1 \circ R = p_2 \circ R \circ sym
  • p 2R=p 1Rsymp_2 \circ R = p_1 \circ R \circ sym
  • ii=id Ei \circ i = id_E

where p 1,p 2:V×VVp_1, p_2:V \times V \to V are the unique projection morphisms of the binary product.

In a general category

A loop graph object in a category 𝒞\mathcal{C} is a loop digraph object (V,E,s:EV,t:EV)(V, E, s:E \to V, t:E \to V) with a morphism i:EEi:E \to E such that

  • sR=tsyms \circ R = t \circ sym
  • tR=ssymt \circ R = s \circ sym
  • ii=id Ei \circ i = id_E

Properties

A loop graph object is equivalently an object with an internal symmetric binary endorelation.

Created on May 13, 2022 at 22:16:53. See the history of this page for a list of all contributions to it.