nLab loop graph object

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 $\mathcal{C}$ with finite products is a loop digraph object $(V, E, R:E \to V \times V)$ with a morphism $i:E \to E$ such that

• $p_1 \circ R = p_2 \circ R \circ sym$
• $p_2 \circ R = p_1 \circ R \circ sym$
• $i \circ i = id_E$

where $p_1, p_2:V \times V \to V$ are the unique projection morphisms of the binary product.

In a general category

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

• $s \circ R = t \circ sym$
• $t \circ R = s \circ sym$
• $i \circ i = id_E$

Properties

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

Related concepts

• graph

• symmetric relation

• loop digraph object

• internal relation