nLab loop digraph object

Idea

A loop digraph object internal to a category is an object that behaves in that category like loop digraphs do in Set.

 Definition

In a category with finite products

A loop digraph object in a category $\mathcal{C}$ with finite products is

• a object $V$ in $\mathcal{C}$

• an object $E$ in $\mathcal{C}$

• a morphism $R:E \to V \times V$

such that $R$ is monic: for every object $A$ in $\mathcal{C}$ and morphisms $f: A \to E$ and $g: A \to E$, $R \circ f = R \circ g$ implies that $f = g$.

In general categories

A loop digraph object in a category $\mathcal{C}$ is

• a object $V$ in $\mathcal{C}$

• an object $E$ in $\mathcal{C}$

• a morphism $s:E \to V$

• a morphism $t:E \to V$

such that $s$ and $t$ are jointly monic: for every object $A$ in $\mathcal{C}$ and morphisms $f: A \to E$ and $g: A \to E$, $s \circ f = s \circ g$ and $t \circ f = t \circ g$ imply that $f = g$.

Properties

A loop digraph object is equivalently an object with an internal binary endorelation.

Related concepts

• graph

• digraph

• relation

• loop graph object

• internal relation