Definition:Graph (Category Theory)

Definition
A graph is an interpretation of a metagraph within set theory.

Let $\mathfrak U$ be a class of sets.

A metagraph $\mathcal D$ is a graph if:


 * 1. The objects form a subset $\mathcal G_0$ or $\operatorname{ob}\mathcal G \subseteq \mathfrak U$


 * 2. The morphisms form a subset $\mathcal G_1$ or $\operatorname{mor}\mathcal G$ or $\operatorname{Hom}\mathcal G \subseteq \mathfrak U$

If the class $\mathfrak U$ is a set, then morphisms are functions, and the domain and codomain in the definition of a morphism are those familiar from set theory.

If $\mathfrak U$ is a proper class this is not the case, for example the morphisms of $\mathcal C$ need not be functions.