Definition:Category

Definition
A category is an interpretation of the metacategory axioms within set theory.

Because a metacategory is a metagraph, this means that a category is a graph.

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

A metacategory $\mathcal C$ is a category if:


 * 1. The objects form a subset $\mathcal C_0$ or $\operatorname{ob}\mathcal C \subseteq \mathfrak U$


 * 2. The morphisms form a subset $\mathcal C_1$ or $\operatorname{mor}\mathcal C$ or $\operatorname{Hom}\mathcal C \subseteq \mathfrak U$

If the class $\mathfrak U$ is a set, then morphisms are functions, and the domain, codomain and composition in the definition of a metacategory 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.