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:

$(1): \quad$ The objects form a subset $\mathcal C_0$ or $\operatorname{ob} \ \mathcal C \subseteq \mathfrak U$

$(2): \quad$ 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.