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 $\mathbf C$ is a category :


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


 * $(2): \quad$ The morphisms form a subset $\mathbf C_1$ or $\operatorname{mor} \ \mathbf C$ or $\operatorname{Hom} \ \mathbf C \subseteq \mathfrak U$

A category is what one needs to define in order to define a functor.

Also see

 * Definition:Small Category

Generalizations

 * Definition:Monoidally Enriched Category