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 if and only if:

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

$(2): \quad$ 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

Sources

• 1964: Peter Freyd: Abelian Categories ... (previous) ... (next): Introduction