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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 $\operatorname{Hom} \ \mathbf C \subseteq \mathfrak U$

Also see

  • Results about category theory can be found here.