Definition:Category
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Definition
A category is an interpretation of the metacategory axioms within set theory.
It consists of objects and morphisms.
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.
Generalizations
Sources
- 1964: Peter Freyd: Abelian Categories ... (previous) ... (next): Introduction
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): category: 2.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): category: 2.