Definition:Locally Small Category
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Definition
Let $\mathbf C$ be a metacategory.
Then $\mathbf C$ is said to be locally small if and only if all of its hom classes are sets.
That is, if and only if for all objects $X, Y \in \mathbf C_0$ of $\mathbf C$:
- $\operatorname{Hom}_{\mathbf C} \left({X, Y}\right) = \set {f \in \mathbf C_1: f: X \to Y}$
is a set.
Sources
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.8$: Definition $1.12$