Small category

A category $\mathfrak{K}$ whose class of morphisms $\text{Mor}\,\mathfrak{K}$ is a set. A small category $\mathfrak{K}$ is called a $U$-category if $\text{Mor}\,\mathfrak{K} \subset U$, where $U$ is a universe. For a small category $\mathfrak{K}$ and an arbitrary locally small category $\mathfrak{C}$ the category of covariant (contravariant) functors (cf. Functor) from $\mathfrak{K}$ to $\mathfrak{C}$ is locally small. In particular, the small categories form the closed category $\textsf{Cat}$ of small categories, one of the basic categories of mathematics [1].

References

[1]	F.W. Lawvere, "The category of categories as a foundation for mathematics" S. Eilenberg (ed.) et al. (ed.) , Proc. conf. categorical algebra (La Jolla, 1965) , Springer (1966) pp. 1–20

Comments

A category is called locally small if, for any pair of objects $A$ and $B$, the class of morphisms from $A$ to $B$ is a set. (Some authors assume this condition as part of the definition of a category.) A locally small category is small if and only if its class of objects is a set.

Cf. also Universe.