Let $\mathcal{U}$ be a universe (so is, in particular, a set of sets).

A set $S$ is said to be $\mathcal{U}$ -small if it is isomorphic to an element of $\mathcal{U}$ (i.e., there is a bijection between $S$ and some element of $\mathcal{U}$ ).

A category $C$ is $\mathcal{U}$ -small (or just small, if no confusion is likely to arise) if the set of objects of $C$ is isomorphic to a set in $\mathcal{U}$ , and is a $\mathcal{U}$ -category if for every pair of objects $A$ , $B$ in $C$ , $\Hom(A,B)$ is isomorphic to a set in $\mathcal{U}$ .

These definitions amount to restrictions on the cardinality of the objects involved, and are intended to provide a condition that will allow operations such as extracting the category of functors or taking the direct limit to give results that are reasonable, that is, either isomorphic to an object of $\mathcal{U}$ or made up of objects of $\mathcal{U}$ .

Observe that the category of subsets of $\mathcal{U}$ is a $\mathcal{U}$ -category but is not $\mathcal{U}$ -small.

Bibliography

SGA4

Grothendieck et al., Séminaires en Gèometrie Algèbrique 4, tomes 1, 2, and 3.

Mur68

Murphy, O. Some modern methods in the theory of lion hunting, American Mathematical Monthly 75 (2), Feb., 1968, 185-187.