Small set (category theory)

In category theory, a small set is one in a fixed universe of sets (as the word universe is used in mathematics in general).

This is used when one does not wish to bother with set-theoretic concerns of what is and what is not considered a set, which concerns would arise if one tried to speak of the category of "all sets".

A small set is not to be confused with a small category, which is a category in which the collection of arrows (and therefore also the collection of objects) is a set.

We gain an intermediate notion of moderate set: a subset of the universe, which may be small or large.

Since in many cases the choice of foundations is irrelevant, it makes sense to always say “small set” for emphasis even if one has in mind a foundation where all sets are small.