nLab locally small category

Definition

A category is said to be locally small if each of its hom-sets is a small set, i.e., is a set instead of a proper class. Local smallness is included by some authors in the definition of “category.”

In other words, a locally small category is a $Set$-category, i.e. a category enriched in the category Set.

If Grothendieck universes are used to handle size issues, then one speaks of a locally $U$-small category if all hom-sets are elements of $U$, or of a $U\Set$-category or simply of a $U$-category.

Compare with small category; a category is small if it is locally small and its set of objects is also a set. (Some care must be taken if you want this definition to be equivalence-invariant.)

Remarks

Local smallness is an instance of a general scheme by which a category may be called “locally $P$” if all hom-sets satisfy property $P$. This is more commonly used in enriched category theory where the hom-objects have more structure than a set and can support more interesting properties.

For instance, a topologically enriched category may be said to be locally discrete if its hom-spaces have discrete topologies (hence it is essentially just an ordinary category). Likewise, a 2-category is said to be locally discrete if its hom-categories are discrete (so it is essentially an ordinary category), locally groupoidal if its hom-categories are groupoids, and so on.

However, this use of the word “locally” does not really have anything to do with the intuitive geometrical meaning of “local,” so it should not be taken too literally, especially when one is dealing with internal categories in spaces. Furthermore, in other contexts a category is often said to be “locally $P$” if it has the completely unrelated property that all its slice categories satisfy property $P$; see for instance locally cartesian closed category. Confusion is rarely created, however, because the properties $P$ that one is interested in applying to hom-sets are usually quite different from those that one applies to slice categories.

Related concepts

• small category , locally small category

• large category