What are the categories whose objects form a set, called?

Question

Small categories are ones whose objects and all the morphisms form sets.

Locally small categories are those whose hom-sets are small.

Is there a name for categories whose objects form a set, but not necessarily the morphisms?

Answer 1

As far as I'm aware, there is not an established term for large categories whose class of objects forms a set; certainly there isn't a term in common use. Presumably this is because, while it is easy to come up with artificial examples (e.g. taking small full subcategories of large categories), natural examples are harder to come by.

One way to view these different variants of categories is via enriched category theory. We can speak of small and large $V$-enriched categories (where $V$ is a monoidal category): a small $V$-category has a set of objects, whereas a large $V$-category has a class of objects. There are two (cartesian) monoidal categories of interest here: the category $\mathbf{Set}$ of sets, and the category $\mathbf{Class}$ of classes. (Exactly how one sets up these definitions depends on one's foundation.)

Then:

• A small category is precisely a small $\mathbf{Set}$-enriched category.
• A locally small category is a large $\mathbf{Set}$-enriched category.
• A large category is a large $\mathbf{Class}$-enriched category.

The concept you mention is the missing combination: a small $\mathbf{Class}$-enriched category. This would be a concise and unambiguous term to use if you need one.