Let U be a Grothendieck universe, and U+ its successor universe (assume Grothendieck's universe axiom).
Everybody agrees that a U-small category is a category whose sets of objects and morphisms are both elements of U. For the "next larger size" of categories which are not necessarily even locally small, call them just U-categories, there are two possible definitions:
- a category whose set of objects and Hom-sets are all subsets of U;
- a category whose set of objects and Hom-sets are all elements of U+ (U+-small categories).
I quite prefer the second notion, so that the category of U-categories is cartesian closed and we can form localizations. This is the usage of Dwyer-Hirschhorn-Kan-Smith, "Homotopy Limit Functors on Model Categories and Homotopical Categories". I think the first more closely corresponds to non-Grothendieck universe-based treatments of category theory using sets and classes, but I might be wrong about that.
For U-locally small categories there are again two possible definitions:
- a category whose set of objects is a subset of U and whose Hom-sets are elements of U,
- a category whose set of objects is an element of U+ and whose Hom-sets are elements of U.
I don't see a strong reason to prefer one over the other, except that the second is more parallel with my preference for U-categories. DHKS uses the first. As an example of the difference between them, if I have a U-locally small category C, I can form the category (poset) of full subcategories of C; this is U-locally small under the second definition, but not the first. Is this a good thing or a bad thing? Or are there no theorems I would care about that are affected by this difference? Does anyone have an opinion about these two definitions?