# How should we define “locally small”?

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?

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In the definition of U-locally small, shouldn't it be "Hom-sets" instead of "set of morphisms"? – Peter Arndt Oct 30 '09 at 17:32
Yes, thank you! Fixed. – Reid Barton Oct 30 '09 at 19:12