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?
 A: You are correct that the first notion of U-category corresponds more
closely to non-Grothendieck-universe-based treatments, e.g. using NBG
or MK set-class theory.  To be precise, if U is a universe, defining
"set" to mean "element of U" and "class" to mean "subset of U" gives a
model of MK set-class theory (and hence also NBG, which is weaker than
MK).  A comparison of the relationships between different set-theoretic
treatments of large categories can be found in my expository paper
"Set theory for category theory."
Here is an example of one theorem that can (maybe) tell the difference
between the two notions of U-locally-small categories.  Let C be a
U-category whose hom-sets are in U (i.e. a "U-locally-small category"
by your second definition).  Then C has a Yoneda embedding C →
[Cº,Set] where Set is the U-category of U-small sets.  Note that
[Cº,Set] is only a U-category by your second definition (i.e. a
U⁺-small category).  We say that C is lex-total if this Yoneda
embedding has a left adjoint which preserves finite limits.  It is a
theorem of Freyd, which can be found in Ross Street's paper "Notions
of topos," that if C is lex-total and also U-locally-small according
to your first definition (its set of objects is a subset of U), then C
is a Grothendieck topos (i.e. the category of U-small sheaves on some
U-small site).  The converse is not hard to prove, so this gives a
characterization of Grothendieck toposes.  As far as I know, it is
unknown whether there can be lex-total U-categories with very large
object sets that are not Grothendieck toposes.
I would personally be inclined to use your second definition of
"U-locally small," because as you say it matches your preferred
definition of large category relative to U (which I would prefer to
just call a "U⁺-small category", since its definition makes no reference
to U), and also because the term "U-locally small" sounds as if it
only imposes a smallness condition locally.  Street uses "moderate"
for a category with at most a U-small set of isomorphism classes of
objects, so if one wants to state a theorem (such as the above) about
U-locally-small categories according to your first definition, one can
instead say "U-locally-small and U-moderate."
