If you’re working with universes, then the same trick lets you make a vector space in one universe whose basis is all topological spaces in all smaller universes.

Unfortunately, I don’t know enough about the other stuff to really understand why you want these things.

As for the n-category stuff, is it really foundationally any more difficult than doing it for 1-categories? You just need a class of objects, a class of 1-morphisms, …, and a class of n-morphisms, and you just need to write down whatever associativity axioms they need to satisfy. (That’s a problem for category theorists to solve, but I don’t see how it would mess up the set/class stuff underlying it.) I guess at the omega-category level things might get tougher because of the apparent non-well-foundedness of what’s going on.

Presumably Jacob Lurie has written about these foundational issues in his tome on the subject?

So instead, let me ask when we (pretending that I’m a category theorist) actually care about this. For one, we do want the category of all categories. Almost. In fact, the collection of all categories has more structure than just being a category: it has objects (categories), morphisms (functors), and 2-morphisms (natural transformations). And, in general, we really do want to be able to talk about n-categories, requiring that our classes go up and up and up.

And really we want omega-categories. The natural definition of “omega-category” is (i) a collection of objects (ii) between any two objects there is an omega category of morphisms. I.e. the natural definition of “omega category” is “(weak) category enriched over OmegaCat”. I say “weak category” because the point is that composition, for instance, is not associative in an omega-category, but only associative up to something. So really, in the current language, the definition of an “omega-category” is that it is an omega-category enriched over OmegaCat. Anyway, so the natural structure of OmegaCat is as an omega-category, not as an (omega+1)-category, at least as far as I can tell.

So that’s one thing that these proposals don’t deal with: we really want infinite chains of type-increase.

But is there ever a time when a mathematician would want to really think much about functors that themselves break levels? To MacLane, a group, for instance, is always a small category (as it has only one object) and always a small group: a “category” is defined as “a category enriched over Set”, so there must only be a (small) set of morphisms between any pair of objects. In any case, we do want the talk about the category of representations of a group; i.e. the functor category from (small) Grp to (large) Vect. And in general people ask about representations of a category, meaning functors into Vect; e.g. (T)QFTs. But I don’t know if anyone has really wanted to know about representations of Cat, say. We do talk about functors, say, between Cat to BiCat (the former being a 2-category, latter being the collection (weak) 2-categories, equipped with its natural structure as a 3-category). For instance, there is a functor Cat \to BiCat that introduces only identity 2-morphisms, and its adjoint functor from BiCat to Cat that decategorifies by modding out by all 2-morphisms. (I don’t remember how adjoint these are; in one direction the composition is the identity, and in the other? And, of course, Cat and BiCat are not really in the same n-category, so we have to be precise what kind of functor we mean…)

Certainly we do want to do things like quantizing and q-deforming Cat; doing so may require, e.g., allowing all (N- or C-)linear combinations of categories. So y’all logic people better give us a language that lets us make the vector space with basis _all categories_. Or at least _all topological spaces_ or something.