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I've been reading Abstract and Concrete Categories: The Joy of Cats by Adámek, et al. They take a slightly different approach to defining categories, since they operate with a theory of collections that has three sorts: sets, classes (collections "too big" to be sets), and conglomerates (collections "too big" to be classes).

They don't give a definition of "locally small category" (unless I missed it) but they distinguish between quasicategories and categories. The way they define categories simpliciter seems to coincide with how I've seen locally small categories defined -- a class-sized collection of objects with only hom-sets (no "hom-classes") for the collections of arrows between objects.

A consequence of this (I believe) is that the collection of morphisms of a category will be at most class-sized. This comports with the usual two sorted collection approaches to category theory that I've seen since they don't have any larger-than-class-sized notion of collection so they wouldn't be able to make sense of anything larger than a class-sized collection of morphisms anyway.

This got me to wondering what the nice-making properties of local smallness really were. A few questions on this front:

  1. Is it crucial that the collection of morphisms be at most class-sized? Or is it enough that you have only hom-sets?

  2. Would a quasicategory with a conglomerate sized collection of objects but only hom-sets have the nice-making features of locally small categories? Obviously their collection of morphisms would be conglomerate-sized because there would be conglomerate-many identity arrows on the objects, but is this something that would render them "less nice" to use than locally small categories (i.e., assuming only class-many objects)?

  3. My speculative thought is that the restriction to class-many objects is necessary because you want something like a functorial embedding of the locally small categories into the category of sets, but since the universe of sets is class-size, this wouldn't be possible if you had conglomerate-many objects (and hence conglomerate-many morphisms). Is something along these lines right?

TL;DR What's so great about locally small categories, and if there were more than class-many objects in a (quasi)category with only hom-sets would that be "just as good" as a locally small category?

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    $\begingroup$ one axiomatization of the notion of smallness is given by the notion of an arity class ncatlab.org/nlab/show/arity+class; smallness in this sense is used when constructing (exact) completions of categories (cf. tac.mta.ca/tac/volumes/27/7/27-07abs.html), formulating and proving the adjoint functor theorem (solution set condition should say that each object has a small pre-reflection), proving that if a functor category valued in a locally small category is itself locally small, then the domain is small (cf. tac.mta.ca/tac/volumes/1995/n9/v1n9.pdf)... $\endgroup$ – Vladimir Sotirov Jun 18 '17 at 22:04
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    $\begingroup$ I'm pretty sure class-locally-small and conglomerate-locally-small categories will be indistinguishable, as the universe foundation doesn't distinguish them. $\endgroup$ – Kevin Carlson Jun 18 '17 at 22:23
  • $\begingroup$ @KevinCarlson When you say universe foundation what do you mean? That term is unfamiliar to me. "Universe" as in Grothendieck Universe? I've seen the nCat formulate notions of size relative to different choices of Grothendieck Universe, $U_n$, such that you have $U_n$-smallness for each Grothendieck Universe you countenance. Or is "universe" meant in a narrower sense like one that would be given by a model of ZFC with an assumed fixed size? Something else entirely? $\endgroup$ – Dennis Jun 18 '17 at 22:27
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    $\begingroup$ @Dennis Yes, as in Grothendieck universes. There a locally small category has an arbitrary set of objects and $(U)$-small sets of morphisms. $\endgroup$ – Kevin Carlson Jun 18 '17 at 22:32
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    $\begingroup$ Right, and I'm claiming the answer is "yes", except for properties which explicitly mention a universe. So for instance, any functor category between $U_1$-locally small categories will be $U_2$-small, which is of course not true for $U_2$-locally small categories; you also can't form the category of all $U_2$-locally small categories, say, without adding another universe; but as far as the serious theorems mentioned by Vladimir go, there will be no problems if you define "locally small" to mean "arbitrary object set, $U_1-small$ morphism sets." $\endgroup$ – Kevin Carlson Jun 18 '17 at 22:54

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