Size issues (small/large categories) when defining stacks in the Algebraic/differentiable/topological setting

Question

Angelo Vistoli in the notes Notes on Grothendieck topologies, fibered categories and descent theory starts the section of category theory with the following note:

We will not distinguish between small and large categories. More generally, we will ignore any set-theoretic difficulties. These can be overcome with standard arguments using universes.

Question : Which of the notions introduced in Angelo Vistoli's notes assumes that the category is small? In particular their application to Algebraic/differentiable/topological stacks?

For example, Behrang Noohi puts the following extra condition in his notes on topological stacks:

Throughout the paper, all topological spaces are assumed to be compactly generated.

This could be because, the category $\text{Top}$ of all topological spaces is not a small category.

Are there any places one has to be careful to not allow large categories?

Some references to support this question :

1. nlab says "In technical terms, a site is a small category equipped with a coverage or Grothendieck topology". It also says (Remark $2.3$ at same page) "Often a site is required to be a small category. But also large sites play a role."
2. David Metzler in Topological and Smooth Stacks defines (page $2$) a site as a small category equipped with Grothendieck topology. It further says "We will want to discuss, for example, “the category of stacks on the category of all topological spaces,” but strictly speaking this does not exist, since the category of topological spaces does not have a set of objects, but rather a proper class. To avoid this problem we will consider throughout some fixed category $\mathbb{T}$ of topological spaces which has a set of objects, or at least, is equivalent to such a category".

So, it "looks like", even though one can define a site over a large category, and then a stack over a site (which was defined on a large category), one often restricts (for computational purposes or personal interests) to a small categories and stacks on them. Is this what it is or am I misunderstanding something here?

Answer 1

Are there any places one has to be careful to not allow large categories?

No. For the purposes of forming the 2-category of algebraic/topological/differentiable stacks, or more generally, some kind of presentable stacks over a large category there are no size issues. Naively, the 2-category of stacks on $S$ is carved out from the presheaf category $[S^{op},\mathbf{Cat}]$ (or $[S^{op},\mathbf{Gpd}]$), which does present size issues for $S$ not essentially small. However, the 2-category of presentable stacks (of groupoids, say, which is the case you are looking at) is equivalent to the bicategory of internal groupoids and anafunctors (and transformations). This can be defined elementarily from the 2-category of internal groupoids, functors and natural transformations. Given a quite weak size condition on the site structure—that is, the size of generating sets of covering families—this bicategory is even locally essentially small. The only case 'in the wild' that I know of that fails this weak condition is the fpqc topology on categories of schemes, and algebraic geometers are a bit wary of that: see tag 0BBK. They are happy to say a single presheaf (of sets, modules, groupoids) is a stack for the fpqc topology, but generally talk about sheaves/stacks for the fppf topology at the finest: see the definition in tag 026O.

Added For a large site not satisfying the condition WISC, the sheafification or stackification functors might not exist. This problem, however, does not impact considering presentable stacks, only when one is wanting to think about arbitrary stacks. For an example of how bad this can get, Waterhouse's paper

Basically bounded functors and flat sheaves, Pacific Journal of Mathematics 57 (1975), no. 2, 597–610 (Project Euclid)

gives an example of a presheaf on the fpqc site that does not admit any sheafification. The following quote from the Stacks Project is relevant:

The fpqc topology cannot be treated in the same way as the fppf topology. Namely, suppose that R is a nonzero ring. We will see in Lemma 34.9.14 that there does not exist a set $A$ of fpqc-coverings of $Spec(R)$ such that every fpqc-covering can be refined by an element of $A$. If $R=k$ is a field, then the reason for this unboundedness is that there does not exist a field extension of $k$ such that every field extension of $k$ is contained in it.

If you ignore set theoretic difficulties, then you run into presheaves which do not have a sheafification, see [Theorem 5.5, Waterhouse-fpqc-sheafification]. A mildly interesting option is to consider only those faithfully flat ring extensions $R\to R'$ where the cardinality of $R'$ is suitably bounded. (And if you consider all schemes in a fixed universe as in SGA4 then you are bounding the cardinality by a strongly inaccessible cardinal.) However, it is not so clear what happens if you change the cardinal to a bigger one. (Tag 022A)