Remark 7.6.3 (00VI)—The Stacks project

Remark 7.6.3. (On set theoretic issues – skip on a first reading.) The main reason for introducing sites is to study the category of sheaves on a site, because it is the generalization of the category of sheaves on a topological space that has been so important in algebraic geometry. In order to avoid thinking about things like “classes of classes” and so on, we will not allow sites to be “big” categories, in contrast to what we do for categories and $2$-categories.

Suppose that $\mathcal{C}$ is a category and that $\text{Cov}(\mathcal{C})$ is a proper class of coverings satisfying (1), (2) and (3) above. We will not allow this as a site either, mainly because we are going to take limits over coverings. However, there are several natural ways to replace $\text{Cov}(\mathcal{C})$ by a set of coverings or a slightly different structure that give rise to the same category of sheaves. For example:

In Sets, Section 3.11 we show how to pick a suitable set of coverings that gives the same category of sheaves.
Another thing we can do is to take the associated topology (see Definition 7.48.2). The resulting topology on $\mathcal{C}$ has the same category of sheaves. Two topologies have the same categories of sheaves if and only if they are equal, see Theorem 7.50.2. A topology on a category is given by a choice of sieves on objects. The collection of all possible sieves and even all possible topologies on $\mathcal{C}$ is a set.
We could also slightly modify the notion of a site, see Remark 7.48.4 below, and end up with a canonical set of coverings.

Each of these solutions has some minor drawback. For the first, one has to check that constructions later on do not depend on the choice of the set of coverings. For the second, one has to learn about topologies and redo many of the arguments for sites. For the third, see the last sentence of Remark 7.48.4.

Our approach will be to work with sites as in Definition 7.6.2 above. Given a category $\mathcal{C}$ with a proper class of coverings as above, we will replace this by a set of coverings producing a site using Sets, Lemma 3.11.1. It is shown in Lemma 7.8.8 below that the resulting category of sheaves (the topos) is independent of this choice. We leave it to the reader to use one of the other two strategies to deal with these issues if he/she so desires.

Comments (2)

Comment #7054 by nkym on February 13, 2022 at 06:56

The underlying category of the affine crystalline site 07HL, the big crystalline site 07I5, and the crystalline site 07IF seem to be big in the sense disallowed here.

Comment #7244 by Johan on April 30, 2022 at 16:36

Yep. So what needs to be done is to follow the outline given in Section 34.2 and worked out in particular for the big Zariski site on schemes in Section 34.3 in the case of the crystalline site. Let me discuss this a bit more and we can add it to the Stacks project in more detailed form later.

First, in Section 60.5 we only define a category and not a site, so the problem doesn't arise there.

Next, in Section 60.8, more precisely in Definition 60.8.4, we need to choose a sufficiently big cardinal $\alpha$ as in Lemma 3.9.2 and then define $\text{CRIS}(X/S)$ as those objects $(U, T, \delta)$ such that $U$ and $T$ are contained in $\textit{Sch}_\alpha$ . Next, one needs to additionally choose a set of coverings defining the structure of a site on $\text{CRIS}(X/S)$ such that every covering in $\text{CRIS}(X/S)$ is combinatorially equivalent to one in the given set.

In Section 60.9 we then automatically obtain a valid site.

All of this is easy. The only small point is that in taking fibre products for example, or when taking divided power hulls the result doesn't all of a sudden produce an object which is too large (and therefore not isomorphic to an object of the category of objects chosen). This is a minor issue but also the discussion in the chapter on Sets shows, by a meta-mathematical argument, that this never(!) leads to mathematical problems as one may simply replace the function $Bound$ used in Lemma 3.9.2 by a faster growing function when chosing the cardinal $\alpha$ above to obtain a solution. And indeed, in the case of the pro-etale topology, we did this exact thing to get through.

Experts often do a replacement of this kind without ever mentioning it.

There are also:

2 comment(s) on Section 7.6: Sites

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