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I've been learning about sites from the Stacks Project, which is generally very precise in its terminology, but I've found some of their conventions very confusing in this part. Their definition of a site is given here. Below the definition, they immediately make a remark about set theoretic issues (Remark 6.3 in the link). In particular, they address the issue of big categories, and when such categories give rise to a proper class of covering families. They fix the convention that they will allow big categories, and then use the result here to choose a set of covering families that gives the same category of sheaves as the proper class of covering families.

From the outset, this seems ok. But reading the statements of these theorems, the entire idea seems redundant. For example, in the second link, they define a site to be a small category. Indeed the second link above begins:

"Suppose that $\mathcal{C}$ is a category (as in Categories, Definition 4.2.1) and that Cov($\mathcal{C}$) is a proper class of coverings satisfying properties (1), (2), and (3) of Sites, Definition 7.6.2."

The definition of category they refer to in Definition 4.2.1 requires that it only have a set of objects. But then surely it is impossible to choose a proper class of coverings.

Am I misunderstanding the notion of a small category, and the notion of sites completely? Or is this just an inconsistency in the Stacks Project (this seems unlikely since it specifically references the Stacks Project definition) or is there something else going on I am not understanding?

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    $\begingroup$ I'm guessing it's just imperfect coordination on the parts of the authors of separate pages. $\endgroup$ Nov 8, 2018 at 6:17
  • $\begingroup$ You should email the maintainer of the stacks project. $\endgroup$ Nov 8, 2018 at 16:47

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