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It seems to me that frequently when discussing stack conditions and descent, people consider only singleton covering families, i.e. there is some single covering map $U\to X$, for which one constructs a category of descent data out of $U$ and $U\times_X U$, and so on. But many sites have non-singleton covering families $(U_i \to X)_{i\in I}$; why are those ignored? Is there some deep reason why it suffices to consider singleton families?

My efforts to understand this so far have led me to think about "superextensive sites", which are sites whose covering families are generated by singleton covers together with inclusions into coproducts (the "extensive topology"). In particular, a fibered category is a stack on a superextensive site iff it is a stack for the singleton covers and also for the extensive topology. But while stack conditions for the extensive topology have an exceptionally simple form (the compatibility conditions being mostly vacuous), they are still not automatic. So why are they often not mentioned?

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I have the feeling that practically these two viewpoints only make a difference when there are not arbitrary coproducts or when there is a type of covers you care about that is not closed under arbitrary coproducts. Regarding the last point: The category of schemes is closed under arbitrary coproducts, but the infinite coproduct of affine schemes is not affine. If one considers only quasi-compact objects (like one often does in the scheme case), considering finite coproducts is practically enough and these are usually unproblematic. –  Lennart Meier Jan 5 at 9:24
Even when covers are closed under coproducts, you still have to include the sheaf/stack condition for non-singleton covers, or else you get a different notion of sheaf/stack. –  Mike Shulman Jan 6 at 21:11
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1 Answer

In applications I've seen, what matters is the topos, not the site. If this is true for your applications, you should feel free to replace your site by any site that produces the same topos. I think you can always make the following conventions without changing the topos (edit: not true, you need some hypothesis on the site; see comments):

  1. If $\{U_i\to X\}_{i\in I}$ is a covering, then so is the singleton $\bigsqcup_I U_i\to X$.
  2. $\{U_i\to \bigsqcup_{j\in I} U_j\}_{i\in I}$ is always a covering.

After that, you can use convention 1 to replace any non-singleton covering by a singleton covering, which is easier to think about (at least easier to symbolically manipulate). But you do have to keep 2 around to be able to prove, for example, that $F(\bigsqcup_{I} U_i)=\prod_{I} F(U_i)$ for any sheaf $F$. As far as I can tell, everybody I know adopts these two conventions and then just talks about singleton covers.

An alternative (probably better) explanation comes from the sieve-theoretic formulation of Grothendieck topologies. All that matters about a covering is the sieve that it generates, and $\bigsqcup_I U_i\to X$ generates the same sieve as $\{U_i\to X\}_{i\in I}$ (edit: also not true; perhaps someone who understands the sieve approach could say what the right statement is). I think the canonical reference for this approach is SGA 4 ("sieve"="crible").

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Unless I am greatly mistaken, $\coprod_I U_i \to X$ does not generate the same sieve as $\{U_i \to X\}_{i\in I}$. For one thing, the map $\coprod_I U_i \to X$ itself is not in the sieve generated by $\{U_i \to X\}_{i\in I}$. –  Mike Shulman Dec 25 '09 at 1:16
By the way, I do only care about the topos, but I don't think you can change the Grothendieck topology on a given category without changing the topos of sheaves, since for any category C, there is a bijection between Grothendieck topologies on C and subtoposes of the presheaf topos Psh(C). I suppose that in theory, two topologies on C might produce sheaf toposes that are abstractly equivalent, but not equivalent as subtoposes of Psh(C), but this seems unlikely. –  Mike Shulman Dec 25 '09 at 1:21
For a concrete counterexample, consider the category of finite sets with its trivial topology. Its topos of sheaves is of course the presheaf topos Psh(FinSet). But if you modify it according to your convention 2, then every jointly surjective family becomes covering, and hence the topos of sheaves collapses back down to Set, which is certainly not equivalent. –  Mike Shulman Dec 25 '09 at 1:29
You're right, they don't generate the same sieve, and I agree with your counterexample. It must be that in examples I come across, the property of being a covering is somehow local on domain, or something else is happening which allows these tricks to work. –  Anton Geraschenko Dec 25 '09 at 20:02
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