Question

Let us work over the etale site $\mbox{Aff}/S$ (for the sake of definiteness) for some fixed base scheme $S$, where the covers are jointly surjective etale maps $\{ U_i \rightarrow U\}_{i\in I}$ (and $I$ is finite if you like). Let us also consider a prestack $F$ fibred in groupoids. Recall the definition of a category of descent data $F(\{ U_i \rightarrow U\}_{i \in I})$ associated to a cover $\{ U_i \rightarrow U\}_{i \in I}$. The objects of this category are collections of elements $\xi_i\in F(U_i)$ together with morphisms $\phi_{ij}$ between their appropriate pullbacks satisfying the cocycle condition. This definition is the one appearing on p. 15 of "Champs algebriques" by Laumon & Moret-Bailly (among other sources, e.g., Vistoli's notes in "FGA explained").

However, one can also consider coverings $U^\prime \rightarrow U$ consisting of one element only. In $\mbox{Aff}/S$ starting with any covering one can obtain one of such form by taking $U^\prime := \bigsqcup U_i$. However, it is not clear to me how this passage to a cover with a single morphism interacts with the associated categories of descent data. I suspect, they should be equivalent (in "Champs algebriques" for instance, the authors switch to the latter when exhibiting the stackification of a prestack in (3.2)) but on the other hand I don't see how is one supposed to get a single $\xi \in F(U^\prime)$ starting off with the $\xi_i$ as above, let alone an equivalence of categories between $F(\{ U_i \rightarrow U\}_{i \in I})$ and $F(\{ U^\prime \rightarrow U\})$. Are these categories equivalent and what is a functor exhibiting this equivalence? And if not, why is one allowed to consider only coverings of the form $U^\prime \rightarrow U$ when constructing the stackification?

Answer 1

This mistake seems to be made all over the place; see also this question. As David has said, the context in which this is usually done is an extensive category with a topology which includes the extensive topology (whose covering families are those of the form $(U_i \to \coprod_i U_i)$).

However, even in this case, the categories of descent data for a covering family and for the associated single cover are only equivalent if you already know that your prestack is a stack for the extensive topology.

At the nLab page on superextensive sites there is a proof that if you have a presheaf that is a sheaf for the extensive topology, and you sheafify it with respect to the singleton covers, then you get a sheaf with respect to all the covering families. I expect that this generalizes to stacks as well. But if your presheaf is not a sheaf for the coproduct families, then sheafifying it for singleton covers is not sufficient.

I'm not necessarily saying that any particular reference says something wrong; I don't have "Champs algebriques" in front of me so I can't check whether they have prestacks that are extensive-stacks already or are otherwise avoiding the issue. But in general it is something you have to worry about.