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?


1 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.