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Let $(C,J)$ be a category with a grothendieck topology. For every object $X \in C$ there's (I hope) a little site which is the full subcategory of the slice category $C_{/X}$ whose objects are the morphisms that appear in some $J$-covering family of $X$ and whose morphisms are commuting triangles s.t. all edges appear in some $J$-covering family of their terminal vertex. The topology on this category is the one induced from $J$ (the topology is irrelevant to the question but it might be relevant to the solution).

In other words we have a (pseudo-)functor $J_{/(-)} : C \to \mathsf{Cat}$ which sends an object to the little $J$-site over it. Suppose now that $J_{/(-)}$ is a stack for some other grothedieck topology $I$ on $C$.

Let $\mathcal{F}$ be a presheaf on $C_{/X}$ s.t. the following conditions hold:

  1. It is a sheaf for the $I$-topology.
  2. There exists an $I$-covering family $\{U_i\}$ of $X$ s.t. for every $U_i$ the pullback $\mathcal{F}_{U_i}$ is a sheaf on the big site $C_{/U_i}$ equipped with the $J$-topology.

Question: Is $\mathcal{F}$ a sheaf on $C_{/X}$ for the $J$-topology?

In informal terms: If $J$-coverings can be glued $I$-locally is $J$-locality an $I$-local property?

Example: Putting $C=Sch$, $J=Etale$ and $I=Zariski$ we get: If $\mathcal{F}$ is both a Zariski sheaf and Zariski locally an Etale sheaf is it an Etale sheaf?

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  • $\begingroup$ I had to do something along these lines in the context of schemes to prove descent for the shape of the infinity topos. If J is finer than I then it is true, but otherwise I don't know. Does your "J is a stack for the I topology" condition tell you that J is finer than I? $\endgroup$ – Joe Berner Feb 18 '17 at 13:51
  • $\begingroup$ @JoeBerner I don't see an immediate reason for this. Looks to me like $J$ could potentially be coarser than $I$. I'll be happy to be proven otherwise. $\endgroup$ – Saal Hardali Feb 18 '17 at 14:01
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The answer is yes. In fact the assumption that $X\mapsto J_{/X}$ is a sheaf is not even relevant. First, let me point out that strictly speaking a Grothendieck topology $J$ does not determine a "small site" over an object $X$, because every morphism belongs to some $J$-covering sieve (the maximal sieve on any object is always $J$-covering). For example, the étale topology on the category of schemes does not determine the small étale site of a scheme, at least not in any obvious way. The reason small sites are so common is that the corresponding topologies come with bases which determine small sites, e.g., jointly surjective families of étale maps form a basis for the étale topology etc.

Now let me explain why the result holds, assuming only 1 and 2; in particular the "small site" will not matter. Let $U$ be the $I$-covering sieve on $X$ generated by $\{U_i\to X\}$. I will view sieves as subpresheaves of representable presheaves, like so: $U \hookrightarrow X$. Let $V$ be a $J$-covering sieve on $Y\in C_{/X}$. Since $I$-local isomorphisms are stable under pullbacks (sheafification is left exact), $V\times_XU\hookrightarrow V$ is an $I$-local isomorphism, and similarly $V\times_XU \hookrightarrow Y\times_XU$ is a $J$-local isomorphism. Since $F$ is an $I$ sheaf on $C$ by 1 and a $J$-sheaf on $C_{/U}$ by 2, it sends all three morphisms $Y\times_XU\hookrightarrow Y$, $V\times_XU\hookrightarrow V$, $V\times_XU \hookrightarrow Y\times_XU$ to isomorphisms. By 2-out-of-3, $F$ also sends $V\hookrightarrow Y$ to an isomorphism. This shows that $F$ is a $J$-sheaf.

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  • $\begingroup$ Thanks! Is this true for stacks of $\infty$-groupoids as well? It seems to me like a similar proof is possible, but i'm still learning this stuff. $\endgroup$ – Saal Hardali Feb 19 '17 at 11:55
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    $\begingroup$ Yes, the same proof applies word for word for presheaves valued in any ∞-category. $\endgroup$ – Marc Hoyois Feb 19 '17 at 15:35
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    $\begingroup$ And $C$ could be an ∞-category as well. $\endgroup$ – Marc Hoyois Feb 19 '17 at 15:53
  • $\begingroup$ Thanks! This is very useful and i'm a bit surprised I never saw it mentioned anywhere. Do you have a reference culminating these kind of results (general stuff about descent theory) or is it the sort of thing one just picks up as he goes? $\endgroup$ – Saal Hardali Feb 19 '17 at 15:57
  • $\begingroup$ I don't know. SGA4 is a good reference for basic descent theory. They use sieves as much as possible, so many proofs translate directly to higher categories. $\endgroup$ – Marc Hoyois Feb 19 '17 at 16:44

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