If $J$-coverings can be glued $I$-locally is $J$-locality an $I$-local property? (Reducing descent problems to simpler ones)

Question

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?

Answer 1

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.