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Suppose that $(C,J)$ and $(D,K)$ are two Grothendieck (possibly $\infty$-)sites and $f:C \to D$ is a functor such that $$f^*:Psh(D) \to Psh(C)$$ sends sheaves to sheaves. Under what conditions will $$f_!:Psh(C) \to Psh(D)$$ send sheaves to sheaves? Here, $f_!$ is the unique colimit preserving functor that sends each representable $y(c)$ to $y(f(c)).$ I am not looking for degenerate cases, but more for useful criteria to check to see if it holds in certain non-trivial examples.

Note: I am not assuming either Grothendieck topology is subcanonical, but, I am interested in this case as well.

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Can you give one nontrivial example where the hypothesis on $f^{\ast}$ (which is very restrictive in the classical setting of continuous maps of topological spaces) does hold and the answer is affirmative? The hypothesis holds for open embeddings of topological spaces yet the conclusion fails for that case. What is the motivation for this question? – user27920 Jul 7 '14 at 15:14
@user52824: Not really, I have instead situations where I would like this to hold, and do not know if it does. If I can prove that it holds, then I'll have some examples. I was hoping for a formal answer. – David Carchedi Jul 7 '14 at 15:29
I am skeptical that you have a situation where the hypothesis actually holds (just because in essentially all situations which I can think of, the hypothesis on $f^{\ast}$ fails). Can you give an interesting case where the hypothesis can actually be verified and the conclusion is not obviously false? – user27920 Jul 7 '14 at 20:44
Let $:\pi:\mathit{Mfd} \to \mathit{Mfd}[W^{-1}]_\infty$ be natural functor from the category of manifolds to the $\infty$-category of manifolds with homotopy equivalences weakly inverted (and endow the latter with the induced Grothendieck topology). – David Carchedi Jul 7 '14 at 21:06
I don't know anything about $\infty$-categories, but in more traditional settings the only cases which come to mind when $f^{\ast}$ satisfies your hypotheses is when $f$ is an open embedding (in which case the desired conclusion is readily seen to be false). So do you mean that the easy counterexamples for open embeddings don't readily adapt to your fancier situation? Anyway, I'll stop here since the context for this question is way over my head. – user27920 Jul 7 '14 at 21:33

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