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Let $F\colon\mathcal{C}\to\mathcal{D}$ be a functor between small categories.

Question: Under what conditions is the induced functor $$F^*\colon\mathsf{Set}^\mathcal{D}\to\mathsf{Set}^\mathcal{C}$$ a logical functor between presheaf toposes?

(I prefer to avoid contravariance if possible, so the "presheaf toposes" I'm referring to here are $\mathrm{Psh}(\mathcal{C}^{op})=\mathsf{Set}^\mathcal{C}$ and $\mathrm{Psh}(\mathcal{D}^{op})=\mathsf{Set}^\mathcal{D}$.)

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    $\begingroup$ A few not very deep remarks: 1) the condition that $F^*$ preserve power object is relatively easy to write down explicitely and give an explicit criterion on $f$ equivalent to your question. 2) The answer of John below can be extended to the case where the the foncteur $F$ is (equivalent to) a fibration in groupoid. 3) the criterion obtained in (1) is very strong and I found it very surprising that there is an many example as (2) suggest. In fact I wouldn't be surprised if those were the only examples. $\endgroup$ – Simon Henry Nov 7 '16 at 11:48
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Not really an answer, but maybe a hint as to where to look for one: since $F^\ast$ is the left adjoint part of a geometric morphism $F^\ast \dashv \mathrm{Ran}_F$, it is logical precisely when this geometric morphism is atomic (this is essentially the definition of "atomic geometric morphism"). Atomic morphisms are studied in section C3.5 of Sketches of an Elephant; I just glanced through that section and nothing jumped out at me as answering your question directly, but perhaps there is something useful in there or the references given.

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Very partial answer: if C is a groupoid then $Set^{C}$ inherits elementary topos structure pointwise from $Set$. Accordingly, if $F:C \to D$ is a morphism of groupoids then $F^{\star}:Set^{D} \to Set^{C}$ preserves the elementary topos structure -- ie. is a logical functor.

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As Mike Shulman pointed out, this is the same as requiring the geometric morphism induced by $f$ to be atomic. In Atomic toposes 7.2 you'll find that in the case of $! = F \colon C \to 1$, $F^*$ is logical iff $C$ is a groupoid.

What about the general case? Not an answer in any way, but two (UPDATE 2. went nowhere, added 3.) possible paths:

  1. Atomic morphisms are locally connected (Elephant C3.5). What do we know about functors for which the induced geometric morphism is locally connected? looking at the Elephant again, Lemma C3.3.5 which gives a sufficient condition, and On functors which are lax epimorphisms, which characterizes connected ones, one is tempted to conjecture that the induced geometric morphism is locally connected iff it is absolutely dense into its Cauchy-image, or something like that.
  2. In the spirit of Simon Henry comment above, and given the characterization of atomic sites in Atomic toposes 7.3, or C3.5.8 in the Elephant, maybe it is enough for $F$ to be a fibration in categories where (i) every morphism is an effective epi and (ii) every pair of morphisms with common codomain can be completed to a commutative square (right Ore condition); note that a groupoid obviously satisfies these two conditions. UPDATE not the case, see comment below
  3. Look at its hyperconnected-localic factorization, which in this case coincides with the connected-light factorization (or "comprehensive") of a locally-connected morphism. For essential geometric morphisms between presheaf categories, this yields the comprehensive factorization of a functor into a final functor followed by a discrete fibration; Factorization theorems for geometric morphisms II is a reference (at the end of the paper). Now, a geometric morphism is atomic iff both halves of this factorization are atomic (Elephant C3.5.4), so we have reduced it to when both the final and the discrete fibration components induce an atomic morphism.
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  • $\begingroup$ I'm not sure I understand your condition $(i)$ and $(ii)$ in $2.$, which morphism are you talking ? it is a condition on the fiber of the fibrations ? $\endgroup$ – Simon Henry Nov 8 '16 at 12:35
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    $\begingroup$ In which case I don't think it works: in C3.5.8 you also need a topology (and a rather big one every arrow is a covering ! ) so for example over the terminal category your condition is not going to give back groupoids, but only categories in which every arrow is an effective epi and which satisitifes the right Ore condition. $\endgroup$ – Simon Henry Nov 8 '16 at 12:43
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    $\begingroup$ In fact, if I'm correct, the result of atomic topos 7.2 can be extended to prove that if $F :C \rightarrow D$ is a fibration then $F^*$ is logical if and only $F$ is a fibration in groupoids. What is not clear is what happen when $F$ is not a fibration. $\endgroup$ – Simon Henry Nov 8 '16 at 12:47
  • $\begingroup$ Thanks @Simon, you are completely right. Need more coffee :) $\endgroup$ – Eduardo Pareja Tobes Nov 8 '16 at 18:03
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    $\begingroup$ Interesting... the discrete fibration part corresponds to an etale morphism right ? (or Am I wrong on the covariance because I'm confused by David Spivak convention ? ). If so it is always atomic and we just have to treat the final case... $\endgroup$ – Simon Henry Nov 8 '16 at 18:50

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