Conditions for a functor to induce a logical functor between presheaf toposes?

Question

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 $\mathrm{Psh}(\mathcal{D}^{op})=\mathsf{Set}^\mathcal{D}$.)

Answer 1

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.

Answer 2

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.

Answer 3

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.