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This may be a naïve question but I've been unable to locate a reference that addresses it. Any thoughts are appreciated!

Let $f:\mathcal{E}\to\mathcal{S}$ be a cohesive morphism of toposes. That is, we have an adjoint string of functors $f_!\dashv f^\ast \dashv f_\ast \dashv f^!$.

Are there interesting and/or nondegenerate examples where we have $f^! \dashv f_!$? I'm willing to consider higher toposes, toposes with additional structure, etc.

This is partially motivated by a notational coincidence: in the setting of derived algebraic geometry, where $f_!$ and $f^!$ denote operations with compact support, we do have such an adjunction, albeit in the opposite direction.

More concretely, I am considering certain kinds of computation that share many properties with local geometric morphisms $(f_\ast, f^\ast, f_!)$, and for chaining such computations it would be advantageous to have some $f_! \dashv g \dashv f_\ast$.

It may be that I'm asking too much. Groth and Shulman construct a similar-looking adjoint cycle of period 6 in studying stable derivators, so additional "stitching" may be necessary for my approach to work.

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    $\begingroup$ In the even more restrictive case, when $f_!=f_*$ and $f^!=f^*$, you already get something nontrivial - quintessential localizations. They were studied by Johnstone in 1996. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Aug 4, 2022 at 16:01
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    $\begingroup$ @მამუკაჯიბლაძე Thanks for this reference! In my problem I think there's no hope of having these equalities, but there are really important techniques and examples from this paper that may generalize. For example, Johnstone constructs examples from monoids; using semirings may give examples of periodic adjoint quadruples. $\endgroup$
    – Andrew Dudzik
    Commented Aug 5, 2022 at 11:13
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    $\begingroup$ Concerning chains of adjoints you might find something useful in the questions mathoverflow.net/q/242390/41291 and mathoverflow.net/q/46877/41291 $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Aug 5, 2022 at 13:27
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    $\begingroup$ About that most restrictive case, there is also the paper Bireflectivity by Freyd, O'Hearn, Power, Takeyama, Street and Tennent (1999) $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Aug 5, 2022 at 13:30

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