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Consider a fixed proper simplicial combinatorial model category $\mathcal{M}$. Consider a functor $F:I\to J$ between small categories. It induces a right Quillen functor $F^*:\mathcal{M}^J \to \mathcal{M}^I$ between the projective model structures.

Suppose that $F:I\to J$ induces a homotopy equivalence between the classifying spaces of $I$ and $J$. Can we say something more about $F^*:\mathcal{M}^J \to \mathcal{M}^I$ or its left adjoint ?

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    $\begingroup$ It would be best to clarify what kind of answer is desirable here, since many things could be said about $F^*$. Since taking the classifying space of $I$ inverts all morphisms in $I$ up to a homotopy, it is natural to perform a similar operation on simplicial presheaves, by left Bousfield localizing at the morphisms of representable presheaves induced by morphisms of I. After such a localization, the functor $F^*$ becomes a Quillen equivalence. This is a theorem of Cisinski: arxiv.org/abs/0803.4342. $\endgroup$
    – Dmitri Pavlov
    Commented May 27, 2023 at 16:47
  • $\begingroup$ @DmitriPavlov Thanks for the reference. $\mathcal{M}$ is a fixed model category which is not the category of simplicial sets. So $F$ induces a right Quillen equivalence between the local projective model structures. I don't know if it will be useful for me but it is the kind of information I had in mind. I am also almost sure that a close question was already asked in Mathoverflow but I can't find it. $\endgroup$
    – Philippe Gaucher
    Commented May 29, 2023 at 5:54
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    $\begingroup$ I agree with Dmitri that it's really not clear what sort of conclusion you're hoping for. Certainly you shouldn't expect to get a Quillen equivalence, for example. On the other hand, you might upgrade the hypotheses to something like assuming that $F$ is homotopy final (i.e. satisfies the hypotheses of Quillen's Theorem A)... $\endgroup$
    – Tim Campion
    Commented Jun 3, 2023 at 13:26

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