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This question is a follow-up of the question I have asked today : left quillen functor and fibrant objects Suppose that we have $$ L :C\leftrightarrow D: R$$ an adjoint Quillen pair. We assume that both model categories are combinatorial model categories.

Suppose that

1- the functor $L$ (left adjoint) takes fibrant-cofibrant objects to fibrant-cofibrant objects.

2- the functor $R$ (right adjoint) takes cofibrations to cofibrations and takes acyclic cofibrations to acyclic cofibrations.

Is it true that $L$ sends fibrant objects to fibrant objects ?

Maybe a more general question would be the following: Under which additional (non trivial) conditions the functor $L$ would takes fibrant objects to fibrant objects ?

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    $\begingroup$ Have you tried looking for a counterexample amongst the nine model structures on Set? matem.unam.mx/~omar/notes/modelcatsets.html#org8419997 $\endgroup$ – Dylan Wilson Nov 27 '18 at 1:56
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    $\begingroup$ @DylanWilson Unless I'm missing something, I did not find a counterexample in your link! :) $\endgroup$ – ABC Nov 27 '18 at 11:12
  • $\begingroup$ @ABC I think the idea would be to make up a bunch of Quillen pairs among those model structures and come up with a counterexample of your own. $\endgroup$ – Kevin Carlson Nov 28 '18 at 0:01
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    $\begingroup$ @KevinCarlson That is exactly what I checked, I was not able to detect a counterexample... $\endgroup$ – ABC Nov 28 '18 at 9:59
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    $\begingroup$ @ABC Do you have an example of a Quillen adjunction satisfying these hypotheses in mind? At first glance, the hypotheses sound extremely restrictive to me, to the point of being rather artificial. $\endgroup$ – Tim Campion Dec 3 '18 at 22:22

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