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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 the functor $L$ (left adjoint) takes fibrant-cofibrant objects to fibrant-cofibrant objects. I was wondering if it follows that $L$ sends fibrant objects to fibrant objects ?

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To find a counterexample, we should choose $C$ to have a lot of fibrant objects, but few cofibrant objects. So let $D$ be a combinatorial model category and let $C$ be the model category structure on the underlying category of $D$ in which every morphism is an acyclic fibration. This is again a combinatorial model category (with empty sets of generating (acyclic) cofibrations), and the identity functor $C \to D$ is a left Quillen functor. In $C$ every object is fibrant, but only the initial object is cofibrant.

In this case the question becomes: if the initial object of $D$ is fibrant, then is every object of $D$ fibrant? And of course there are many counterexamples.

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  • $\begingroup$ To make the last sentence concrete, I think a counterexample $D$ is the injective model structure on $Ch(R)$ where $R$ is a self-injective ring. Or, the category of pointed simplicial sets. $\endgroup$ Nov 26, 2018 at 13:21
  • $\begingroup$ The counterexample you gave is extreme :) Is there still a "reasonable" additional condition that we could impose on the functor $L$ such that under my assumptions, the functor L will takes fibrant objects to fibrant objects? $\endgroup$
    – ABC
    Nov 26, 2018 at 14:57

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