# 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 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 ?

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.
• 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. – David White Nov 26 '18 at 13:21
• 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? – ABC Nov 26 '18 at 14:57