TITLE: In a model category, is the full subcategory of fibrant objects a reflective subcategory?
QUESTION [6 upvotes]: I apologize in advance if my question is utterly stupid, but I can't resist asking it. So...
Is it true that in a model category ( - for example $\mbox{Set}_\Delta$ with the Joyal model structure - ) the full subcategory of fibrant objects is a reflective subcategory? More concretely, is the fibrant replacement functor a left adjoint to the inclusion functor?
Thanks.

REPLY [2 votes]: Although most model structures appearing "in practice" do not have a reflective subcategory of fibrant objects (unless every object is fibrant), the following result of A. Salch provides many examples that do.
Theorem. Let $\mathcal{C}$ be a complete and cocomplete category, and let $\mathcal{A}$ be a full subcategory which is reflective and replete (i.e., closed under isomorphisms). Then there exists a model structure on $\mathcal{C}$ such that $\mathcal{A}$ consists of the fibrant objects.
See Corollary 6.3 here:
http://arxiv.org/abs/1501.00508v1
The model structure is obtained as a (left) Bousfield localization of the discrete model structure on $\mathcal{C}$. The weak equivalences are the maps that become isomorphisms upon applying the reflector $\mathcal{C} \to \mathcal{A}$.