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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.

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Hmmm. This seems unlikely. Adjointness would say that, for each object $X$, there exists a unique (up to isomorphism) fibrant object $L X$ equipped with a universal morphism $X \to L X$ such that all morphisms $X \to A$ with $A$ fibrant factor through $X \to L X$ uniquely. But fibrant replacements are only unique up to weak equivalence, not isomorphism. –  Zhen Lin Sep 28 '12 at 1:55
    
@ZhenLin This sounds like minimal fibrations. –  Baby Dragon Sep 28 '12 at 1:57
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...or you can just say everything using the language of $\infty$-categories. –  Aaron Mazel-Gee Nov 3 '12 at 5:22

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