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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.2012-09-28
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    @ZhenLin This sounds like minimal fibrations.2012-09-28
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    ...or you can just say everything using the language of $\infty$-categories.2012-11-03

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