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in reaction to an email discussion I had, I have finally added to the section Derived hom-spaces at category of fibrant objects the definition and theorem that had been alluded to there all along.
I added an observation of Karol Szumiło that $sSet$ is a category of fibrant (!) objects with the sharp maps as the “fibrations”. More generally this trick can be applied to (suitable full subcategories of) right proper model categories.
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