Constructing Model Categories with Prescribed Fibrant Objects

We present a weak form of a recognition principle for Quillen model categories due to J.H. Smith. We use it to put a model category structure on the category of small categories enriched over a suitable monoidal simplicial model category. The proof uses a part of the model structure on small simplicial categories due to J. Bergner. We give an application of the weak form of Smith’s result to left Bousfield localizations of categories of monoids in a suitable monoidal model category.