Constructing model categories with prescribed fibrant objects
A theorem of Jeff Smith (cf. Theorem 1.7 of [\textit{T. Beke}, Math. Proc. Camb. Philos. Soc. 129, No. 3, 447--475 (2000; Zbl 0964.55018)]) gives a criterion for detecting closed model categories on locally presentable categories, but it is by no means practical, giving no description of fibrations in the detected model structure. A theorem of D. M. Kan (cf. Theorem 11.3.1 of [\textit{P. S. Hirschhorn}, Model categories and their localizations. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1017.55001)] and Theorem 2.1.19 of [\textit{M. Hovey}, Model categories. Providence, RI: American Mathematical Society (1999; Zbl 0909.55001)]) gives another recognition principle putting small object arguments to use, which yields a full description of fibrations in the recognized model structure. This paper is to make use of the former in order to detect a model category on the category of small categories enriched over a suitable monoidal simplicial model category.NEWLINENEWLINEThe paper under review consists of 5 sections. \S 1 is an introduction, and \S 2 is a review of a proof of Théorème 1.3.22 of [\textit{D.-C. Cisinski}, Les préfaisceaux comme modèles des types d'homotopie. Paris: Société Mathématique de France (2006; Zbl 1111.18008)], where two out of the six properties of a class of maps in [\textit{W. G. Dwyer} et al., Homotopy limit functors on model categories and homotopical categories. Providence, RI: American Mathematical Society (AMS) (2004; Zbl 1072.18012)] play a significant role. \S 3 gives the main result of this paper. The proof makes use of a proof of the similar result for categories enriched over the category of simplicial sets in [\textit{J. E. Bergner}, Trans. Am. Math. Soc. 359, No. 5, 2043--2058 (2007; Zbl 1114.18006)], but the author modifies one of the few steps in Bergner's proof crucially. \S 4 is devoted to an extension of a result in [\textit{R. Fritsch} and \textit{D. M. Latch}, Math. Z. 177, 147--179 (1981; Zbl 0456.55014)] claiming that the pushout of a full and faithful functor is full and faithful. The extension is necessary in the previous section. The last section applies the technique in \S 2 to left Bousfield localizations of a monoidal model category.
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