Stacks and sheaves of categories as fibrant objects. I. (Q2927674)

\textit{A. Joyal} and \textit{M. Tierney} [Lect. Notes Math. 1488, 213--236 (1991; Zbl 0748.18009); C. R. Math. Acad. Sci., Soc. R. Can. 13, No. 1, 2--6 (1991; Zbl 0733.18011)] have shown, by using internal groupoids and categories in a Grothendieck topos, that stacks are fibrant objects of a model category, which are called strong stacks (of groupoids or categories). Using some elaborate results from the homotopy theory of simplicial presheaves on a site, \textit{S. Hollander} [Isr. J. Math. 163, 93--124 (2008; Zbl 1143.14003)] has constructed a model category that is Quillen-equivalent to the model category for strong stacks of groupoids. The principal objective in this paper is to extend [Zbl 1143.14003] to general stacks and then to show that the category of internal categories in a Grothendieck topos admits another model category Quillen-equivalent to the model category for strong stacks of categories. The author's approach, distinct from [Zbl 1143.14003] and [Zbl 0748.18009], is inspired by the famous volume [\textit{J. Giraud}, Cohomologie non abelienne. Berlin etc.: Springer-Verlag (1971; Zbl 0226.14011)], whose \S 1 and \S 2 of Chapitre II has suggested a connection of stacks with left Bousfield localizations of model categories (cf. Chapter 3 of [\textit{P. S. Hirschhorn}, Model categories and their localizations. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1017.55001)]).