On certain 2-categories admitting localisation by bicategories of fractions (Q726711)

The meaning of localise a bicategory at a class of morphisms and the introduction of a bicategory of fractions that permits to construct this localisation was introduced by \textit{D. A. Pronk} in [Compos. Math. 102, No. 3, 243--303 (1996; Zbl 0871.18003)]. For example, 2-categories of differentiable, topological and algebraic stacks are localisations of suitable 2-categories of groupoids. Pronk's results can be applied to other settings where the 2-category and the class of morphisms at which one wants to localise, satisfy some properties that permits the use of a much simpler calculus of fractions linked with the notion of anafunctor introduced by \textit{M. Makkai} [J. Pure Appl. Algebra 108, No. 2, 109--173 (1996; Zbl 0859.18001)] and \textit{T. Bartels} in [``Higher gauge theory I: 2-bundles'', Ph. D. thesis, University of California Riverside (2006), \url{arXiv:math/0410328}]. In [Theory Appl. Categ. 26, 788--829 (2012; Zbl 1275.18023)], the author of the paper under review proved that a sub-2-category of the 2-category of categories internal to a subcanonical site, admit a bicategory of fractions. In this case anafunctors calculate this localization. The main contribution of this paper is to show that for a 2-category with the structure of a 2-site of certain form (all covering maps must be representable fully faithful) Pronk's bicategory of fractions exists. To prove this result, the author does not assume existence of enough fibrant objects or projectives to prove local (essential) smallness. The theorems obtained in this work are sufficient to imply the applications of the abstract framework proposed by \textit{D. A. Pronk} and \textit{M. A. Warren} in [Theory Appl. Categ. 29, 836--873 (2014; Zbl 1361.18003)] where, as in the paper of \textit{O. Abbad} and \textit{E. M. Vitale} [Cah. Topol. Géom. Différ. Catég. 54, No. 3, 221--239 (2013; Zbl 1305.18013)], the localizations was constructed via fibrancy/projectivity conditions.