A model structure on the category of small categories for coverings (Q2839628)

This article provides two points of view on a model structure on the category of small categories. The homotopical one goes as follows. There is a model structure on \textbf{Cat}, due to \textit{A. Joyal} and \textit{M. Tierney} in a more general context [C. R. Math. Acad. Sci., Soc. R. Can. 13, No. 1, 2--6 (1991; Zbl 0733.18011)], where the weak equivalences are the categorical equivalences. This category is actually combinatorial, simplicial, and left proper so that one can apply the formalism of Bousfield localization with respect to the ``groupoidification'' \(\{ 0 \rightarrow 1\} \longrightarrow \{ 0 \leftrightarrows 1\}\). Weak equivalences are then \(1\)-equivalences, i.e. functors inducing isomorphisms on connected components and fundamental groups. This model category structure on small categories is in fact Quillen equivalent to the analogous one on simplicial sets via the nerve.NEWLINENEWLINEThe second point of view is categorical as all constructions can be made very explicitly. The groupoidification of a category is constructed and coverings are defined as having the unique right lifting property with respect to both inclusions of the trivial category \(*\) into \(\{ 0 \rightarrow 1\}\) so that a covering of categories corresponds exactly to coverings of simplicial sets via the nerve. In fact, coverings are fibrations with discrete fibers, the fibrant replacement of a category \(C\) is its groupoidification \(\pi C\), and the cofibrant replacement of a \textit{pointed} and connected category \(C\) is its universal cover \(\widetilde C\).