Lusternik-Schnirelmann category for categories and classifying spaces (Q1709053)

The Lusternik-Schnirelmann category (in short LS-category) of a topological space \(X\) is the minimal number of open subsets, \(U_{i}\), that cover \(X\) and such that the inclusions \(U_{i}\hookrightarrow X\) are null homotopic. This integer is a homotopical invariant. In the paper under review, the author defines the LS-category of a small category and relates it to the LS-category of its classifying space. This number turns out to be invariant under homotopy equivalences based on natural transformations. The author also compares it to the simplicial LS-category of a finite simplicial complex, introduced by \textit{D. Fernández-Ternero} et al. [Topology Appl. 194, 37--50 (2015; Zbl 1327.55004)]. The behaviour of the classical LS-category in the case of a Hurewicz fibration is adapted to this context in terms of fibered and cofibered category.