Lusternik-Schnirelmann category for categories and classifying spaces

This paper presents a notion of Lusternik-Schnirelmann category for small categories, which is an invariant under homotopy equivalences based on natural transformations. We focus on the relationship between this categorical Lusternik-Schnirelmann category and the classical one via the classifying space. We provide a combinatorial method to calculate the classical Lusternik-Schnirelmann category of the classifying space of a finite acyclic category, taking the barycentric subdivision into account. Moreover, we establish the product inequality for fibered and cofibered categories as an analogue of the inequality of the classical Lusternik-Schnirelmann category for Hurewicz fibrations.