Lusternik–Schnirelmann category for categories and classifying spaces

Abstract

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.