On the Lusternik-Schnirelmann category of a simplicial map

TL;DR

This paper introduces a new invariant called the Lusternik-Schnirelmann category of a simplicial map, extending classical concepts to simplicial complexes and exploring its properties and applications.

Contribution

It generalizes the simplicial category to maps, relates it to classical categories, and connects it to the essential category weight in a simplicial setting.

Findings

Properties of the simplicial category of a map are established.

The relation to classical Lusternik-Schnirelmann category is demonstrated.

Applications to products, fibrations, and essential category weight are shown.

Abstract

In this paper, we study the Lusternik-Schnirelmann category of a simplicial map between simplicial complexes, generalizing the simplicial category of a complex to that of a map. Several properties of this new invariant are shown, including its relevance to products and fibrations. We relate this category of a map to the classical Lusternik-Schnirelmann category of a map between finite topological spaces. Finally, we show how the simplicial category of a map may be used to define and study a simplicial version of the essential category weight of J. Strom.