The Lusternik-Schnirelmann category of a Lie groupoid

TL;DR

This paper introduces a new homotopy invariant for Lie groupoids, extending the classical Lusternik-Schnirelmann category to orbifolds, and proves an orbifold version of the LS-theorem for critical points.

Contribution

It develops a bicategorical approach to define a homotopy invariant for Lie groupoids that generalizes the classical LS-category and is invariant under Morita equivalence.

Findings

Defines a new homotopy invariant for Lie groupoids.

Establishes invariance under Morita equivalence.

Proves an orbifold version of the LS-theorem for critical points.

Abstract

We propose a new homotopy invariant for Lie groupoids which generalizes the classical Lusternik-Schnirelmann category for topological spaces. We use a bicategorical approach to develop a notion of contraction in this context. We propose a notion of homotopy between generalized maps given by the 2-arrows in a certain bicategory of fractions. This notion is invariant under Morita equivalence. Thus, when the groupoid defines an orbifold, we have a well defined LS-category for orbifolds. We prove an orbifold version of the classical Lusternik-Schnirelmann theorem for critical points.