The Lusternik-Schnirelmann category of a Lie groupoid

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.

💡 Research Summary

The paper introduces a homotopy invariant for Lie groupoids that extends the classical Lusternik‑Schnirelmann (LS) category from topological spaces to a much broader categorical setting. The authors begin by recalling the structure of Lie groupoids and organizing them into a 2‑category 𝔾pd whose objects are groupoids, 1‑morphisms are “spans’’ (i.e. diagrams H ← K → G), and 2‑morphisms are natural transformations between such spans. To capture Morita equivalence—a central notion of equivalence for groupoids—they localize this 2‑category with respect to the class W of essential equivalences, obtaining a bicategory of fractions 𝔾pd