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

The Lusternik-Schnirelmann category of a Lie groupoid