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.