The Lusternik-Schnirelmann category of a Lie groupoid

Abstract

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.

Internal Absolute Geometry

“…Proof. The assertion follows mutatis mutandis, given our assumptions, from the considerations of section 3 in [5].…”

“…To any co-span of internal groupoids we define an associated homotopy pullback of the n-th degree. If these homotopy pullbacks fulfill a certain vertical filtration condition we meet the requirements to generalize the construction of the Morita invariant homotopy type constructed for the case of Lie groupoids in [5].…”

“…The proof relies essentially on associativity which is furnished by the existence of vertically filtered homotopy pullbacks. The remainder of the argument follows mutatis mutandis the discussion of [5]. The resulting bicategory fulfills the following universal property: There is a bifunctor U : AHG 2 → AHG 2 [W −1 ] that sends essential homotopy equivalences (∼ eh ) to weak generalized equivalences (∼ W ) and is universal with that property, i.e.…”

“…[15]. Recently, the invariant has been extended to the case of differentiable stacks [1] and Lie groupoids [5]. As it seems more suitable, we refer to the generalized version of the invariant henceforth as geometric complexity.…”

The Lusternik-Schnirelmann Category for a Differentiable Stack

“…We have the following important property of the groupoid Lusternik-Schnirelmann category (see [Co2]). The groupoid Lusternik-Schnirelmann category also generalizes the ordinary Lusternik-Schnirelmann category of a smooth manifold.…”