Taher Abualrub Abdul Salam Jarrah Sadok Kallel - Mathematics across contemporary sciences AUS-ICMS, Sharjah, UAE, April 2015

(1)

(2)

The Lusternik-Schnirelmann category or LS-category of a manifold is a numerical invariant introduced by Lusternik and Schnirelmann [].

Fundamental in the definition of LS-category of a smooth manifold or topological space is the concept of a categorical set. A subset of a space is said to be categorical if it is contractible in the space. The Lusternik-Schnirelmann category of a smooth manifold X is defined to be the least number of categorical open sets required to cover X , if that number is finite, otherwise the category is said to be infinite.

In this article, we generalize the notion of Lusternik-Schnirelmann category to differentiable stacks with the intention of providing a useful tool and invariant to study homotopy theory, the theory of geodesics and Morse theory of differentiable stacks. Differentiable stacks naturally generalize smooth manifolds and orbifolds and are therefore of interest in many areas of geometry, topology and mathematical physics. They are basically generalized smooth spaces where its points also have automorphism groups. For example, they often appear as an adequate replacement of quotients for general Lie group actions on smooth manifolds, especially when the naive quotient does not exist as a smooth manifold. Many moduli and classification problems like for example the classification of Riemann surfaces or vector bundles on Riemann surfaces naturally lead to the notion of a differentiable stack. It can be expected that the stacky LS-category will be a very useful topological invariant for these kind of generalized smooth spaces which appear naturally in geometry and physics. We aim to study the geometrical and topological aspects of the stacky LS-category and its applications in a follow-up article [].

The new notion of stacky LS-category for differentiable stacks presented here employs the notion of a categorical substack and is again an invariant of the homotopy type of the differentiable stack, in fact of the underlying topological stack. It generalizes the classical LS-category for manifolds [] for Lie groupoids.

The material of this article is organised as follows: In the first section we collect the basic definitions of differentiable stacks and Lie groupoids and establish some fundamental properties. In particular we exhibit the various connections between differentiable stacks and Lie groupoids. The second section recalls the foundations of Lusternik-Schnirelmann category for groupoids and its Morita invariance. In the third section we introduce the new notion of stacky Lusternik-Schnirelmann category and establish its relationship with the groupoid LS-category of the various groupoids introduced in the Sect..

In this section we will collect in detail the notions and some of the fundamental properties of differentiable stacks and Lie groupoids, which we will use later. We refer the reader to various resources on differentiable stacks [] for more details and specific examples and their interplay.

Differentiable stacks are defined over the category of smooth manifolds. A smooth manifold here will always mean a finite dimensional second countable smooth manifold, which need not necessarily be Hausdorff. We denote the category of smooth manifolds and smooth maps by .

A submersion is a smooth map such that the derivative is surjective for all points . The dimension of the kernel of the linear map is a locally constant function on U and called the relative dimension of the submersion f . An tale morphism is a submersion of relative dimension 0. This means that a morphism f between smooth manifolds is tale if and only if f is a local diffeomorphism.

The tale site on the category is given by the following Grothendieck topology on . We call a family of morphisms in with target X a covering family of X , if all smooth maps are tale and the total map is surjective. This defines a pretopology on generating a Grothendieck topology, the tale topology on (see [], Expos II).

As remarked in [], not all fibre products for two morphisms and exist in , but if at least one of the two morphisms is a submersion, then the fibre product