3 - Sheaves on Lie groupoids

Summary

In Chapter 2 we have introduced the notion of a Lie groupoid, and we have shown how singular spaces (such as the space of leaves of a foliation) can be represented by weak equivalence classes of Lie groupoids.

The purpose of the present chapter is two-fold. First, we will introduce the category of equivariant sheaves for a Lie groupoid G, as well as its derived category. As will become apparent, these categories are particularly relevant for étale groupoids, and more generally for foliation groupoids. Secondly, we will give an explicit construction of the fundamental groupoid π₁(G) associated to a Lie groupoid G. These constructions are related in a Grothendieck style way, by the fact that the fundamental groupoid is determined by the locally constant sheaves.

We begin in Section 3.1 with an explicit description of equivariant sheaves for a groupoid, and give some examples. In Section 3.2 we study how the category of G-sheaves behaves with respect to homomorphisms of Lie groupoids. We show that the category of G-sheaves is in fact functorial with respect to the generalized morphisms, and that a weak (or Morita) equivalence between Lie groupoids induces an equivalence of the associated categories of sheaves.

In Section 3.3 we discuss the fundamental group of a Lie groupoid. More precisely, we construct for each Lie groupoid G a new Lie groupoid π₁(G). This groupoid is weakly equivalent to a discrete groupoid. In particular, the isotropy groups of π₁(G) are discrete. If G is connected, these isotropy groups of π₁(G) are all isomorphic to each other, and will be referred to as ‘the’ fundamental group of G.