We start by considering spaces with singularities: in these spaces, the local structure is not homogeneous, but varies throughout the space, with certain points associated with certain singularity types. Lie groupoids provide a way of encoding these spaces. However, this encoding is not unique: different Lie groupoids may represent the same underlying space. This leads to the idea of 'Morita equivalence' of Lie groupoids: two groupoids are Morita equivalent if they encode the same underlying geometric structure. To make this formal, we turn to the notation of 'localization', where we take a class of arrows and create a 'bicategory of fractions' in which these are formally inverted, becoming isomorphisms. This is the framework we use to build a category that represents our spaces: we will wind up with a bicategory of fractions of Lie groupoids in which all Morita equivalences become isomorphisms.
The goal of this talk is to explain this categorical construction and convince you that it is geometrically motivated and meaningful. I will start with the geometric intuition of how Lie groupoids are used to encode singular spaces, and how Morita equivalence arises as a natural consequence, with examples and pictures. I will then explain the construction, starting from the familiar idea of localizing a prime ideal in a commutative ring and extending to more complicated contexts, culminating in the bicategory of fractions we are interested in. At the end I will mention applications and current work in this area. No background with Lie groupoids or bicategories is assumed and all scary commutative diagrams will be thoroughly explained and motivated.
Morita Equivalence of Lie Groupoids Sponsored by the Meyer Fund