Title: Morita equivalence and bicategories of fractions: a simple geometric idea gets out of hand.
Abstract: We start by considering the geometry of spaces with singularities: in these spaces, the local geometry is not homogeneous as in a manifold, but varies from point to point. Lie groupoids provide a way of encoding this information. However, this encoding is not unique: different Lie groupoids may represent the same singular 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 formalize this equivalence we consider the ‘localization’ of Lie groupoids, which creates a ‘bicategory of fractions’ in which the Morita equivalences become isomorphisms.
The goal of this talk is to describe this bicategory of fractions construction in all its diagrammatic glory, 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 localization 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 diagrams will be thoroughly explained and motivated.
This event will be in a hybrid format. Besides the option to attend in person the presentation will be streamed on zoom. For the zoom link and password, please contact firstname.lastname@example.org