## 15/07/2010, Thursday, 16:15–17:15

Jeffrey Morton, University of Western Ontario
2-Linearization, Discrete and Smooth

This talk will introduce the construction "2-linearization", an extension of the "groupoidification" program of Baez and Dolan which seeks to interpret constructions in linear algebra in terms of groupoids and spans of groupoids. The 2-linearization construction finds this as a special case of a 2-functor $\Lambda :\mathrm{Span}\left(\mathrm{Gpd}\right)\to 2\mathrm{Vect}$, where $2\mathrm{Vect}$ is the 2-category of Kapranov-Voevodsky 2-vector spaces. This 2-functor is constructed in terms of pairs of ambi-adjoint functors associated to each groupoid homomorphism, the "push" and "pull" operations, closely related to restriction and induction maps in representation theory, Grothendieck's 6-operation framework for sheaves, among other examples. We will begin with the discrete case, and consider generalizations to smooth groupoids. Finally we will consider some applications.
Support: FCT, CAMGSD, New Geometry and Topology