Invertible Fusion Categories
Organizers
Speaker
Sean Sanford
Time
Tuesday, December 3, 2024 9:40 AM - 11:15 AM
Venue
A3-2-301
Online
Zoom 468 248 1222
(BIMSA)
Abstract
A tensor category $\mathcal C$ over a field $\mathbb K$ is said to be invertible if there's a tensor category $\mathcal D$ such that $\mathcal C \boxtimes \mathcal D$ is Morita equivalent to $\mathbf{Vec}_{\mathbb K}$. When $\mathbb K$ is algebraically closed, it is well-known that the only invertible fusion category is $\mathbf{Vec}_{\mathbb K}$, and any invertible multi-fusion category is Morita equivalent to $\mathbf{Vec}_{\mathbb K}$. By contrast, we show that for general $\mathbb K$ the invertible multi-fusion categories over a field $\mathbb K$ are classified (up to Morita equivalence) by $H^3(\mathbb K;\mathbb G_m)$, the third Galois cohomology of the absolute Galois group of $\mathbb K$. In this talk, we will provide some motivation for the study of these invertible categories, and discuss the main technique that allows for this classification: categorical inflation. If there is extra time, we will talk about the implications this classification has for invertible fusion 2-categories.