Putting the Brauer back in Brauer-Picard

Abstract

We establish a 6-term left exact sequence, involving Galois cohomology of the base field $\mathbb K$, and the Brauer-Picard groupoid of a fusion category. This generalizes a result of Etingof, Nikshych, and Ostrik to the setting where $\mathbb K$ is not algebraically closed. Following their example, we use this exact sequence to compute examples of graded extensions of fusion categories over $\mathbb R$. Along the way, we establish several structural theorems regarding the duality morphisms for a fusion category as an object in the 4-category of braided tensor categories. The paper ends with a speculative look at a potential higher categorical explanation of the main result.