Drinfeld centers for bicategories

TL;DR

This paper extends the concept of Drinfeld centers from tensor categories to bicategories, providing computational tools and an obstruction theory to understand their invariants across various examples.

Contribution

It introduces a spectral sequence for computing invariants of Drinfeld centers in bicategories and develops an obstruction theory to distinguish them from classifying category centers.

Findings

Spectral sequence for invariants computation

Obstruction theory for Drinfeld centers

Applications to bicategories of groups, rings, and fusion categories

Abstract

We generalize Drinfeld's notion of the center of a tensor category to bicategories. In this generality, we present a spectral sequence to compute the basic invariants of Drinfeld centers: the abelian monoid of isomorphism classes of objects, and the abelian automorphism group of its identity object. There is an associated obstruction theory that explains the difference between the Drinfeld center and the center of the classifying category. For examples, we discuss bicategories of groups and bands, rings and bimodules, as well as fusion categories.