Tensor categories attached to double groupoids

Abstract

The construction of a quantum groupoid out of a double groupoid satisfying a filling condition and a perturbation datum is given. This extends previous work that appeared in math.QA/0308228. Several important classes of examples of tensor categories are shown to fit into this construction. Certain invariants such as a pivotal group-like element and quantum and Frobenius-Perron dimensions of simple objects are computed.

Revised version to appear in Adv. Math., 45 pages. Added examples where the square of the antipode is not the identity, necessary and sufficient conditions for the representation category to be fusion, and the computation of quantum and Frobenius-Perron dimensions of simple representations