A Picard–Brauer exact sequence of categorical groups

A categorical group is a monoidalgroupoid in which each object has a tensorial inverse. Two main examples are the Picard categorical group of a monoidal category and the Brauer categorical group of a braided monoidal category with stable coequalizers. After discussing the notions of kernel, cokernel and exact sequence for categorical groups, we show that, given a suitable monoidalfunctor between two symmetric monoidal categories with stable coequalizers, it is possible to build up a five-term Picard–Brauer exact sequence of categorical groups. The usual Units-Picard and Picard–Brauer exact sequences of abelian groups follow from this exact sequence of categorical groups. We also discuss the direct sum decomposition of the Brauer–Long group.