A Picard-Brauer exact sequence of categorical groups

Vitale, Enrico
(2002) Journal of Pure and Applied Algebra — Vol. 175, n° 1-3, p. 383-408 (2002)

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Abstract
A categorical group is a monoidal groupoid 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, cokemel and exact sequence for categorical groups, we show that, given a suitable monoidal functor 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. (C) 2002 Elsevier Science B.V. All rights reserved.
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Vitale, E. (2002). A Picard-Brauer exact sequence of categorical groups. Journal of Pure and Applied Algebra, 175(1-3), 383-408. https://doi.org/10.1016/S0022-4049(02)00142-1 (Original work published 2002)