Braided categorical groups and strictifying associators

A key invariant of a braided categorical group is its quadratic form, introduced by Joyal and Street. We show that the categorical group is braided equivalent to a simultaneously skeletal and strictly associative one if and only if the quadratic form comes from a bilinear form. This generalizes the result of Johnson–Osorno that all Picard groupoids can simultaneously be strictified and skeletalized, except that in the braided case there is a genuine obstruction.

Keywords

braided categorical group, Picard groupoid, strictification, skeletalization, associator

2010 Mathematics Subject Classification

18D10, 19D23

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The author was supported by DFG GK1821 “Cohomological Methods in Geometry”.

Received 27 November 2019

Accepted 10 February 2020

Published 13 May 2020