Homology, Homotopy and Applications

Volume 22 (2020)

Number 2

Braided categorical groups and strictifying associators

Pages: 295 – 321

DOI: https://dx.doi.org/10.4310/HHA.2020.v22.n2.a19


Oliver Braunling (Mathematical Institute, University of Freiburg, Freiburg im Breisgau, Germany)


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.


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

2010 Mathematics Subject Classification

18D10, 19D23

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

Received 27 November 2019

Accepted 10 February 2020

Published 13 May 2020