Braided categorical groups and strictifying associators

TL;DR

This paper characterizes when braided categorical groups can be strictified and skeletalized, linking this to properties of their quadratic forms and revealing an obstruction absent in Picard groupoids.

Contribution

It generalizes Johnson-Osorno's result by identifying a specific obstruction for strictifying braided categorical groups based on quadratic form polarization.

Findings

Braided categorical groups are equivalent to strict and skeletal ones if polarization equals symmetrization.

A genuine obstruction exists for strictification in the braided case.

The result extends the understanding of strictification beyond Picard groupoids.

Abstract

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 polarization of this quadratic form is the symmetrization of 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.