Crossed modules and symmetric cohomology of groups

This paper links the third symmetric cohomology (introduced by Staic [10] and Zarelua [12]) to crossed modules with certain properties. The equivalent result in the language of $2$‑groups states that an extension of $2$-groups corresponds to an element of $HS^3$ iff it possesses a section which preserves inverses in the $2$‑categorical sense. This ties in with Staic’s (and Zarelua’s) result regarding $HS^2$ and abelian extensions of groups.

Keywords

group cohomology, crossed modules, symmetric cohomology

2010 Mathematics Subject Classification

18D05, 20J06

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This research was supported by the EPSRC grant EP/N014189/1 “Joining the dots: from data to insight”.

Received 1 April 2019

Received revised 25 July 2019

Accepted 28 August 2019

Published 15 April 2020