An alternative description of symmetric monoidal categories, and symmetric 2-groups

TL;DR

The paper presents a new way to describe symmetric monoidal categories using combined associo-commutator isomorphisms, simplifying coherence conditions and enabling a cohomological classification of symmetric 2-groups.

Contribution

It introduces an alternative description of symmetric monoidal categories and symmetric 2-groups, connecting them to Eilenberg-MacLane cubical cohomology.

Findings

New description simplifies coherence laws

Provides cohomological classification of symmetric 2-groups

Establishes links to Eilenberg-MacLane cubical cohomology

Abstract

An equivalent description of a symmetric monoidal category is introduced in which, instead of separate associator and commutator isomorphisms satisfying the usual coherence axioms, we simply have associo-commutator isomorphisms satisfying their own coherence laws. In particular, this yields an alternative description of a symmetric 2-group and leads to a cohomological classification of these objects in terms of Eilenberg-MacLane cubical cohomology for abelian groups.