On the third cohomology group of commutative monoids

TL;DR

This paper explores the third cohomology group of commutative monoids, providing a classification that extends known results for Picard categories to a broader context involving symmetric monoidal abelian groupoids.

Contribution

It introduces a new interpretation of third cohomology classes for commutative monoids and generalizes classification results for symmetric monoidal structures.

Findings

Provides a classification of symmetric monoidal abelian groupoids related to third cohomology

Extends the classification of Picard categories to a more general setting

Connects cohomology classes with categorical structures in monoids

Abstract

We interpret Grillet's symmetric thrid cohomology classes of commutative monoids in terms of strictly symmetric monoidal abelian groupoids. We state and prove a classification result that generalizes the well-known one for strictly commutative Picard categories by Deligne and Fr\"ohlich and Wall, and Sinh.