Higher cohomologies of commutative monoids

TL;DR

This paper extends cohomology theories to commutative monoids, establishing new relationships with existing theories and providing classifications and computations for specific monoids.

Contribution

It introduces higher cohomologies for commutative monoids, connecting them with established theories and applying them to classify symmetric monoidal groupoids.

Findings

Cohomological classification of symmetric monoidal groupoids

Computations for cyclic monoids

Relationships with Leech and Grillet cohomologies

Abstract

Extending Eilenberg-Mac Lane's methods, higher level cohomologies for commutative monoids are introduced and studied. Relationships with pre-existing theories (Leech, Grillet, ...) are stated. The paper includes a cohomological classification for symmetric monoidal groupoids and several computations for cyclic monoids.