Cohomological classification of braided $Ann$-categories

TL;DR

This paper classifies braided $Ann$-categories using cohomology of commutative rings, showing they are equivalent to certain algebraic structures, thus extending the understanding of their categorical and algebraic properties.

Contribution

It provides a cohomological classification theorem for braided $Ann$-categories, linking their structure to algebraic cohomology, and demonstrates their equivalence to $(R,M)$-type categories.

Findings

Every braided $Ann$-category is equivalent to a braided $(R,M)$-category.

The classification is achieved via cohomology of commutative rings.

The paper establishes a structure transport theorem for these categories.

Abstract

A braided $Ann$-category $\mathcal A$ is an $Ann$-category $\mathcal A$ together with a braiding $c$ such that $(\mathcal A, \otimes, a, c, (1,l,r))$ is a braided tensor category, moreover $c$ is compatible with the distributivity constraints. According to the structure transport theorem, the paper shows that each braided $Ann$-category is equivalent to a braided $Ann$-category of the type $(R,M)$, hence the proof of the classification theorem for braided $Ann$-categories by the cohomology of commutative rings is presented.