On the braiding of an Ann-category

TL;DR

This paper explores the structure of braided Ann-categories, analyzing how distributivity constraints relate to other axioms and connecting these concepts to distributivity and ring-like categories, with the center construction as an example.

Contribution

It clarifies the dependence of distributivity constraints on axioms and relates braided Ann-categories to distributivity and ring-like categories, providing new insights into their structure.

Findings

Distributivity constraints depend on other axioms.

Braided Ann-categories relate to distributivity and ring-like categories.

Center construction yields an example of an unsymmetric braided Ann-category.

Abstract

A braided Ann-category $\A$ is an Ann-category $\A$ together with the braiding $c$ such that $(\A, \otimes, a, c, (I,l,r))$ is a braided tensor category, and $c$ is compatible with the distributivity constraints. The paper shows the dependence of the left (or right) distributivity constraint on other axioms. Hence, the paper shows the relation to the concepts of {\it distributivity category} due to M. L. Laplaza and {\it ring-like category} due to A. Frohlich and C.T.C Wall. The center construction of an almost strict Ann-category is an example of an unsymmetric braided Ann-category.