Structure of Ann-categories

Abstract

Each Ann-category $\A$ is equivalent to an Ann-category of the type $(R,M),$ where $M$ is an $R$-bimodule. The family of constraints of $A$ induces a {\it structure} on $(R,M).$ The main result of the paper is: 1. {\it There exists a bijection between the set of structures on $(R,M)$ and the group of Mac Lane 3-cocycles $Z^{3}_{MaL}(R, M).$} 2. {\it There exists a bijection between $C(R,M)$ of congruence classes of Ann-categories whose pre-stick is of the type $(R,M)$ and the Mac Lane cohomology group $H^3_{\textrm{MaL}}(R,M).$}