Structure of Ann-Categories

TL;DR

This paper explores the structure of Ann-categories, showing how they can be reduced to simpler forms and characterized by three key invariants including a cohomology class, thus advancing the understanding of their algebraic properties.

Contribution

It introduces a method to convert Ann-categories into reduced forms and identifies three invariants that uniquely determine their structure.

Findings

Ann-categories can be reduced to a form of type (R,M) with five structure functions.

Each Ann-category is uniquely determined by a ring, a bimodule, and a cohomology class.

The structure is characterized by invariants including MacLane's ring cohomology.

Abstract

This paper presents the structure conversion by which from an Ann-category $\A,$ we can obtain its reduced Ann-category of the type $(R,M)$ whose structure is a family of five functions $k=(\xi,\eta,\alpha,\lambda,\rho)$. Then we will show that each Ann-category is determined by three invariants: 1. The ring $\Pi_0(\A)$ of the isomorphic classes of objects of $\A$, 2. $\Pi_0(\A)$-bimodule $\Pi_1(\A) = \Aut_{\A}(0),$ 3. The element $ \bar{k}\in H^{3}_{M}(\Pi_0(\A), \Pi_1(\A))$ (the ring cohomology due to MacLane).