Structure of Ann-categories of type \((R,N)\) (Q817071)

The paper under review is the fifth article of a series devoted to the study of Ann-categories [\textit{N.T. Quang}, Tap Chí Toán Hoc 20, 41--47 (1992; Zbl 0940.18502), ibid. 16, 17--26 (1988; Zbl 0940.18501), ibid. 15, 14--24 (1987; Zbl 0940.18503)]. After short recalls on Ann-categories and the definition of Ann-categories of type \((R,N)\), the author proves the main theorem of this article: ``Any Ann-categories \(\mathcal{A}\) is Ann-equivalent to an almost strict Ann-category of type \((R,N)\)''. In fact it comes from the definition of an Ann-category and the reduced Ann-category that \(N\) is an abelian group and \(R\) is a ring. It follows from this theorem that each structure of a reduced Ann-category of type \((R,N)\) is a pair of function: \(\eta : R\times R \rightarrow N\) and \(\lambda \text{ : } R\times R\times R \rightarrow N\) which must verify nine conditions. The author emphasizes that if the Ann-category of type \((R,N)\) is regular (the commutativity constraint on \(\oplus\) is the identity) then the structure of this category is an element of the cohomology group \(H^3(R,N)\).