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Register a new userStructure of Ann-categories and Mac Lane - Shukla cohomology
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Authors
Nguyen Tien Quang
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In this paper we study the structure of a class of categories having two operations which satisfy axioms analoguos to that of rings. Such categories are called Ann - categories. We obtain the classification theorems for regular Ann - categories and Ann - functors by using Mac Lane - Shukla cohomology of rings. These results give new interpretations of the cohomology groups and of the rings
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Every ring extension of $A$ by $R$ induces a pair of group homomorphisms $mathcal{L}^{*}:Rto End_Z(A)/L(A);mathcal{R}^{*}:Rto End_Z(A)/R(A),$ preserving multiplication, satisfying some certain conditions. A such 4-tuple $(R,A,mathcal{L}^{*},mathcal{R}^{*})$ is called a ring pre-extension. Each ring pre-extension induces a $R$-bimodule structure on bicenter $K_A$ of ring $A,$ and induces an obstruction $k,$ which is a 3-cocycle of $Z$-algebra $R,$ with coefficients in $R$-bimodule $K_A$ in the sense of Shukla. Each obstruction $k$ in this sense induces a structure of a regular Ann-category of type $(R,K_A).$ This result gives us the first application of Ann-category in extension problems of algebraic structures, as well as in cohomology theories.
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 $H^3_{textrm{MaL}}(R,M).$}
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).
In this paper we present some applications of Ann-category theory to classification of crossed bimodules over rings, classification of ring extensions of the type of a crossed bimodule.
In this paper, we have studied the axiomatics of {it Ann-categories} and {it categorical rings.} These are the categories with distributivity constraints whose axiomatics are similar with those of ring structures. The main result we have achieved is proving the independence of the axiomatics of Ann-category definition. And then we have proved that after adding an axiom into the definition of categorical rings, we obtain the new axiomatics which is equivalent to the one of Ann-categories.
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