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Nguyen Tien Quang
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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).
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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 $H^3_{textrm{MaL}}(R,M).$}
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
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.
This paper presents the proof of the coherence theorem for Ann-categories whose set of axioms and original basic properties were given in [9]. Let $$A=(A,{Ah},c,(0,g,d),a,(1,l,r),{Lh},{Rh})$$ be an Ann-category. The coherence theorem states that in the category $ A$, any morphism built from the above isomorphisms and the identification by composition and the two operations $tx$, $ts$ only depends on its source and its target. The first coherence theorems were built for monoidal and symmetric monoidal categories by Mac Lane [7]. After that, as shown in the References, there are many results relating to the coherence problem for certain classes of categories. For Ann-categories, applying Hoang Xuan Sinhs ideas used for Gr-categories in [2], the proof of the coherence theorem is constructed by faithfully ``embedding each arbitrary Ann-category into a quite strict Ann-category. Here, a {it quite strict} Ann-categogy is an Ann-category whose all constraints are strict, except for the commutativity and left distributivity ones. This paper is the work continuing from [9]. If there is no explanation, the terminologies and notations in this paper mean as in [9].
In this paper, we study tensor (or monoidal) categories of finite rank over an algebraically closed field $mathbb F$. Given a tensor category $mathcal{C}$, we have two structure invariants of $mathcal{C}$: the Green ring (or the representation ring) $r(mathcal{C})$ and the Auslander algebra $A(mathcal{C})$ of $mathcal{C}$. We show that a Krull-Schmit abelian tensor category $mathcal{C}$ of finite rank is uniquely determined (up to tensor equivalences) by its two structure invariants and the associated associator system of $mathcal{C}$. In fact, we can reconstruct the tensor category $mathcal{C}$ from its two invarinats and the associator system. More general, given a quadruple $(R, A, phi, a)$ satisfying certain conditions, where $R$ is a $mathbb{Z}_+$-ring of rank $n$, $A$ is a finite dimensional $mathbb F$-algebra with a complete set of $n$ primitive orthogonal idempotents, $phi$ is an algebra map from $Aotimes_{mathbb F}A$ to an algebra $M(R, A, n)$ constructed from $A$ and $R$, and $a={a_{i,j,l}|1< i,j,l<n}$ is a family of invertible matrices over $A$, we can construct a Krull-Schmidt and abelian tensor category $mathcal C$ over $mathbb{F}$ such that $R$ is the Green ring of $mathcal C$ and $A$ is the Auslander algebra of $mathcal C$. In this case, $mathcal C$ has finitely many indecomposable objects (up to isomorphisms) and finite dimensional Hom-spaces. Moreover, we will give a necessary and sufficient condition for such two tensor categories to be tensor equivalent.
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