arXiv subjects

Nguyen Tien Quang

Publications and source records attributed to Nguyen Tien Quang.

24 records · Page 2Linked to original sources

arXiv2009

On Gr-Functors between Gr-Categories: Obstruction theory for Gr-Functors of the type $(φ,f)$

Each Gr-functor of the type $(φ,f)$ of a Gr-category of the type $(Π,\C)$ has the obstruction be an element $\overline{k}\in H^3(Π,\C).$ When this obstruction vanishes, there exists a bijection between congruence classes of Gr-functors of the type $(φ,f)$ and the cohomology group $H^2(Π,\C).$ Then the relation of Gr-category theory and the group extension problem can be established and used to prove that each Gr-category is Gr-equivalent to a strict one.

math.CT↗

arXiv2009

The Relation between Ann-Categories and Ring Categories

There are different categorizations of the definition of a {\it ring} such as {\it Ann-category} (see N. T. Quang [6]), {\it ring category} (see M. Kapranov and V.Voevodsky [2]),... The main result of this paper is to prove that every axiom of the definition of a {\it ring category}, without the axiom $x_{0}=y_{0},$ can be deduced from the axiomatics of an {\it Ann-category}.

math.CT↗

arXiv2007

On the Axiomatics of Ann-Categories

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.

math.CT↗

arXiv2007

Structure of Ann-categories and Mac Lane - Shukla cohomology

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

math.CT↗

arXiv2007

The Coherence Theorem for 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 Sinh's 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].

math.CT↗

arXiv2007

Ring extension problem, Shukla cohomology and Ann-category theory

Every ring extension of $A$ by $R$ induces a pair of group homomorphisms $\mathcal{L}^{*}:R\to End_\Z(A)/L(A);\mathcal{R}^{*}:R\to End_\Z(A)/R(A),$ 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.

math.CT↗