arXiv subjects
Publications and source records attributed to Nguyen Tien Quang.
Prolongations of a group extension can be studied in a more general situation that we call group extensions of the co-type of a crossed module. Cohomology classification of such extensions is obtained by applying the obstruction theory of monoidal functors.
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 $k=(ξ,η,α,λ,ρ)$. Then we will show that each Ann-category is determined by three invariants: 1. The ring $Π_0(\A)$ of the isomorphic classes of objects of $\A$, 2. $Π_0(\A)$-bimodule $Π_1(\A) = \Aut_{\A}(0),$ 3. The element $ \bar{k}\in H^{3}_{M}(Π_0(\A), Π_1(\A))$ (the ring cohomology due to MacLane).
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 state the notion of morphisms in the category of abelian crossed modules and prove that this category is equivalent to the category of strict Picard categories and regular symmetric monoidal functors. The theory of obstructions for symmetric monoidal functors and symmetric cohomology groups are applied to show a treatment of the group extension problem of the type of an abelian crossed module.
The aim of this paper is to study co-prolongations of central extensions. We construct the obstruction theory for co-prolongations and classify the equivalence classes of these by kernels of a homomorphisms between 2-dimensional cohomology groups of groups.
If $Γ$ is a group, then braided $Γ$-crossed modules are classified by braided strict $Γ$-graded categorial groups. The Schreier theory obtained for $Γ$-module extensions of the type of an abelian $Γ$-crossed module is a generalization of the theory of $Γ$-module extensions.
The notion of Ann-categories is a categorification of the ring structure. Regular Ann-categories were classified by Shukla algebraic cohomology. In this article, we state and prove the precise theorem on classification for the general case due to Mac Lane cohomology for rings. And an application for classification problem of ring extensions is also introduced.
In this paper we study equivariant crossed modules in its link with strict graded categorical groups. The resulting Schreier theory for equivariant group extensions of the type of an equivariant crossed module generalizes both the theory of group extensions of the type of a crossed module and the one of equivariant group extensions.
The aim of this paper is to study the $(α, γ)$-prolongation of central extensions. We obtain the obstruction theory for $(α, γ)$-prolongations and classify $(α, γ)$-prolongations thanks to low-dimensional cohomology groups of groups.
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 state some applications of Gr-category theory on the classification of crossed modules and on the classification of extensions of groups of the type of a crossed module.
In this paper, we state and prove precise theorems on the classification of the category of (braided) categorical groups and their (braided) monoidal functors, and some applications obtained from the basic studies on monoidal functors between categorical groups.
In this paper, we show another proof of the problem by constructing a strict monoidal category M(C) consisting of M-functors and M-morphisms of a category C and we prove C is equivalent to it. The proof is based on a basic character of monoidal equivalences. Ideas and techniques of these proofs can been used to prove the equivalence between an Ann-category and an almost strict Ann-category.
A braided $Ann$-category $\mathcal A$ is an $Ann$-category $\mathcal A$ together with a braiding $c$ such that $(\mathcal A, \otimes, a, c, (1,l,r))$ is a braided tensor category, moreover $c$ is compatible with the distributivity constraints. According to the structure transport theorem, the paper shows that each braided $Ann$-category is equivalent to a braided $Ann$-category of the type $(R,M)$, hence the proof of the classification theorem for braided $Ann$-categories by the cohomology of commutative rings is presented.
Dual monoidal category $\mathcal C^\ast$ of a monoidal functor $F:\mathcal C\to \mathcal V$ has been constructed by S. Majid. In this paper, we extend the construction of dual structures for an Ann-functor $F:\mathcal B\to \mathcal A$. In particular, when $F=id_{\mathcal A}$, then the dual category $\mathcal A^{\ast}$ is indeed the center of $\mathcal A$ and this is a braided Ann-category.
Any $Γ$-graded categorical group is determined by a factor set of a categorical group. This paper studies the factor set of the group $Γ$ with coefficients in the categorical group of the type $(Π,A).$ Then, an interpretation of the notion of $Γ-$operator $3-$cocycle is presented and the proof of cohomological classification theorem for the a $Γ-$graded Gr-category is also presented.
A braided Ann-category $\A$ is an Ann-category $\A$ together with the braiding $c$ such that $(\A, \otimes, a, c, (I,l,r))$ is a braided tensor category, and $c$ is compatible with the distributivity constraints. The paper shows the dependence of the left (or right) distributivity constraint on other axioms. Hence, the paper shows the relation to the concepts of {\it distributivity category} due to M. L. Laplaza and {\it ring-like category} due to A. Frohlich and C.T.C Wall. The center construction of an almost strict Ann-category is an example of an unsymmetric braided Ann-category.
Regular Ann-functor classification problem has been solved with Shukla cohomology. In this paper, we would like to present a solution to the above problem in the general case and in the case of strong Ann-functors with, respectively, Mac Lane cohomology and Hochschild cohomology.