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.
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-ca
tegory 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 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.
If $Gamma $ is a group, then braided $Gamma $-crossed modules are classified by braided strict $Gamma $-graded categorial groups. The Schreier theory obtained for $Gamma $-module extensions of the type of an abelian $Gamma $-crossed module is a gener
alization of the theory of $Gamma $-module extensions.
The aim of this paper is to study the $(alpha, gamma)$-prolongation of central extensions. We obtain the obstruction theory for $(alpha, gamma)$-prolongations and classify $(alpha, gamma)$-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 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 equiv
alences. Ideas and techniques of these proofs can been used to prove the equivalence between an Ann-category and an almost strict Ann-category.
Each Gr-functor of the type $(varphi,f)$ of a Gr-category of the type $(Pi,C)$ has the obstruction be an element $overline{k}in H^3(Pi,C).$ When this obstruction vanishes, there exists a bijection between congruence classes of Gr-functors of the type
$(varphi,f)$ and the cohomology group $H^2(Pi,C).$ 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.
