Graded extensions of categories (original) (raw)

Some algebraic applications of graded categorical group theory

Theory and Applications of Categories

The homotopy classification of graded categorical groups and their homomorphisms is applied, in this paper, to obtain appropriate treatments for diverse crossed product constructions with operators which appear in several algebraic contexts. Precise classification theorems are therefore stated for equivariant extensions by groups either of monoids, or groups, or rings, or rings-groups or algebras as well as for graded Clifford systems with operators, equivariant Azumaya algebras over Galois extensions of commutative rings and for strongly graded bialgebras and Hopf algebras with operators. These specialized classifications follow from the theory of graded categorical groups after identifying, in each case, adequate systems of factor sets with graded monoidal functors to suitable graded categorical groups associated to the structure dealt with.

Extending Structures I: Unifying Crossed and Bicrossed Products

2010

Let H be a group. We define the concept of an extending structure of the group H which consists of a new algebraic structure (S, *) called H-group, two actions and a generalized cocycle satisfying some compatibility conditions. The extending structure unifies two dual notions from group theory: the crossed system from the extension problem and the matched pair from the factorization problem. A general product, which we called the unified product, associated to an extending structure is defined such that both the crossed product and the bicrossed product of two groups are special cases of it. The main properties of the unified product are proven. The above construction is related to the existence of hidden symmetries of H-principal bundles.

Fibred categorical theory of obstruction and classification of morphisms

2021

We set up a fibred categorical theory of obstruction and classification of morphisms that specializes to the one of monoidal functors between categorical groups and also to the Schreier-Mac Lane theory of group extensions. Further applications are provided, as for example a classification of unital associative algebra extensions with non-abelian kernel in terms of Hochschild cohomology.

Some Results on Strict Graded Categorical Groups Nguyen

2015

The group extension problem has an important significance in the development of modern algebra. Some notions of this problem such as crossed product, factor set, and obstruction (see [1]) are not only applied to rings or to algebraic types but also are raised to a categorical level. The theory of graded categorical groups studied by Cegarra et al. [2] can be viewed as a generalization of both the categorical group theory of Sinh [3] and the graded category theory of Fröhlich and Wall [4]. The equivariant group extension problem is one of applications of this theory. Strict graded categorical groups, with their simple structures compared to the general case, aremore likely to give a lot of interesting applications. In [5] we presented an application of this notion to the classification of equivariant crossed modules. In this paper we continue to introduce some other applications. Firstly, we show that if [h] is the third invariant of the strict graded categorical group Hol Γ G and p ...

Graded Extensions of Monoidal Categories

Journal of Algebra, 2001

The long-known results of Schreier᎐Eilenberg᎐Mac Lane on group extensions are raised to a categorical level, for the classification and construction of the manifold of all graded monoidal categories, the type being given group ⌫ with 1-component a given monoidal category. Explicit application is made to the classification of strongly graded bialgebras over commutative rings.

Categorification and group extensions

Applied Categorical Structures, 2002

We review several known categorification procedures, and introduce a functorial categorification of group extensions (Section 4.1) with applications to non-Abelian group cohomology (Section 4.2). The obstruction to the existence of group extensions (Section 4.2.4, Equation (9)) is interpreted as a "coboundary" condition (Proposition 4.5).

On Crossed Modules in Modified Categories of Interest

arXiv: Category Theory, 2016

We introduce some algebraic structures such as singularity, commutators and central extension in modified categories of interest. Additionally, we introduce the cat$^{1}$-objects with their connection to crossed modules in these categories which gives rise to unify many notions about (pre)crossed modules in various algebras of categories.

Semidirect products of categories and applications

Journal of Pure and Applied Algebra, 1999

This paper offers definitions for the semidirect product of categories, Cayley graphs for categories, and the kernel of a relational morphism of categories which will allow us to construct free (profinite) objects for the semidirect product of two (pseudo)varieties of categories analogous to the monoid case. The main point of this paper is to prove g(V * W) = gV * gW. Previous attempts at this have contained errors which, the author feels, are due to an incorrect usage of the wreath product. In this paper we make no use of wreath products, but use instead representations of the free objects. Analogous results hold for semigroups and semigroupoids.