Homotopy classification of braided graded categorical groups (original) (raw)
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On Graded Categorical Groups and Equivariant Group Extensions
Canadian Journal of Mathematics, 2002
In this article we state and prove precise theorems on the homotopy classification of graded categorical groups and their homomorphisms. The results use equivariant group cohomology, and they are applied to show a treatment of the general equivariant group extension problem.
On some groups related to the Braid Groups of type A
Annals of the University of Craiova - Mathematics and Computer Science Series, 2010
We prove that a family of groups R(n) forms the algebraic structure of an operad and that they admit a presentation similar to that of the Braid groups of type A. This result provides a new proof that the Braid Groups form an operad, a topic emphasized in ~\cite{16}~\cite{ulrike}. These groups proved to be useful in several problems which belong to different areas of Mathematics. Representations of R(n) came from a system of mixed Yang-Baxter type equations. We define the Hopf equation in braided monoidal categories and we prove that representations for our groups came from any braided Hopf algebra with invertible antipode. Using this result, we prove that there is a morphism from R(n) to the mapping class group Gamman,1\Gamma_{n,1}Gamman,1, using some results from 3-dimensional topology.
On Braided Linear Gr-categories
We provide explicit and unified formulae for the normalized 3-cocycles on arbitrary finite abelian groups. As an application, we compute all the braided monoidal structures on linear Gr-categories.
Cohomological classification of braided AnnAnnAnn-categories
2010
A braided AnnAnnAnn-category mathcalA\mathcal AmathcalA is an AnnAnnAnn-category mathcalA\mathcal AmathcalA together with a braiding ccc such that (mathcalA,otimes,a,c,(1,l,r))(\mathcal A, \otimes, a, c, (1,l,r))(mathcalA,otimes,a,c,(1,l,r)) is a braided tensor category, moreover ccc is compatible with the distributivity constraints. According to the structure transport theorem, the paper shows that each braided AnnAnnAnn-category is equivalent to a braided AnnAnnAnn-category of the type (R,M)(R,M)(R,M), hence the proof of the classification theorem for braided AnnAnnAnn-categories by the cohomology of commutative rings is presented.
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.
Homotopical presentations of braid groups via reduced lifts
2021
In [De2], Deligne showed that the reduced lift presentation of a finite type generalized braid group remains correct if it is (suitably) interpreted as a presentation of a topological monoid. In this expository paper, we point out that Deligne’s argument does not require the ‘finite type’ hypothesis, so it gives a different proof of [Do, Thm. 5.1]. We also review how to use this result to construct an action of the braid group on the finite or affine Hecke ∞-category via intertwining functors.
Canadian Journal of Mathematics, 1991
The general problem of what should be a non-abelian cohomology, what is it supposed to do, and what should be the coefficients, form a set of interesting questions which has been around for a long time. In the particular setting of groups, a comprehensible and well motivated cohomology theory has been so far stated in dimensions ≤ 2, the coefficients for being crossed modules. The main effort to define an appropriate for groups has been done by Dedecker [16] and Van Deuren [40]; they studied the obstruction to lifting non-abelian 2-cocycles and concluded with first approach for , which requires “super crossed groups” as coefficients. However, as Dedecker said “some polishing work remains necessary” for his cohomology.
On catn-groups and homotopy types
Journal of Pure and Applied Algebra, 1993
Bullejos, M., A.M. Cegarra and J. Duskin, On cat"-groups and homotopy types, Journal of Pure and Applied Algebra 86 (1993) 135-154. We give an algebraic proof of Loday's 'Classification theorem' for truncated homotopy types. In particular we give a precise construction of the homotopy cat"-group associated to a pointed topological space which is based on the use of the internal fundamental groupoid functor together with Illusie's 'total Dee'. modules.
Braided equivariant crossed modules and cohomology of Γ-modules
Indian Journal of Pure and Applied Mathematics, 2014
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.
Crossed modules and quantum groups in braided categories II
arXiv (Cornell University), 1994
Let A be a Hopf algebra in a braided category C. Crossed modules over A are introduced and studied as objects with both module and comodule structures satisfying a compatibility condition. The category DY (C) A A of crossed modules is braided and is a concrete realization of a known general construction of a double or center of a monoidal category. For a quantum braided group (A, A, R) the corresponding braided category of modules C O(A,A) is identified with a full subcategory in DY (C) A A. The connection with cross products is discussed and a suitable cross product in the class of quantum braided groups is built. Majid-Radford theorem, which gives equivalent conditions for an ordinary Hopf algebra to be such a cross product, is generalized to the braided category. Majid's bosonization theorem is also generalized.