Equivariant Extensions of Categorical Groups

Garzón, Antonio R.; Río, A. del

doi:10.1007/s10485-005-4383-1

Cited by 4 publications

(4 citation statements)

References 9 publications

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“…, an extension G of Q by K and an element of Der(Q, REP × (K)) the abelian 3-group of derivations [9] from Q to its (via Q → B(K)) module REP × (K). This 3-group is given by the truncated relative group cochain complex…”

Section: More Concretely If We Have An Extensionmentioning

confidence: 99%

Extensions and duality

Shapiro

2018

Preprint

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For a fixed finite group Q and semi-simple finite dimensional algebra S, we examine an equivalence between strongly Q-graded algebras (extensions) with identity component S and S 1 -gerbes on action groupoids of Q on the set of isomorphism classes of simple objects of the category of S-modules. This clarifies the nature of the map considered in [14]. Motivated by this and [5] we suggest and study a notion of extensions suitable to the case when S is replaced by a Hopf algebra, in the sense that there is a bijection between extensions with "fiber" H and H * . In particular we focus on the case of H equal to the group algebra of a finite group. When K is abelian, the answer is particularly symmetric as duality of Hopf algebras does not take us outside of the category of groups.

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Section: More Concretely If We Have An Extensionmentioning

confidence: 99%

Extensions and duality

Shapiro

2018

Preprint

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“…Then, in this case, one has that the exterior homotopy category of exterior N-Eilenberg-Mac Lane spaces which are first countable at infinity is equivalent to the category of global towers of groups. (d) For S = {1, 2}, it is well known that the fundamental categorical group functor ρ 2 : Top * → CG and the classifying functor B : CG → Top * induce an equivalence between the category of 2-types of pointed 0-connected spaces and categorical groups up to weak equivalence, see [3,13]. Therefore, in this case, the functor (tow + ρ 2 )ε and the construction Tel (tow + B) induce an equivalence of categories…”

Section: Remarkmentioning

confidence: 99%

S-types of Global Towers of Spaces and Exterior Spaces

Río

Hernández

Rivas

2008

Appl Categor Struct

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The closed model category of exterior spaces, that contains the proper category, is a useful tool for the study of non compact spaces and manifolds. The notion of exterior weak N-S-equivalences is given by exterior maps which induce isomorphisms on the k-th N-exterior homotopy groups π N k for k ∈ S, where S is a set of non negative integers. The category of exterior spaces with a base ray localized by exterior weak N-S-equivalences is called the category of exterior N-Stypes. The existence of closed model structures in the category of exterior spaces permits to establish equivalences between homotopy categories obtained by dividing by exterior homotopy relations, and categories of fractions (localized categories) given by the inversion of classes of week equivalences. The family of neighbourhoods 'at infinity' of an exterior space can be interpreted as a global prospace and under the condition of first countable at infinity we can consider a global tower instead of a prospace. The objective of this paper is to use localized categories to find the connection between S-types of exterior spaces and S-types of global towers of spaces. The main result of this paper establishes an equivalence between the category of Stypes of rayed first countable exterior spaces and the category of S-types of global towers of pointed spaces. As a consequence of this result, categories of global towers of algebraic models localized up to weak equivalences can be used to give some algebraic models of S-types.The authors acknowledge the financial support given by the projects FOMENTA 2007/03 and MTM2007-65431.

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“…A more general situation for Picard categories was given by A. Fröhlich and C. T. C. Wall with the name graded categorical groups [6] (which is later called graded Pic-categories by A. Cegarra and E. Khmaladze [2]). Homotopical classification theorems for the variety of graded categorical groups, the variety of braided graded categorical groups, and its particular case, the variety of graded Picard categories have been presented, respectively, in [7], [2], [3]. Each category raises to a 3-cocycle in some sense that each congruence class of the same categories is corresponding to a third-dimensional cohomology class.…”

Section: Introductionmentioning

confidence: 99%

“…In order to obtain descriptions on structures as well as cohomological classification, N. T. Quang introduced the concept of Ann-categories, as a categorification of the concept of rings, with requirements of invertibility of objects and morphisms of the underlying category, similar to the case of categorical groups (see [7]). These additional requirements are not too special, because if P is a Picard category, then the category End(P) of Pic-functors over P is an Anncategory (see [18]), this was repeated in [9].…”

Section: Introductionmentioning

confidence: 99%