Adjunction between crossed modules of groups and algebras

Casas, J. M.; Inassaridze, Nick; Khmaladze, Emzar; Ladra, M.

doi:10.1007/s40062-013-0073-0

Cited by 10 publications

(9 citation statements)

References 15 publications

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“…In the case of crossed modules, these internal actions agree with the concept of an action studied by Norrie [33] and Forrester-Barker [20]. Here we recall the presentation given in [11]. An action of a crossed module pH, G, µq on another crossed module pM, P, νq is completely determined by an action of G (and so H) on M and P together with a map ξ : HˆP Ñ M such that the conditions…”

Section: Crossed Modules Their Nerves and Their Actionssupporting

confidence: 66%

A comonadic interpretation of Baues–Ellis homology of crossed modules

Donadze

Linden

2018

J. Homotopy Relat. Struct.

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We introduce and study a homology theory of crossed modules with coefficients in an abelian crossed module. We discuss the basic properties of these new homology groups and give some applications. We then restrict our attention to the case of integral coefficients. In this case we regain the homology of crossed modules originally defined by Baues and further developed by Ellis. We show that it is an instance of Barr-Beck comonadic homology, so that we may use a result of Everaert and Gran to obtain Hopf formulae in all dimensions.

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Section: Crossed Modules Their Nerves and Their Actionssupporting

confidence: 66%

A comonadic interpretation of Baues–Ellis homology of crossed modules

Donadze

Linden

2018

J. Homotopy Relat. Struct.

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“…Actions of crossed modules of different structures have also been described in terms of equations, as it can be checked in [10] for crossed modules of groups, in [8], [11] or [31] for crossed modules of Lie algebras, and in [7] for crossed modules of Leibniz algebras. Bearing those examples in mind, it is seemingly reasonable to address the problem by following the next steps: firstly, we consider a homomorphism from (M, P, η) to Act(L, D, µ) = (Tetra(D, L), Tetra(L, D, µ), ∆), and translate into equations every property satisfied by that homomorphism.…”

Section: The Actormentioning

confidence: 99%

“…In [10], it is given an extension to crossed modules of the adjunction between the unit group functor and the group algebra functor. Additionally, the 2-dimensional generalizations of the corresponding adjunctions for Lie vs Alg and Lb vs Dias are presented in [9,13], where the resulting commutative squares of categories and functors are assembled into four parallelepipeds containing the original adjunctions and their natural generalizations:…”

Section: Introductionmentioning

confidence: 99%

Actor of a crossed module of dialgebras via tetramultipliers

Casas¹,

Fernández-Casado²,

García-Martínez³

et al. 2021

Preprint

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We study the representability of actions in the category of crossed modules of dialgebras via tetramultipliers. We deduce a pair of dialgebras in order to construct an object which, under certain circumstances, is the actor (also known as the split extension classifier). Moreover, we give give a full description of actions in terms of equations. Finally, we check that under the aforementioned circumstances, the center coincides with the kernel of the canonical map from a crossed module to its actor.

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“…This is a working definition, completely bypassing its (unknown, and possibly not-existing) relation with simplicial algebras with Moore complex of length one. We note that crossed modules of associative algebras are defined in [Arv04,CIKL14,DIKL12], and our definition is more restrictive. Notice that, in the case of commutative algebras (where left and right actions are the same thing) our definition of crossed modules coincides with the one in [AP96].…”

Section: Crossed Modules Of Bare Algebrasmentioning

confidence: 99%

Crossed modules of Hopf algebras and of associative algebras and two-dimensional holonomy

Martins

2016

Journal of Geometry and Physics

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Article:Faria Martins, J orcid.org/0000-0001- 8113-3646 (2016) ReuseUnless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version -refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher's website. TakedownIf you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. Abstract After a thorough treatment of all algebraic structures involved, we address two dimensional holonomy operators with values in crossed modules of Hopf algebras and in crossed modules of associative algebras (called here crossed modules of bare algebras.) In particular, we will consider two general formulations of the two-dimensional holonomy of a (fully primitive) Hopf 2-connection (exact and blur), the first being multiplicative the second being additive, proving that they coincide in a certain natural quotient (defining what we called the fuzzy holonomy of a fully primitive Hopf 2-connection).

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Adjunction between crossed modules of groups and algebras

Abstract: We construct a pair of adjoint functors between the categories of crossed modules of groups and associative algebras and establish an equivalence of categories between module structures over a crossed module of groups and its respective crossed module of associative algebras.

Cited by 10 publications

References 15 publications

A comonadic interpretation of Baues–Ellis homology of crossed modules

A comonadic interpretation of Baues–Ellis homology of crossed modules

Actor of a crossed module of dialgebras via tetramultipliers

Crossed modules of Hopf algebras and of associative algebras and two-dimensional holonomy

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