Universal Central Extensions of Internal Crossed Modules via the Non-abelian Tensor Product
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Universal Central Extensions of Internal Crossed Modules via the Non-abelian Tensor Product Davide di Micco1
· Tim Van der Linden2
Received: 9 January 2020 / Accepted: 14 February 2020 © Springer Nature B.V. 2020
Abstract In the context of internal crossed modules over a fixed base object in a given semi-abelian category, we use the non-abelian tensor product in order to prove that an object is perfect (in an appropriate sense) if and only if it admits a universal central extension. This extends results of Brown and Loday (Topology 26(3):311–335, 1987, in the case of groups) and Edalatzadeh (Appl Categ Struct 27(2):111–123, 2019, in the case of Lie algebras). Our aim is to explain how those results can be understood in terms of categorical Galois theory: Edalatzadeh’s interpretation in terms of quasi-pointed categories applies, but a more straightforward approach based on the theory developed in a pointed setting by Casas and Van der Linden (Appl Categ Struct 22(1):253–268, 2014) works as well. Keywords Semi-abelian category · Crossed module · Crossed square · Commutator · Non-abelian tensor product · Universal central extension Mathematics Subject Classification 17B99 · 18D35 · 18E10 · 18G60 · 20J15
1 Introduction The aim of this article is to study a result on universal central extensions of crossed modules due to Brown and Loday ([7], in the case of groups) and Edalatzadeh ([13], in the case of Lie algebras). We prove, namely, that a crossed module over a fixed base object is perfect (in an appropriate sense) if and only if it admits a universal central extension. We first follow
Communicated by George Janelidze. Van der Linden is a Research Associate of the Fonds de la Recherche Scientifique—FNRS.
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Davide di Micco [email protected] Tim Van der Linden [email protected]
1
Università degli Studi di Milano, Via Saldini 50, 20133 Milan, Italy
2
Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, chemin du cyclotron 2 bte L7.01.02, 1348 Louvain-la-Neuve, Belgium
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D. Micco, T. Linden
an ad-hoc approach, extending the result to the context of Janelidze–Márki–Tholen semiabelian categories [29] by using a general version, developed in [12] of the non-abelian tensor product of Brown and Loday [7]. We then provide two interpretations from the perspective of categorical Galois theory. The first one follows the line of Edalatzadeh [13] in the context of quasi-pointed [3] categories (which have an initial object 0 and a terminal object 1 such that 0 → 1 is a monomorphism). This allows us to capture centrality, but we could not find a natural way to treat perfectness in this setting. We then switch to the pointed context (0 ∼ = 1) where the theory developed by Casas and the second author [10] can be used. In this simpler environment we find a convenient interpretation both of centrality and perfectness. The text is structured as follows. In Sect. 2 we give an overview of basic definitions and results of categorical Galois theory, with particular emphasis on
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