DI, MICCO DAVIDE. "AN INTRINSIC APPROACH TO THE NON-ABELIAN TENSOR PRODUCT." Doctoral thesis, Università degli Studi di Milano, 2020. http://hdl.handle.net/2434/703934.
Abstract:
The notion of a non-abelian tensor product of groups first appeared in a paper where Brown and Loday generalised a theorem on CW-complexes by using the new notion of non-abelian tensor product of two groups acting on each other, instead of the usual tensor product of abelian groups. In particular, they took two groups acting on each other and they defined their non-abelian tensor product via an explicit presentation. This led to the development of an algebraic theory based on this construction. Many results were obtained treating the properties which are satisfied by this non-abelian tensor product as well as some explicit calculations in particular classes of groups. In order to state many of their results regarding this tensor product, Brown and Loday needed to require, as an additional condition, that the two groups M and N acted on each other compatibly: these amount to the existence of a group L and of two crossed modules structures of M and N on L such that the original actions are induced from these crossed module structures. Furthermore, they proved that the non-abelian tensor product is part of a so-called crossed square of groups: this particular crossed square is the pushout of a specific diagram in the category of crossed squares of groups. Note that crossed squares are a 2-dimensional version of crossed modules of groups. Following the idea of generalising the algebraic theory arising from the study of the non-abelian tensor product of groups, Ellis gave a definition of non-abelian tensor product of Lie algebras, and obtained similar results. Further generalisations have been studied in the contexts of Leibniz algebras, restricted Lie algebras, Lie-Rinehart algebras, Hom-Lie algebras, Hom-Leibniz algebras, Hom-Lie-Rinehart algebras, Lie superalgebras and restricted Lie superalgebras. The aim of our work is to build a general version of non-abelian tensor product, having the specific definitions in the categories of groups and Lie algebras as particular instances. In order to do so we first extend the concept of a pair of compatible actions (introduced in the case of groups by Brown and Loday and in the case of Lie algebras by Ellis) to semi-abelian categories. This is indeed the most general environment in which we are able to talk about actions, due to the concept of internal actions. In this general context, we give a diagrammatic definition of the compatibility conditions for internal actions, which specialises to the particular definitions known for groups and Lie algebras. We then give a new construction of the Peiffer product in this setting and we use these tools to show that in any semi-abelian category satisfying the "Smith-is-Huq" condition, asking that two actions are compatible is the same as requiring that these actions are induced from a pair of internal crossed modules over a common base object. Thanks to this equivalence, in order to deal with the generalisation to the semi-abelian context of the non-abelian tensor product, we are able to use a pair of internal crossed modules over a common base object instead of a pair of compatible internal actions, whose formalism is far more intricate. Now we fix a semi-abelian category A satisfying "Smith-is-Huq" and we show that, for each pair of internal L-crossed modules, it is possible to construct an internal crossed square which is the pushout (in the category of crossed squares) of the general version of the diagram used by Brown and Loday in the groups case. The non-abelian tensor product is then defined as a piece of this internal crossed square. We show that if A is the category of groups or the category of Lie algebras, this general construction coincides with the specific notions of non-abelian tensor products already known in these settings. We construct an L-crossed module structure on this non-abelian tensor product, some additional universal properties are shown and by using these we prove that this tensor product is a bifunctor. Once we have the non-abelian tensor product among our tools, we are also able to state the new definition of "weak crossed square": the idea behind this is to generalise the explicit presentations of crossed squares known for groups and for Lie algebras. These equivalent definitions, which (contrarily to the semi-abelian one) do not rely on the formalism of internal groupoids but include some set-theoretic constructions, are shown to be equivalent to the implicit ones, where, by definition, crossed squares are crossed modules of crossed modules and hence normalisations of double groupoids. Our idea is to give an alternative explicit description of crossed squares of groups (resp. Lie algebras) using the non-abelian tensor product, so that it does not involve anymore the so-called emph{crossed pairing} (resp. emph{Lie pairing}), which is not a morphism in the base category but only a set-theoretic function; in its place we use a morphism from the non-abelian tensor product which is more suitable for generalisations. Doing so, the explicit definitions can be summarised by saying that a crossed square is a commutative square of crossed modules, compatible with an additional crossed module structure on the diagonal, and endowed with a morphism out of the non-abelian tensor product. Our definition of weak crossed squares is based on the one of non-abelian tensor product and plays the role of the explicit version of the definition of internal crossed squares: in particular we proved that it restricts to the explicit definitions for groups and Lie algebras and hence that in these cases weak crossed squares are equivalent to crossed squares. So far we have shown that any internal crossed square is automatically a weak crossed square, but we are currently missing precise conditions on the base category under which the converse is true: this means that any internal crossed square can be described explicitly as a particular weak crossed square, but this is not a complete characterisation. In order to give a direct application of our non-abelian tensor product construction, we focus on universal central extensions in the category of L-crossed modules: Casas and Van der Linden studied the theory of universal central extensions in semi-abelian categories, using the general notion of central extension (with respect to a Birkhoff subcategory) given by Janelidze and Kelly. We are mainly interested in one of their results, namely that, given a Birkhoff subcategory B of a semi-abelian category X with enough projectives, an object of X is B-perfect if and only if it admits a universal B-central extension. Edalatzadeh considered the category of L-crossed modules of Lie algebras and crossed modules with vanishing aspherical commutator as Birkhoff subcategory B. Since the first one is not a semi-abelian category the existing theory does not apply in this situation: nevertheless he managed to prove the same result, and furthermore he gave an explicit construction of the universal B-central extensions by using the non-abelian tensor product of Lie algebras. Using our general definition of non-abelian tensor product of L-crossed modules as given in the third chapter, we are able to extend Edalatzadeh's results to the category of L-crossed modules in any semi-abelian category A satisfying the "Smith-is-Huq" condition: this is a useful application of the construction of the non-abelian tensor product, which again manages to express in this more general setting exactly the same properties as in its known particular instances. Furthermore, taking the subcategory of abelian objects as Birkhoff subcategory of the category of crossed modules in A, we are able to show that, whenever the category A has enough projectives, our generalisation of Edalatzadeh's work is partly a consequence of Casas' and Van der Linden's theorem, reframing Edalatzadeh's result within the standard theory of universal central extensions in the semi-abelian context. There are two non-trivial consequences of this fact. First of all, besides the existence of the universal B-central extension for each B-perfect crossed module in A, we are also able to give its explicit construction by using the non-abelian tensor product: notice that this construction is completely unrelated to what has been done by Casas and Van der Linden. Secondly, this construction of universal B-central extensions is valid even when A does not have enough projectives, whereas within the general theory this is a key requirement for the result to hold.
