Academic literature on the topic 'Semi-abelian categorie'

AbstractWe study Quillen's model category structure for homotopy of simplicial objects in the context of Janelidze, Márki and Tholen's semi-abelian categories. This model structure exists as soon as is regular Mal'tsev and has enough regular projectives; then the fibrations are the Kan fibrations of S. When, moreover, is semi-abelian, weak equivalences and homology isomorphisms coincide.

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.

The main theme of the thesis is to present and compare three different viewpoints on semi-abelian homology, resulting in three ways of defining and calculating homology objects. Any two of these three homology theories coincide whenever they are both defined, but having these different approaches available makes it possible to choose the most appropriate one in any given situation, and their respective strengths complement each other to give powerful homological tools. The oldest viewpoint, which is borrowed from the abelian context where it was introduced by Barr and Beck, is comonadic homology, generating projective simplicial resolutions in a functorial way. This concept only works in monadic semi-abelian categories, such as semi-abelian varieties, including the categories of groups and Lie algebras. Comonadic homology can be viewed not only as a functor in the first entry, giving homology of objects for a particular choice of coefficients, but also as a functor in the second variable, varying the coefficients themselves. As such it has certain universality properties which single it out amongst theories of a similar kind. This is well-known in the setting of abelian categories, but here we extend this result to our semi-abelian context. Fixing the choice of coefficients again, the question naturally arises of how the homology theory depends on the chosen comonad. Again it is well-known in the abelian case that the theory only depends on the projective class which the comonad generates. We extend this to the semi-abelian setting by proving a comparison theorem for simplicial resolutions. This leads to the result that any two projective simplicial resolutions, the definition of which requires slightly more care in the semi-abelian setting, give rise to the same homology. Thus again the homology theory only depends on the projective class. The second viewpoint uses Hopf formulae to define homology, and works in a non-monadic setting; it only requires a semi-abelian category with enough projectives. Even this slightly weaker setting leads to strong results such as a long exact homology sequence, the Everaert sequence, which is a generalised and extended version of the Stallings-Stammbach sequence known for groups. Hopf formulae use projective presentations of objects, and this is closer to the abelian philosophy of using any projective resolution, rather than a special functorial one generated by a comonad. To define higher Hopf formulae for the higher homology objects the use of categorical Galois theory is crucial. This theory allows a choice of Birkhoff subcategory to generate a class of central extensions, which play a big role not only in the definition via Hopf formulae but also in our third viewpoint. This final and new viewpoint we consider is homology via satellites or pointwise Kan extensions. This makes the universal properties of the homology objects apparent, giving a useful new tool in dealing with statements about homology. The driving motivation behind this point of view is the Everaert sequence mentioned above. Janelidze's theory of generalised satellites enables us to use the universal properties of the Everaert sequence to interpret homology as a pointwise Kan extension, or limit. In the first instance, this allows us to calculate homology step by step, and it removes the need for projective objects from the definition. Furthermore, we show that homology is the limit of the diagram consisting of the kernels of all central extensions of a given object, which forges a strong connection between homology and cohomology. When enough projectives are available, we can interpret homology as calculating fixed points of endomorphisms of a given projective presentation.