Tim Van der Linden

Higher Hopf formulae for homology via Galois Theory

We use Janelidzes Categorical Galois Theory to extend Brown and Elliss higher Hopf formulae for homology of groups to arbitrary semi-abelian monadic categories. Given such a category Alpha and a chosen Birkhoff subcategory Beta of Alpha, thus we describe the Barr–Beck derived functors of the reflector of Alpha onto Beta in terms of centralization of higher extensions. In case Alpha is the category Gp of all groups and Beta is the category Ab of all abelian groups, this yields a new proof for Brown and Elliss formulae. We also give explicit formulae in the cases of groups vs. k-nilpotent groups, groups vs. k-solvable groups and precrossed modules vs. crossed modules.

On the second cohomology group in semi-abelian categories

We develop some new aspects of cohomology in the context of semi-abelian categories: we establish a Hochschild�Serre 5-term exact sequence extending the classical one for groups and Lie algebras; we prove that an object is perfect if and only if it admits a universal central extension; we show how the second Barr�Beck cohomology group classifies isomorphism classes of central extensions; we prove a universal coefficient theorem to explain the relationship with homology.

A Note on the "Smith is Huq" Condition

We show that two known conditions which arose naturally in commutator theory and in the theory of internal crossed modules coincide: every star-multiplicative graph is multiplicative if and only if every two effective equivalence relations commute as soon as their normalisations do. This answers a question asked by George Janelidze.

On satellites in semi-abelian categories: Homology without projectives

Working in a semi-abelian context, we use Janelidze’s theory of generalised satellites to study universal properties of the Everaert long exact homology sequence. This results in a new definition of homology which does not depend on the existence of projective objects. We explore the relations with other notions of homology, and thus prove a version of the higher Hopf formulae. We also work out some examples.

The ternary commutator obstruction for internal crossed modules

Abstract In finitely cocomplete homological categories, co-smash products give rise to (possibly higher-order) commutators of subobjects. We use binary and ternary co-smash products and the associated commutators to give characterisations of internal crossed modules and internal categories, respectively. The ternary terms are redundant if the category has the Smith is Huq property, which means that two equivalence relations on a given object commute precisely when their normalisations do. In fact, we show that the difference between the Smith commutator of such relations and the Huq commutator of their normalisations is measured by a ternary commutator, so that the Smith is Huq property itself can be characterised by the relation between the latter two commutators. This allows us to show that the category of loops does not have the Smith is Huq property, which also implies that ternary commutators are generally not decomposable into nested binary ones. Thus, in contexts where Smith is Huq need not hold, we obtain a new description of internal categories, Beck modules and double central extensions, as well as a decomposition formula for the Smith commutator. The ternary commutator now also appears in the Hopf formula for the third homology with coefficients in the abelianisation functor.

Relative Commutator Theory in Semi-Abelian Categories

Basing ourselves on the concept of double central extension from categorical Galois theory, we study a notion of commutator which is defined relative to a birkhoff subcategory Beta of a semi-abelian category Alpha. This commutator characterises Janelidze and Kellys Beta-central extensions ; when the subcategory Beta is determined by the abelian objects in Alpha, it coincides with Huqs commutator ; and when the category Alpha is a variety of Omega-groups, it coincides with the relative commutator introduced by the first author.

Universal central extensions in semi-abelian categories

Basing ourselves on Janelidze and Kelly’s general notion of central extension, we study universal central extensions in the context of semi-abelian categories. We consider a new fundamental condition on composition of central extensions and give examples of categories which do, or do not, satisfy this condition.

RESOLUTIONS, HIGHER EXTENSIONS AND THE RELATIVE MAL'TSEV AXIOM

Abstract We study how the concept of higher-dimensional extension which comes from categorical Galois theory relates to simplicial resolutions. For instance, an augmented simplicial object is a resolution if and only if its truncation in every dimension gives a higher extension, in which sense resolutions are infinite-dimensional extensions or higher extensions are finite-dimensional resolutions . We also relate certain stability conditions of extensions to the Kan property for simplicial objects. This gives a new proof of the fact that a regular category is Malʼtsev if and only if every simplicial object is Kan, using a relative setting of extensions.

Higher central extensions and cohomology

Abstract We establish a Galois-theoretic interpretation of cohomology in semi-abelian categories: cohomology with trivial coefficients classifies central extensions, also in arbitrarily high degrees. This allows us to obtain a duality, in a certain sense, between “internal” homology and “external” cohomology in semi-abelian categories. These results depend on a geometric viewpoint of the concept of a higher central extension, as well as the algebraic one in terms of commutators.

PERI-ABELIAN CATEGORIES AND THE UNIVERSAL CENTRAL EXTENSION CONDITION

We study the relation between Bourns notion of peri-abelian category and conditions involving the coincidence of the Smith, Huq and Higgins commutators. In particular, we show that a semi-abelian category is peri-abelian if and only if for each normal subobject K◁XK◁X, the Higgins commutator of K with itself coincides with the normalisation of the Smith commutator of the denormalisation of K with itself. We show that if a category is peri-abelian, then the condition (UCE), which was introduced and studied by Casas and the second author, holds for that category. In addition, we show, using amongst other things a result by Cigoli, that all categories of interest in the sense of Orzech are peri-abelian and therefore satisfy the condition (UCE).