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Tim Van der Linden

Publications and source records attributed to Tim Van der Linden.

At least 55 records · Page 3Linked to original sources
arXiv2014

On the normality of Higgins commutators

In a semi-abelian context, we study the condition (NH) asking that Higgins commutators of normal subobjects are normal subobjects. We provide examples of categories that do or do not satisfy this property. We focus on the relationship with the "Smith is Huq" condition (SH) and characterise those semi-abelian categories in which both (NH) and (SH) hold in terms of reflection and preservation properties of the change of base functors of the fibration of points.

math.CT↗
arXiv2014

Further remarks on the "Smith is Huq" condition

We compare the 'Smith is Huq' condition (SH) with three commutator conditions in semi-abelian categories: first an apparently weaker condition which arose in joint work with Bourn and turns out to be equivalent with (SH), then an apparently equivalent condition which takes commutation of non-normal subobjects into account and turns out to be stronger than (SH). This leads to the even stronger condition that weighted commutators (in the sense of Gran, Janelidze and Ursini) are independent of the chosen weight, which is known to be false for groups but turns out to be true in any two-nilpotent semi-abelian category.

math.CT↗
arXiv2013

An observation on n-permutability

We prove that in a regular category all reflexive and transitive relations are symmetric if and only if every internal category is an internal groupoid. In particular, these conditions hold when the category is n-permutable for some n.

math.CT↗
arXiv2013

Approximate Hagemann-Mitschke co-operations

We show that varietal techniques based on the existence of operations of a certain arity can be extended to n-permutable categories with binary coproducts. This is achieved via what we call approximate Hagemann-Mitschke co-operations, a generalisation of the notion of approximate Mal'tsev co-operation. In particular, we extend characterisation theorems for n-permutable varieties due to J. Hagemann and A. Mitschke to regular categories with binary coproducts.

math.CT↗
arXiv2013

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.

math.CT↗
arXiv2012

A relative theory of universal central extensions

Basing ourselves on Janelidze and Kelly's general notion of central extension, we study universal central extensions in the context of semi-abelian categories. Thus we unify classical, recent and new results in one conceptual framework. The theory we develop is relative to a chosen Birkhoff subcategory of the category considered: for instance, we consider groups vs. abelian groups, Lie algebras vs. vector spaces, precrossed modules vs. crossed modules and Leibniz algebras vs. Lie algebras. We consider a fundamental condition on composition of central extensions and give examples of categories which do, or do not, satisfy this condition.

math.AT↗
arXiv2012

The ternary commutator obstruction for internal crossed modules

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 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.

math.CT↗
arXiv2012

Resolutions, higher extensions and the relative Mal'tsev axiom

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 Mal'tsev if and only if every simplicial object is Kan, using a relative setting of extensions.

math.CT↗
arXiv2011

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 B of a semi-abelian category A. This commutator characterises Janelidze and Kelly's B-central extensions; when the subcategory B is determined by the abelian objects in A, it coincides with Huq's commutator; and when the category A is a variety of omega-groups, it coincides with the relative commutator introduced by the first author.

math.CT↗
arXiv2011

Galois theory and commutators

We prove that the relative commutator with respect to a subvariety of a variety of Omega-groups introduced by the first author can be described in terms of categorical Galois theory. This extends the known correspondence between the Froehlich-Lue and the Janelidze-Kelly notions of central extension. As an example outside the context of Omega-groups we study the reflection of the category of loops to the category of groups where we obtain an interpretation of the associator as a relative commutator.

math.RA↗
arXiv2010

Some results on homology of Leibniz and Lie n-algebras

From the viewpoint of semi-abelian homology, some recent results on homology of Leibniz n-algebras can be explained categorically. In parallel with these results, we develop an analogous theory for Lie n-algebras. We also consider the relative case: homology of Leibniz n-algebras relative to the subvariety of Lie n-algebras.

math.AT↗
arXiv2010

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.

math.CT↗
arXiv2009

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.

math.AT↗
arXiv2007

A comparison theorem for simplicial resolutions

It is well known that Barr and Beck's definition of comonadic homology makes sense also with a functor of coefficients taking values in a semi-abelian category instead of an abelian one. The question arises whether such a homology theory has the same convenient properties as in the abelian case. Here we focus on independence of the chosen comonad: conditions for homology to depend on the induced class of projectives only.

math.AT↗
arXiv2007

Higher Hopf formulae for homology via Galois Theory

We use Janelidze's Categorical Galois Theory to extend Brown and Ellis's higher Hopf formulae for homology of groups to arbitrary semi-abelian monadic categories. Given such a category A and a chosen Birkhoff subcategory B of A, thus we describe the Barr-Beck derived functors of the reflector of A onto B in terms of centralization of higher extensions. In case A is the category Gp of all groups and B is the category Ab of all abelian groups, this yields a new proof for Brown and Ellis's 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.

math.AT↗
arXiv2007

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.

math.KT↗