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Mikhailov: A colimit of classifying spaces
"... We recall a grouptheoretic description of the first nonvanishing homotopy group of a certain (n+1)ad of spaces and show how it yields several formulae for homotopy and homology groups of specific spaces. In particular we obtain an alternative proof of J. Wu’s grouptheoretic description of the ho ..."
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Cited by 7 (3 self)
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We recall a grouptheoretic description of the first nonvanishing homotopy group of a certain (n+1)ad of spaces and show how it yields several formulae for homotopy and homology groups of specific spaces. In particular we obtain an alternative proof of J. Wu’s grouptheoretic description of the homotopy groups of a 2sphere. 1
Higher central extensions and Hopf formulae
, 2009
"... Higher extensions and higher central extensions, which are of importance to nonabelian homological algebra, are studied, and some fundamental properties are proven. As an application, a direct proof of the invariance of the higher Hopf formulae is obtained. ..."
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Cited by 3 (1 self)
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Higher extensions and higher central extensions, which are of importance to nonabelian homological algebra, are studied, and some fundamental properties are proven. As an application, a direct proof of the invariance of the higher Hopf formulae is obtained.
A COMPARISON THEOREM FOR SIMPLICIAL RESOLUTIONS
 JOURNAL OF HOMOTOPY AND RELATED STRUCTURES, VOL. 2(1), 2007, PP.109–126
, 2007
"... 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 semiabelian category instead of an abelian one. The question arises whether such a homology theory has the same convenient properties as in the abelian case. Her ..."
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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 semiabelian 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.
ON ACTIONS AND STRICT ACTIONS IN HOMOLOGICAL CATEGORIES
"... Abstract. Let G be an object of a finitely cocomplete homological category C. We study actions of G on objects A of C (defined by Bourn and Janelidze as being algebras over a certain monad TG), with two objectives: investigating to which extent actions can be described in terms of smaller data, call ..."
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Abstract. Let G be an object of a finitely cocomplete homological category C. We study actions of G on objects A of C (defined by Bourn and Janelidze as being algebras over a certain monad TG), with two objectives: investigating to which extent actions can be described in terms of smaller data, called action cores; and to single out those abstract action cores which extend to actions corresponding to semidirect products of A and G (in a nonexact setting, not every action does). This amounts to exhibiting a subcategory of the category of the actions of G on objects A which is equivalent with the category of points in C over G, and to describing it in terms of action cores. This notion and its study are based on a preliminary investigation of cosmash products, in which crosseffects of functors in a general categorical context turn out to be a useful tool. The cosmash products also allow us to define higher categorical commutators, different from the ones of Huq, which are not generally expressible in terms of nested binary ones. We use strict action cores to show that any normal subobject of an object E (i.e., the equivalence class of 0 for some equivalence relation on E in C) admits a strict conjugation action of E. If C is semiabelian, we show that for subobjects X, Y of
RELATIVE MAL’TSEV CATEGORIES
, 2013
"... We define relative regular Mal’tsev categories and give an overview of conditions which are equivalent to the relative Mal’tsev axiom. These include conditions on relations as well as conditions on simplicial objects. We also give various examples and counterexamples. ..."
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We define relative regular Mal’tsev categories and give an overview of conditions which are equivalent to the relative Mal’tsev axiom. These include conditions on relations as well as conditions on simplicial objects. We also give various examples and counterexamples.
On satellites in semiabelian categories: . . .
, 2009
"... Working in a semiabelian 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 wit ..."
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Working in a semiabelian 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.
On satellites in semiabelian categories: Homology . . .
, 2009
"... Working in a semiabelian 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 ..."
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Working in a semiabelian 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.
HIGHER CENTRAL EXTENSIONS VIA COMMUTATORS
, 2012
"... We prove that all semiabelian categories with the the Smith is Huq property satisfy the Commutator Condition (CC): higher central extensions may be characterised in terms of binary (Huq or Smith) commutators. In fact, even Higgins commutators suffice. As a consequence, in the presence of enough p ..."
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We prove that all semiabelian categories with the the Smith is Huq property satisfy the Commutator Condition (CC): higher central extensions may be characterised in terms of binary (Huq or Smith) commutators. In fact, even Higgins commutators suffice. As a consequence, in the presence of enough projectives we obtain explicit Hopf formulae for homology with coefficients in the abelianisation functor, and an interpretation of cohomology with coefficients in an abelian object in terms of equivalence classes of higher central extensions. We also give a counterexample against (CC) in the semiabelian category of (commutative) loops.