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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
, 902
"... 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. 0 ..."
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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. 0
THE THIRD COHOMOLOGY GROUP CLASSIFIES DOUBLE CENTRAL EXTENSIONS
, 2010
"... We characterise the double central extensions in a semiabelian category in terms of commutator conditions. We prove that the third cohomology group H3 (Z, A) of an object Z with coefficients in an abelian object A classifies the double central extensions of Z by A. ..."
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Cited by 2 (1 self)
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We characterise the double central extensions in a semiabelian category in terms of commutator conditions. We prove that the third cohomology group H3 (Z, A) of an object Z with coefficients in an abelian object A classifies the double central extensions of Z by A.
Journal of Homotopy and Related Structures, vol. 2(1), 2007, pp.109–126 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 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.
A COMPARISON THEOREM FOR SIMPLICIAL RESOLUTIONS
, 707
"... Abstract. 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 ..."
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Abstract. 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.
HIGHER CENTRAL EXTENSIONS VIA COMMUTATORS
"... Abstract. 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 ..."
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Abstract. 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.
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
Contents
, 808
"... Abstract. 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 relat ..."
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Abstract. 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.
RELATIVE MAL’TSEV CATEGORIES
"... Abstract. 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. 1. ..."
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Abstract. 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. 1.
Report on Research in Teams Project: Universal Higher Extensions
, 2011
"... explain this, we outline below the before/aftereffect which resulted from the opportunity to meet facetoface and to devote ourselves entirely to the task at hand for the duration of the visit. 1 The Situation Before the Visit In one stream of development, one partner in the project had just achie ..."
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explain this, we outline below the before/aftereffect which resulted from the opportunity to meet facetoface and to devote ourselves entirely to the task at hand for the duration of the visit. 1 The Situation Before the Visit In one stream of development, one partner in the project had just achieved an interpretation of group cohomology or Liealgebra cohomology in terms of higherdimensional central extensions, as developed by Rodelo–Van der Linden [14, 15]. It extends the classical interpretation of the second group cohomology in terms of equivalence classes of short exact sequences with central kernel. This development builds upon the notion of semiabelian category as in Janelidze–Márki–Tholen [11], and the concept of higher central extension developed within the framework of categorical Galois theory based on the work of Janelidze et al. (See [2, 5, 9, 10].) In addition, methods from the theory of simplicial groups are used. In a parallel and complementary stream of development, the second partner in the project had just achieved a proof of existence of universal nstep extensions of modules over an arbitrary unitary ring [12]. This development involves certain higher torsion theories, some potentially noncommutative. It immediately has several applications ranging from • a torsion theoretic conceptual hindsight explanation for existing computational results about the effect