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Fibrations of groupoids
 J. Algebra
, 1970
"... theory, and change of base for groupoids and multiple ..."
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Cited by 24 (15 self)
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theory, and change of base for groupoids and multiple
Higher Hopf formulae for homology via Galois Theory, preprint math.AT/0701815
, 2007
"... and Ellis’s higher Hopf formulae for homology of groups to arbitrary semiabelian monadic categories. Given such a category A and a chosen Birkhoff subcategory B of A, thus we describe the BarrBeck derived functors of the reflector of A onto B in terms of centralization of higher extensions. In case ..."
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Cited by 10 (3 self)
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and Ellis’s higher Hopf formulae for homology of groups to arbitrary semiabelian monadic categories. Given such a category A and a chosen Birkhoff subcategory B of A, thus we describe the BarrBeck derived functors of the reflector of A onto B in terms of centralization of higher extensions. In case A
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
Homotopy Theory, and Change of Base for Groupoids and Multiple Groupoids
, 1996
"... This survey article shows how the notion of "change of base", used in some applications to homotopy theory of the fundamental groupoid, has surprising higher dimensional analogues, through the use of certain higher homotopy groupoids with values in forms of multiple groupoids. ..."
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Cited by 5 (5 self)
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This survey article shows how the notion of "change of base", used in some applications to homotopy theory of the fundamental groupoid, has surprising higher dimensional analogues, through the use of certain higher homotopy groupoids with values in forms of multiple groupoids.
NFOLD ČECH DERIVED FUNCTORS AND GENERALISED HOPF TYPE FORMULAS
"... Abstract. In 1988, Brown and Ellis published [3] a generalised Hopf formula for the higher homology of a group. Although substantially correct, their result lacks one necessary condition. We give here a counterexample to the result without that condition. The main aim of this paper is, however, to g ..."
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Cited by 4 (0 self)
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Abstract. In 1988, Brown and Ellis published [3] a generalised Hopf formula for the higher homology of a group. Although substantially correct, their result lacks one necessary condition. We give here a counterexample to the result without that condition. The main aim of this paper is, however, to generalise this corrected result to derive formulae of Hopf type for the nfold Čech derived functors of the lower central series functors Zk. The paper ends with an application to algebraic Ktheory. Introduction and Summary The well known Hopf formula for the second integral homology of a group says that for a given group G there is an isomorphism H2(G) ∼ = R ∩ [F, F]
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
Intersection of subgroups in free groups and homotopy groups, preprint
"... Abstract. We show that the intersection of three subgroups in a free group is related to the computation of the third homotopy group π3. This generalizes a result of GutierrezRatcliffe who relate the intersection of two subgroups with the computation of π2. Let K be a twodimensional CWcomplex with ..."
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Cited by 2 (2 self)
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Abstract. We show that the intersection of three subgroups in a free group is related to the computation of the third homotopy group π3. This generalizes a result of GutierrezRatcliffe who relate the intersection of two subgroups with the computation of π2. Let K be a twodimensional CWcomplex with subcomplexes K1, K2, K3 such that K = K1 ∪ K2 ∪ K3 and K1 ∩ K2 ∩ K3 is the 1skeleton K1 of K. We construct a natural homomorphism of π1(K)modules
More About Homological Properties Of Precrossed Modules
, 2000
"... Homology groups modulo q of a precrossed Pmodule in any dimensions are defined in terms of nonabelian derived functors, where q is a nonnegative integer. The Hopf formula is proved for the second homology group modulo q of a precrossed Pmodule which shows that for q = 0 our definition is a natural ..."
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Cited by 1 (1 self)
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Homology groups modulo q of a precrossed Pmodule in any dimensions are defined in terms of nonabelian derived functors, where q is a nonnegative integer. The Hopf formula is proved for the second homology group modulo q of a precrossed Pmodule which shows that for q = 0 our definition is a natural extension of Conduch'e and Ellis ' definition [CE]. Some other properties of homologies of precrossed Pmodules are investigated. Introduction The homology of precrossed modules was introduced by Conduch'e and Ellis in [CE]. The aim of this paper is to pursue their line of investigation homological properties of precrossed modules. Let P be a group. A precrossed Pmodule (M; ) is a group homomorphism : M ! P together with an action of P on M denoted by p m for p 2 P and m 2 M , which satisfies the following condition: ( p m) = p(m)p \Gamma1 : If in addition the following Peiffer identity holds (m) m 0 = mm 0 m \Gamma1 ; (M; ) is a crossed Pmodule (see e.x. [BH]). A mor...
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.