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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
Homotopical excision, and Hurewicz theorems, for ncubes of spaces
 Proc. London Math. Soc
, 1987
"... The fact that the relative homotopy groups do not satisfy excision makes the computation of absolute homotopy groups difficult in comparison with homology groups. The failure of excision is measured by triad homotopy groups πn(X; A, B), with n � 3 (for n = 2, this gives a based set), which fit into ..."
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Cited by 18 (9 self)
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The fact that the relative homotopy groups do not satisfy excision makes the computation of absolute homotopy groups difficult in comparison with homology groups. The failure of excision is measured by triad homotopy groups πn(X; A, B), with n � 3 (for n = 2, this gives a based set), which fit into an exact sequence.
Quillen Closed Model Structures for Sheaves
, 1995
"... In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2groupoids, the category of bisim ..."
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Cited by 14 (0 self)
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In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2groupoids, the category of bisimplicial sheaves and the category of simplicial sheaves of groupoids. Subsequently, the homotopy theories of these categories are related to the homotopy theory of simplicial sheaves. 1 Introduction There are two ways of trying to generalize the well known closed model structure on the category of simplicial sets to the category of simplicial objects in a Grothendieck topos. One way is to concentrate on the local aspect, and to use the Kanfibrations as a starting point. In [14] Heller showed that for simplicial presheaves there is a local (there called right) closed model structure. In [2] K. Brown showed that for a topological space X the category of "locally fibrant" sheaves of spectra on ...
Hopf formulae for the higher homology of a group
 Bull. London Math. Soc
, 1988
"... In this note we generalise Hopfs formula for the second homology of a group G in terms of a free presentation R> • F^ » G. We prove: THEOREM 1. Let Rlt...,Rn be normal subgroups of a group F such that F/Y\iiiinRi = G, and for each proper subset A of <«> = {1,...,«} the groups Hr(F/Y[ieA R,) are tri ..."
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Cited by 10 (3 self)
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In this note we generalise Hopfs formula for the second homology of a group G in terms of a free presentation R> • F^ » G. We prove: THEOREM 1. Let Rlt...,Rn be normal subgroups of a group F such that F/Y\iiiinRi = G, and for each proper subset A of <«> = {1,...,«} the groups Hr(F/Y[ieA R,) are trivial for r = 2ifA = 0, and for r = \A \ + 1 and \A \ + 2 if A # 0 (for example, the groups F/\\ieARi are free for A£(n)). Then there is an isomorphisml)l /»£<«> ieA ilA s tne Here \A \ denotes the order of A, and ^6^1 * subgroup of F generated by trivial subgroup). Also is understood to mean F. Thus for n = 2 the formula reads the subgroups Ri with ieA (in particular, I~[ie0^i * S ^ e f]ie0Rt H3(G) « {R, n R2 n [F, F]}/{[F, R, 0^) [Rlt R2]}. Note that for any group G and n ^ 1, such an F and Rt can be found: let F l (G) be the free group on G; define inductively Ft = F^^G)), and set F = F n (G); for 1 ^ / ^ n let e(: F n (G)> F n ~\G) denote the canonical homorphisms induced by applying F"' * to the standard 'augmentation ' map F^F 1 " 1 ^))>F l ~\G) (where /ro((7) = Gyf and set ^ ( = Kerfii An alternative method, analogous to methods in [4,5], is best illustrated for n = 2. Choose any surjections F{^G with F { free, / = 1,2. Let P be the pullback of these surjections and choose a surjection F » P with F free. Let Rf be the kernel of the composite F> P> Ft. In general, one constructs inductively an «cube of groups F such that, for A c <«>: (i) /r js free if ^ ^ <«>, (ii) /r js G for ^ = <«>, and (iii) the morphism FA»limB=)/4FB is surjective. Such an /icube might be called afibrant npresentation of G. Again, suppose G = F/HKwhere //and #are normal subgroups of Fsuch that F, F/H and F/KSLTQ free. For example, we might be given a presentation <A ^ £/, V) of
Computing homotopy types using crossed ncubes of groups
 in Adams Memorial Symposium on Algebraic Topology
, 1992
"... Dedicated to the memory of Frank Adams ..."
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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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.
SEMISTRICT TAMSAMANI NGROUPOIDS AND CONNECTED NTYPES
, 2007
"... Tamsamani’s weak ngroupoids are known to model ntypes. In this paper we show that every Tamsamani weak ngroupoid representing a connected ntype is equivalent in a suitable way to a semistrict one. We obtain this result by comparing Tamsamani’s weak ngroupoids and cat n−1groups as models of co ..."
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Cited by 5 (1 self)
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Tamsamani’s weak ngroupoids are known to model ntypes. In this paper we show that every Tamsamani weak ngroupoid representing a connected ntype is equivalent in a suitable way to a semistrict one. We obtain this result by comparing Tamsamani’s weak ngroupoids and cat n−1groups as models of connected ntypes.
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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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]