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From groups to groupoids: a brief survey
 Bull. London Math. Soc
, 1987
"... A groupoid should be thought of as a group with many objects, or with many identities. A precise definition is given below. A groupoid with one object is essentially just a group. So the notion of groupoid is an extension of that of groups. It gives an additional convenience, flexibility and range o ..."
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Cited by 58 (7 self)
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A groupoid should be thought of as a group with many objects, or with many identities. A precise definition is given below. A groupoid with one object is essentially just a group. So the notion of groupoid is an extension of that of groups. It gives an additional convenience, flexibility and range of applications, so that even for purely grouptheoretical work, it can be useful to take a path through the world of groupoids.
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
Crossed Complexes And Homotopy Groupoids As Non Commutative Tools For Higher Dimensional LocalToGlobal Problems
"... We outline the main features of the definitions and applications of crossed complexes and cubical #groupoids with connections. ..."
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Cited by 18 (7 self)
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We outline the main features of the definitions and applications of crossed complexes and cubical #groupoids with connections.
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 ..."
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.
TQFTs from Homotopy ntypes
, 1995
"... : Using simplicial methods developed in [22], we construct topological quantum field theories using an algebraic model of a homotopy ntype as initial data, generalising a construction of Yetter in [23] for n=1 and in [24] for n=2 Introduction In [23], Yetter showed how to construct a topological q ..."
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Cited by 4 (2 self)
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: Using simplicial methods developed in [22], we construct topological quantum field theories using an algebraic model of a homotopy ntype as initial data, generalising a construction of Yetter in [23] for n=1 and in [24] for n=2 Introduction In [23], Yetter showed how to construct a topological quantum field theory with coefficients in a finite group. In [24], he showed that his construction could be extended to handle coefficients in a finite categorical group, or cat 1 group. These objects are algebraic models for certain homotopy 2types. The topological quantum field theories thus constructed are (2+1) TQFTs, but the methods used do not depend on the manifolds being surfaces, except to avoid possible irregularities related to problems of triangulations in low dimensions. Yetter ended that second note with some open questions, the third of which was: can one carry out the same sort of construction for algebraic models of higher homotopy types? In this note we will show that a ...
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]
Effective generalized SeifertVan Kampen: how to calculate ΩX, preprint available at qalg/9710011
"... A central concept in algebraic topology since the 1970’s has been that of delooping machine [4] [23] [29]. Such a “machine ” corresponds to a notion of Hspace, or space with a multiplication satisfying associativity, unity and inverse properties up to homotopy in an appropriate way, including highe ..."
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Cited by 4 (1 self)
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A central concept in algebraic topology since the 1970’s has been that of delooping machine [4] [23] [29]. Such a “machine ” corresponds to a notion of Hspace, or space with a multiplication satisfying associativity, unity and inverse properties up to homotopy in an appropriate way, including higher order coherences as first investigated in [33]. A delooping machine is a specification of the extra homotopical structure carried by the loop space ΩX of a connected basepointed topological space X, exactly the structure allowing recovery of X by a “classifying space ” construction. The first level of structure is that the component set π0(ΩX) has a structure of group π1(X, x). Classically the SeifertVan Kampen theorem states that a pushout diagram of connected spaces gives rise to a pushout diagram of groups π1. The loop space construction ΩX with its delooping structure being the higherorder “topologized ” generalization of π1, an obvious question is whether a similar SeifertVan Kampen statement holds for ΩX. The aim of this paper is to describe the operation underlying pushout of spaces with loop space structure, answering the above question by giving a SeifertVan Kampen statement for delooping machinery. We work with Segal’s machine [28] [36]. Our SeifertVan