theory, and change of base for groupoids and multiple

We outline the main features of the definitions and applications of crossed complexes and cubical #-groupoids with connections.

Dedicated to the memory of Frank Adams

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 isomorphism-l)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 /i-cube might be called afibrant n-presentation 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

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.

: Using simplicial methods developed in [22], we construct topological quantum field theories using an algebraic model of a homotopy n-type 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 2-types. 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 ...

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 n-fold Čech derived functors of the lower central series functors Zk. The paper ends with an application to algebraic K-theory. 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]

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cal structures suggests, then sustained interaction is essential. Bringing people together will result in significant technical progress and significantly greater conceptual understanding of these structures. Despite their complexity, if developed coherently, these structures, like categories themselves, should eventually become part of the standard mathematical culture. PROJECT DESCRIPTION EXPLORATIONS OF HIGHER CATEGORICAL STRUCTURES AND THEIR APPLICATIONS PROPOSED RESEARCH 1. Historical background and higher homotopies Eilenberg and Mac Lane introduced categories, functors, and natural transformations in their 1945 paper [47]. The language they introduced transformed modern mathematics. Their focus was not on categories and functors, but on natural transformations, which are maps between functors. Higher category theory concerns higher level notions of naturality, which can be viewed as maps between natural transformations, and maps between such maps, and so for

d manifolds of a certain dimension and the maps between them are equivalence classes of cobordisms between them, which are manifolds with boundary in the next higher dimension. However, it is in many respects far more natural to deal with an ncobordism "category" constructed from points, edges, surfaces, and so on through n-manifolds that have boundaries with corners. The structure encodes cobordisms between cobordisms between cobordisms. This is an n-category with additional structure, and one needs analogously structured linear categories as targets for the appropriate "functors" that define the relevant TQFT's. One could equally well introduce the basic idea in terms of formulations of programming languages that describe processes between processes between processes. A closely analogous idea has long been used in the study of homotopies between homotopies between homotopies in algebraic topology. Analogous structures appear throughout mathematics. In contrast to the original Eilenb