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]

Introduction \Gamma-spaces were introduced by Segal [S], who proved that they are combinatorial models for connective spectra (see also [A], [BF]). Based on Kan-Thurston theorem we show that any \Gamma-space is stably weak equivalent to a discrete \Gamma-group. By a well-known theorem of Dold-Kan the Moore normalization establishes the equivalence between the category of simplicial abelian groups and the category of chain complexes (see [DP]). mimicking the construction of normalization of simplicial groups, we give a similar construction for \Gamma-groups. This construction is based on the notion of cross-effects of functors [BP], which is a generalizatin of the classical definition of Eilenberg and Mac Lane [EM] to the non-abelian setup. Finally a Dold-Kan type theorem for the category of \Gamma-groups is proved. In abelian case our theorem claims that the category of abelian \Gamma-groups is equivalent to the category of functors Ab\Omega , where\Om

Following Ellis, [9], we investigate the notion of totally free crossed square and related squared complexes. It is shown how to interpret the information in a free simplicial group given with a choice of CW-basis, interms of the data for a totally free crossed square. Results of Ellis then apply to give a description in terms of tensor products of crossed modules. The paper ends with a purely algebraic derivation of a result

The nonabelian tensor square G ⊗ G of the group G is the group generated by the symbols g ⊗ h, where g, h ∈ G, subject to the relations gg ′ ⊗ h = ( g g ′ ⊗ g h)(g ⊗ h) and g ⊗ hh ′ = (g ⊗ h) ( h g ⊗ h h ′) for all g, g, h, h ′ ∈ G, where g g ′ = gg ′ g −1 is conjugation on the left. Following the work of C. Miller [18], R. K. Dennis in [10] introduced the nonabelian tensor square which is a specialization of the more general nonabelian tensor product independently introduced by R. Brown and J.-L. Loday [6]. By computing the nonabelian tensor square we mean finding a standard or simplified presentation for it. In the case of finite groups, the definition of the nonabelian tensor square gives a finite presentation that can be simplified using Tietze transformations. This simplified presentation can then be examined to determine the nonabelian tensor square. This was the approach taken in [3], in which the nonabelian tensor square was computed for each nonabelian group of order up to 30. Creating a presentation from the definition of the nonabelian tensor square, simplifying it using Tietze transformations and computing a structure description from the simplified presentation can be implemented in few lines of GAP [16]. However, this strategy does not scale well to finite groups G having order greater than 100 since the initial presentation has |G | 2 generators and 2|G | 3 relations. The most general method for computing the nonabelian tensor square uses the notion of a crossed pairing (see [3]). Let G and L be

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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

A functor from simplicial algebras to crossed n-cubes is shown to be an embedding on a reflexive subcategory of the category of simplicial algebras that contains representatives for all n types.

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This article investigates the homotopy theory of simplicial commutative algebras with a view to homological applications. 1