Results 1 
9 of
9
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 ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
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]
On algebraic models for homotopy 3types
 J. Homotopy Relat. Struct
"... We explore the relations among quadratic modules, 2crossed modules, crossed squares and simplicial groups with Moore complex of length 2. ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
We explore the relations among quadratic modules, 2crossed modules, crossed squares and simplicial groups with Moore complex of length 2.
Freeness Conditions for Crossed Squares and Squared Complexes.
, 2008
"... 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 CWbasis, interms of the data for a totally free crossed square. Results of Ellis then apply to ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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 CWbasis, 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
form the first 9 chapters of a longer document that is still evolving!)
"... an introduction to crossed gadgetry and cohomology in algebra and topology. (Notes initially prepared for the XVI Encuentro Rioplatense de ..."
Abstract
 Add to MetaCart
an introduction to crossed gadgetry and cohomology in algebra and topology. (Notes initially prepared for the XVI Encuentro Rioplatense de
Scategories, Sgroupoids, Segal categories and quasicategories
, 2008
"... The notes were prepared for a series of talks that I gave in Hagen in late June and early July 2003, and, with some changes, in the University of La Laguña, the Canary Islands, in September, 2003. They assume the audience knows some abstract homotopy theory and as Heiner Kamps was in the audience in ..."
Abstract
 Add to MetaCart
The notes were prepared for a series of talks that I gave in Hagen in late June and early July 2003, and, with some changes, in the University of La Laguña, the Canary Islands, in September, 2003. They assume the audience knows some abstract homotopy theory and as Heiner Kamps was in the audience in Hagen, it is safe to assume that the notes assume a reasonable knowledge of our book, [26], or any equivalent text if one can be found! What do the notes set out to do? “Aims and Objectives! ” or should it be “Learning Outcomes”? • To revisit some oldish material on abstract homotopy and simplicially enriched categories, that seems to be being used in today’s resurgence of interest in the area and to try to view it in a new light, or perhaps from new directions; • To introduce Segal categories and various other tools used by the NiceToulouse group of abstract homotopy theorists and link them into some of the older ideas; • To introduce Joyal’s quasicategories, (previously called weak Kan complexes but I agree with André that his nomenclature is better so will adopt it) and show how that theory links in with some old ideas of Boardman and Vogt, Dwyer and Kan, and Cordier and myself; • To ask lots of questions of myself and of the reader. The notes include some material from the ‘Cubo ’ article, [35], which was itself based on notes for a course at the Corso estivo Categorie e Topologia in 1991, but the overlap has been kept as small as is feasible as the purpose and the audience of the two sets of notes are different and the abstract homotopy theory has ‘moved on’, in part, to try the new methods out on those same ‘old ’ problems and to attack new ones as well. As usual when you try to specify ‘learning outcomes ’ you end up asking who has done the learning, the audience? Perhaps. The lecturer, most certainly! 1
THREE CROSSED MODULES
, 812
"... We introduce the notion of 3crossed module, which extends the notions of 1crossed module (Whitehead) and 2crossed module (Conduché). ..."
Abstract
 Add to MetaCart
We introduce the notion of 3crossed module, which extends the notions of 1crossed module (Whitehead) and 2crossed module (Conduché).