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Algebraic Topology


BROWN, R. & SIVERA, R.

Nonabelian Algebraic Topology

The first draft of Part 1 of a three-part book is now available on the web. From the Preface:
Our aim for this book is to give a connected and we hope readable account of the main features of work on extending to higher dimensions the theory and applications of the fundamental group.


04.01 : BROWN, R., KAMPS, K.H. & PORTER, T.

A homotopy double groupoid of a Hausdorff space II: a van Kampen theorem

Summary:

This paper is the second in a series exploring the properties of a functor which assigns a homotopy double groupoid with connections to a Hausdorff space.

We show that this functor satisfies a version of the van Kampen theorem, and so is a suitable tool for nonabelian, 2-dimensional, local-to-global problems. The methods are analogous to those developed by Brown and Higgins for similar theorems for other higher homotopy groupoids.

An integral part of the proof is a detailed discussion of commutative cubes in a double category with connections, and a proof of the key result that any composition of commutative cubes is commutative. These results have recently been generalised to all dimensions by Philip Higgins.

Published in (and Download):

Theory and Applications of Categories 14 (2005) 200-220.

04.02 : PORTER, T.

S-categories, S-groupoids, Segal categories and quasicategories

Summary:

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 Laguna, 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, 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 Nice-Toulouse group of abstract homotopy theorists and link them in to some of the older ideas;
  • To introduce Joyal's quasicategories, (previously called weak Kan complexes but I agree with Andre 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 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 y ou try to specify `learning outcomes' you end up asking who has done the learning, the audience? Perhaps. The lecturer, most certainly!

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04.05 : FORRESTER-BARKER, M.E.

Representations of crossed modules and cat1-groups

Published in:

University of Wales, Bangor, PhD thesis (2004)

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04.15 : BROWN, R.

Nonabelian Algebraic Topology

Summary:

This is an extended account of a short presentation with this title given at the Minneapolis IMA Workshop on n-categories: foundations and applications, June 7-18, 2004, organised by John Baez and Peter May.

It gives a sketch of the background for the book in preparation with this title.

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04.16 : HIGGINS, P.J.

Thin Elements and Commutative Shells in Cubical omega-categories

Summary:

The relationships between thin elements, commutative shells and connections in cubical omega-categories are explored by a method which does not involve the use of pasting theory or nerves of omega-categories (both of which were previously needed for this purpose; see 00.11, Section 9). It is shown that composites of commutative shells are commutative and that thin structures are equivalent to appropriate sets of connections; this work extends to all dimensions the results proved in dimensions 2 and 3 in 99.13 and 04.01.

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04.20 : BROWN, R. & SIVERA, R.

Acyclic models and crossed complexes

Summary:

Published in:

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