Atlas: Multiple groupoids as a non abelian tool for local-to-global problems by Ronald Brown

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Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras and Semiabelian Categories
September 23-28, 2002
Fields Institute
Toronto, ON, Canada

Organizers
George Janelidze, Georgian Academy of Sciences, Bodo Pareigis, University of Munich, Walter Tholen, York University

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Multiple groupoids as a non abelian tool for local-to-global problems
by
Ronald Brown
Mathematics Division, School of Informatics, University of Wales, Bangor

Local-to-global problems play a central role in mathematics and science. In the period from 1974 the writer, with Philip Higgins and with Jean-Louis Loday, developed the use of higher homotopy groupoids as non abelian structures which could attack some such problems in homotopy theory. The higher homotopy groupoids used were non trivial structures determined by a filtered space (Brown-Higgins) or an n-cube of spaces (Loday). Some complicated equivalences of categories showed these structures to be related to well known relative and n-adic homotopy theory. The Generalised Van Kampen Theorems (GVKTs) which used these homotopy groupoids gave non abelian colimit theorems which yielded new and precise calculations of some homotopical invariants. Computing colimits of certain multiple groupoids also suggested new algebraic constructions, such as a non abelian tensor product of groups.

These GVKTs are not so generally used, and so this lecture will give an introduction to their use and to the intuitions behind them, such as the slogan `algebraic inverses to subdivision'. It is hoped that the workshop will help to clarify issues such as the relation of this work to descent problems.

Date received: May 13, 2002