Homology, Homotopy and Applications

Volume 10 (2008)

Number 1

A new higher homotopy groupoid: the fundamental globular $ω$-groupoid of a filtered space

Pages: 327 – 343

DOI: https://dx.doi.org/10.4310/HHA.2008.v10.n1.a14


Ronald Brown (School of Computer Science, University of Wales, Gwynedd, Wales, United Kingdom)


We show that the graded set of filter homotopy classes rel vertices of maps from the $n$-globe to a filtered space may be given the structure of (strict) globular $ω$-groupoid. The proofs use an analogous fundamental cubical $ω$-groupoid due to the author and Philip Higgins in 1981. This method also relates the construction to the fundamental crossed complex of a filtered space, and this relation allows the proof that the crossed complex associated to the free globular $ω$-groupoid on one element of dimension n is the fundamental crossed complex of the $n$-globe.


filtered space, higher homotopy groupoid, higher homotopy van Kampen theorem, cubical singular complex, free globular groupoid

2010 Mathematics Subject Classification

18D10, 18G30, 18G50, 20L05, 55N10, 55N25

Published 1 January 2008