preprint: A van Kampen theorem for the homotopy double groupoid of a Hausdorff space
Preprint available http://www.informatics.bangor.ac.uk/public/mathematics/research/preprints/04... 04.01 : BROWN, R., KAMPS, K.H. & PORTER, T. A van Kampen theorem for the homotopy double groupoid of a Hausdorff space Summary: We show that the homotopy double groupoid of a Hausdorff space defined by the authors in a previous paper satisfies a version of the van Kampen theorem, and so is a suitable tool for non abelian, 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. There 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. Ronnie Brown Professor Emeritus R. Brown, Department of Mathematics, University of Wales, Bangor Dean St., Bangor, Gwynedd LL57 1UT, United Kingdom Tel. direct:+44 1248 382474|office: 382681 fax: +44 1248 361429 World Wide Web: home page: http://www.bangor.ac.uk/~mas010/ (Links to survey articles: Higher dimensional group theory Groupoids and crossed objects in algebraic topology) Centre for the Popularisation of Mathematics: http://www.cpm.informatics.bangor.ac.uk/ (reorganised site with new sculpture animations)
I have just put on the web two papers that explore constructive localic (and also topos) aspects of Lawvere's generalized metric spaces (see TAC Reprints 1, "Metric spaces, generalized logic and closed categories"). My papers are: "Localic completion of generalized metric spaces I" "Localic completion of generalized metric spaces II: Powerlocales" They can both be found through http://www.cs.bham.ac.uk/~sjv/README.html Lawvere describes generalized metric spaces as categories enriched over the extended half line [0, infinity]. I replace this enrichment category by a locale with the same points, and show how to obtain a completion as a locale. Topos aspects include a proof that the generic localic completion is an opfibration in the 2-category of grothendieck toposes. My second paper shows how the powerlocales of the localic completions are themselves also localic completions, and uses this to analyse questions of compactness. Steve Vickers.
participants (2)
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Ronnie Brown -
Steve Vickers