A Homotopy 2-Groupoid of a Hausdorff Space |
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Authors: | K A Hardie K H Kamps R W Kieboom |
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Institution: | (1) Department of Mathematics, University of Cape Town, 7700 Rondebosch, South Africa;(2) Fachbereich Mathematik, Fernuniversität, Postfach 940, D-58084 Hagen, Germany;(3) Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, F 10, B-1050 Brussels, Belgium |
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Abstract: | If X is a Hausdorff space we construct a 2-groupoid G
2
X with the following properties. The underlying category of G
2
X is the `path groupoid" of X whose objects are the points of X and whose morphisms are equivalence classes f, g of paths f, g in X under a relation of thin relative homotopy. The groupoid of 2-morphisms of G
2
X is a quotient groupoid X / N X, where X is the groupoid whose objects are paths and whose morphisms are relative homotopy classes of homotopies between paths. N X is a normal subgroupoid of X determined by the thin relative homotopies. There is an isomorphism G
2
X(f,f) 2(X, f(0)) between the 2-endomorphism group of f and the second homotopy group of X based at the initial point of the path f. The 2-groupoids of function spaces yield a 2-groupoid enrichment of a (convenient) category of pointed spaces.We show how the 2-morphisms may be regarded as 2-tracks. We make precise how cubical diagrams inhabited by 2-tracks can be pasted. |
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Keywords: | 2-groupoid 2-track track homotopy higher homotopy structures tree fundamental groupoid pasting piecewise linear map Gray tensor product interchange 2-track folding map |
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