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 Π X/NX,where ΠX is the groupoid whose objects are paths and whose morphisms are relative homotopy classes of homotopies between paths. NX is a normal subgroupoid of fIX determined by the thin relative homotopies. There is an isomorphism G 2 X (<f>, <f>) ≈ π2(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 2groupoids 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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Dedicated to Bernhard Banaschewski on his 70th birthday
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Hardie, K.A., Kamps, K.H., Kieboom, R.W. (2000). A Homotopy 2-Groupoid of a Hausdorff Space. In: Brümmer, G., Gilmour, C. (eds) Papers in Honour of Bernhard Banaschewski. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2529-3_11
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DOI: https://doi.org/10.1007/978-94-017-2529-3_11
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