ISSN:
1572-9095
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
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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 / N X, where Π 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 Π X 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 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1008758412196
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