A Homotopy 2-Groupoid of a Hausdorff Space

K. A. Hardie, K. H. Kamps and R. W. Kieboom
Additional contact information
K. A. Hardie: University of Cape Town, Department of Mathematics
K. H. Kamps: Fernuniversität, Fachbereich Mathematik
R. W. Kieboom: Vrije Universiteit Brussel, Department of Mathematics

A chapter in Papers in Honour of Bernhard Banaschewski, 2000, pp 209-234 from Springer

Abstract: 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 , 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 ( , ) ≈ π2(X, f(0)) between the 2-endomorphism group of 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.

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.; 18D05; 20L05; 55Q05; 55Q35 (search for similar items in EconPapers)
Date: 2000
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-017-2529-3_11

Ordering information: This item can be ordered from
http://www.springer.com/9789401725293

DOI: 10.1007/978-94-017-2529-3_11

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().