We associate to a Hausdorff space, $ X $, a double groupoid, $ \mbox{\boldmath $ \rho $}^{\square}_{2} (X) $, the homotopy double groupoid of $ X $. The construction is based on the geometric notion of thin square. Under the equivalence of categories between small $ 2 $-categories and double categories with connection the homotopy double groupoid corresponds to the homotopy 2- groupoid, $ {\bf G}_{2} (X) $. The cubical nature of $ \mbox{\boldmath $ \rho $}^{\square}_{2} (X) $ as opposed to the globular nature of $ {\bf G}_{2} (X) $ should provide a convenient tool when handling `local-to-global' problems as encountered in a generalised van Kampen theorem and dealing with tensor products and enrichments of the category of compactly generated Hausdorff spaces.
Keywords: double groupoid, connection, thin structure, 2-groupoid, double track, 2- track, thin square, homotopy addition lemma.
2000 MSC: 18D05, 20L05, 55Q05, 55Q35.
Theory and Applications of Categories, Vol. 10, 2002, No. 2, pp 71-93.
http://www.tac.mta.ca/tac/volumes/10/2/10-02.dvi
http://www.tac.mta.ca/tac/volumes/10/2/10-02.ps
http://www.tac.mta.ca/tac/volumes/10/2/10-01.pdf
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/10/2/10-02.dvi
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/10/1/10-02.ps