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.
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.psTAC Home