# A homotopy double groupoid of a Hausdorff space

## Ronald Brown, Keith A. Hardie, Klaus Heiner Kamps, Timothy Porter

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.

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