A Homotopy Double Groupoid of a Hausdorff Space

We associate to a Hausdorff space, X, a double groupoid, ρ 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 given in [BM] the homotopy double groupoid corresponds to the homotopy 2-groupoid, G2(X), constructed in [HKK]. The cubical nature of ρ 2 (X) as opposed to the globular nature ofG2(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. Introduction We associate to a Hausdorff space, X, a double groupoid, ρ 2 (X), called the homotopy double groupoid of X. The construction is based on the geometric notion of thin square extending the notion of thin relative homotopy introduced in [HKK]. Roughly speaking, a thin square is a continuous map from the unit square of the real plane into X which factors through a tree. More precisely, the homotopy double groupoid is a double groupoid with connection which, under the equivalence of categories between small 2-categories and double categories with connection given in [BM], corresponds to the homotopy 2-groupoid, G2(X), of X constructed in [HKK]. We make use of the properties of pushouts of trees in the category of Hausdorff spaces investigated in [HKK]. The construction of the 2-cells of the homotopy double groupoid is based on a suitable cubical approach to the notion of thin 3-cube whereas the construction of the 2-cells of the homotopy 2-groupoid can be interpreted by means of a globular notion of thin 3-cube. Why double groupoids with connection? The homotopy double groupoid of a space and the related homotopy 2-groupoid are constructed directly from the cubical singular complex and so remain close to geometric intuition in an almost classical way. Unlike in the globular 2-groupoid approach, however, the resulting structure remains cubical and in particular is symmetrical with respect to the two directions. Cubes subdivide neatly so complicated pasting arguments can be avoided in this context. Composition as against subdivision is easy to handle. The ‘geometry’ is near to the surface and naturally leads to the algebra. Received by the editors 2001 July 17 and, in revised form, 2002 January 7. Transmitted by Robert Paré. Published on 2002 January 17. 2000 Mathematics Subject Classification: 18D05, 20L05, 55Q05, 55Q35.