A homotopy double groupoid of a Hausdorff space II: a van Kampen theorem

R. Brown, K.H. Kamps, and T. Porter

This paper is the second in a series exploring the properties of a functor which assigns a homotopy double groupoid with connections to a Hausdorff space.

We show that this functor satisfies a version of the van Kampen theorem, and so is a suitable tool for nonabelian, 2-dimensional, local-to-global problems. The methods are analogous to those developed by Brown and Higgins for similar theorems for other higher homotopy groupoids.

An integral part of the proof is a detailed discussion of commutative cubes in a double category with connections, and a proof of the key result that any composition of commutative cubes is commutative. These results have recently been generalised to all dimensions by Philip Higgins.

Keywords: double groupoid, double category, thin structure, connections, commutative cube, van Kampen theorem

2000 MSC: 18D05, 20L05, 55Q05, 55Q35

Theory and Applications of Categories, Vol. 14, 2005, No. 9, pp 200-220.


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