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.
http://www.tac.mta.ca/tac/volumes/14/9/14-09.dvi
http://www.tac.mta.ca/tac/volumes/14/9/14-09.ps
http://www.tac.mta.ca/tac/volumes/14/9/14-09.pdf
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/14/9/14-09.dvi
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/14/9/14-09.ps