Brown, Ronald; Kamps, K. H.; Porter, Timothy A homotopy double groupoid of a Hausdorff space. II: A van Kampen theorem. (English) Zbl 1076.18004 Theory Appl. Categ. 14, 200-220 (2005). [For part I, see R. Brown, K. A. Hardie, K. H. Kamps and T. Porter, Theory Appl. Categ. 10, 71–93 (2002; Zbl 0986.18010).]It has been shown that there is a functor assigning a double groupoid with connections to a Hausdorff topological space. The functor is obtained from singular cubes by applying equivalence relations based on certain types of relative homotopy. The functor has a \(1\)-dimensional part with the fundamental groupoid as a quotient, and it has a \(2\)-dimensional part containing the second homotopy groups at all base points. In this paper the authors show that the functor satisfies a van Kampen theorem for open coverings. The proof involves commutative cubes. Reviewer: Richard John Steiner (Glasgow) Cited in 4 Documents MSC: 18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010) 20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms) 55Q05 Homotopy groups, general; sets of homotopy classes 55Q35 Operations in homotopy groups Keywords:homotopy double groupoid; connections; commutative cube; van Kampen theorem Citations:Zbl 0986.18001; Zbl 0986.18010 PDFBibTeX XMLCite \textit{R. Brown} et al., Theory Appl. Categ. 14, 200--220 (2005; Zbl 1076.18004) Full Text: arXiv EuDML Link