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A homotopy double groupoid of a Hausdorff space. II: A van Kampen theorem. (English) Zbl 1076.18004

[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.

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
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