‘Double modules’, double categories and groupoids, and a new homotopical double groupoid

Double categories were introduced by Ehresmann [Ehr65] as an example of structured category. Examples of double groupoids were shown to arise from crossed modules in [BS76b, BS76a] and this result was applied in [BH78] to give a 2-dimensional version of the Van Kampen Theorem for the fundamental group, namely a colimit theorem for the fundamental crossed module of a based pair. These results were generalised to crossed modules of groupoids, and to higher dimensions, in [BH81a]. A more general version of the construction of double groupoids was given in [BM92] in terms of a core diagram. Another construction of double groupoids due to the author was taken up by Lu and Weinstein in [LW89] for purposes of Poisson groupoids. The first aim of this note is to give a generalisation of this last construction. We do not obtain an equivalence of categories, and so in effect the construction shows that double groupoids can be quite complex objects. Perhaps they should be considered among the basic structures in mathematics. A classification of double groupoids is given in [AN]. The second aim (Section 2) is to give a new construction of a double groupoid for a topological space X with three subspaces A,B,C such that C ⊆ A∩B. Here C is thought of as a set of base points, and the double groupoid is well defined if the two induced morphisms π2(A, c) → π2(X, c), π2(B, c) → π2(X, c) have the same image. Under this condition, the double groupoid also contains the two relative homotopy groups π2(X,A, c), π2(X,B, c) for all c ∈ C. This is an extension of results of [BH78]. Thus this construction has the advantage of generality, symmetry, and multiple compositions in either directions, advantages not available for the traditional relative homotopy groups. The relation between the two constructions is unclear, and it is hoped that this paper will encourage further study of the area. The bibliography gives other uses of double categories and groupoids, [Spe77, BM99, DP93, Ehr63b, BJ04, Mac99], but is not intended to be exhaustive.