`Double modules, double categories and groupoids, and a new homotopical double groupoid

We give a rather general construction of double categories and so, under further conditions, double groupoids, from a structure we call a `double module’. We also give a homotopical construction of a double groupoid from a triad consisting of a space, two subspaces, and a set of base points, under a condition which also implies that this double groupoid contains two second relative homotopy groups.

💡 Research Summary

The paper introduces a unifying algebraic framework called a “double module” from which one can systematically construct double categories and, under additional invertibility hypotheses, double groupoids. A double module consists of a set (M) equipped with two structure maps to sets (H) (horizontal) and (V) (vertical) together with two binary operations—horizontal composition (\circ_H) and vertical composition (\circ_V). The crucial requirement is a interchange law
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