Determination of a double Lie groupoid by its core diagram

Abstract

In a double groupoid S, we show that there is a canonical groupoid structure on the set of those squares of S for which the two source edges are identities; we call this the core groupoid of S. The target maps from the core groupoid to the groupoids of horizontal and vertical edges of S are now base-preserving morphisms whose kernels commute, and we call the diagram consisting of the core groupoid and these two morphisms the core diagram of S. If S is a double Lie groupoid, and each groupoid structure on S satisfies a natural double form of local triviality, we show that the core diagram determines S and, conversely, that a locally trivial double Lie groupoid may be constructed from an abstractly given core diagram satisfying some natural additional conditions. In the algebraic case, the corresponding result includes the known equivalences between crossed modules, special double groupoids with special connection (Brown and Spencer), and cat 1-groups (Loday). These cases correspond to core diagrams for which both target morphisms are (compatibly) split surjections.