The structure of double groupoids

Details The structure of double groupoids The structure of double groupoids

2006-02-22

arxiv.org/abs/math/0602497v4

J. Pure Appl. Algebra Volume 213, Issue 6, June 2009, Pages 1031-1045

Mathematics

Category Theory

amslatex, 28 pages, revised version to appear in J. Pure Appl. Algebra

Scientific paper

We give a general description of the structure of a discrete double groupoid (with an extra, quite natural, filling condition) in terms of groupoid factorizations and groupoid 2-cocycles with coefficients in abelian group bundles. Our description goes as follows: To any double groupoid, we associate an abelian group bundle and a second double groupoid, its frame. The frame satisfies that every box is determined by its edges, and thus is called a `slim' double groupoid. In a first step, we prove that every double groupoid is obtained as an extension of its associated abelian group bundle by its frame. In a second, independent, step we prove that every slim double groupoid with filling condition is completely determined by a factorization of a certain canonically defined `diagonal' groupoid.

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