The structure of double groupoids

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Abstract

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 the first step, we prove that every double groupoid is obtained as an extension of its associated abelian group bundle by its frame. In the second, independent, step we prove that every slim double groupoid with a filling condition is completely determined by a factorization of a certain canonically defined 'diagonal' groupoid. © 2008 Elsevier B.V. All rights reserved.

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Andruskiewitsch, N., & Natale, S. (2009). The structure of double groupoids. Journal of Pure and Applied Algebra, 213(6), 1031–1045. https://doi.org/10.1016/j.jpaa.2008.11.003

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