Construction of higher groupoids via matched pairs actions

In this work, we construct a relationship between matched pairs and triples of groupoids. Given two 3-groupoids with a common edge, we construct a triple groupoid by using the matched pairs actions.

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[1] Brandt H. Ubereine verallgemeinerung des gruppenbegriffes. Mathematische Annalen 1926; 96: 360-366 (in German).
[2] Brown R. Groupoids and Van Kampen’s theorem. Proceedings of the London Mathematical Society 1967; 3: 385- 400.
[3] Brown R. Fibration of groupoids. Journal of Algebra 1970; 15: 103-132.
[4] Brown R. Groupoids as coefficients. Proceedings of the London Mathematical Society 1972; 3 (25): 413-426.
[5] Brown R. Topology and Groupoids. Charleston, SC, USA: BookSurge Publishing, 2006.
[6] Brown R. Double groupoids, matched pairs and then matched triples. https://arxiv.org/abs/1104.1644, 2011.
[7] Brown R, Janelidze G. Galois theory and a new homotopy double groupoid of a map of spaces. Applied Categorical Structures 2004; 12: 63-80.
[8] Brown R, Spencer C. Double groupoids and crossed modules. Cahiers de Topologie et Géométrie Différentielle Catégoriques 1976; 17: 343-364.
[9] Ehresmann C. Catégories doubles et catégories structurées. Comptes Rendus Mathematique Academie des Sciences Paris 1963; 2569: 1198-1201 (in French).
[10] Gateva-Ivanova T, Majid S. Matched pairs approach to set theoretic solutions of the Yang–Baxter equation. Journal of Algebra 2008; 319: 1462–1529.
[11] Higgins PJ. Notes on Categories and Groupoids. London, UK: Van Nostrand Reinhold, 1971.
[12] Loday JL. Spaces with finitely many nontrivial homotopy groups. Journal of Pure and Applied Algebra 1982; 24: 179-202.
[13] Majard D. N-tuple groupoids and optimally coupled factorizations. Theory and Application of Categories 2013; 28: 304-331.
[14] Majid S. Matched pairs of Lie groups associated to solutions of the Yang-Baxter equations. Pacific Journal of Mathematics 1990; 140: 311-332.
[15] Majid S. Foundations of Quantum Group Theory. Cambridge, UK: Cambridge University Press, 1995.
[16] Singer WM. Extension theory for connected Hopf algebras. Journal of Algebra 1972; 21: 1-16.
[17] Takeuchi M. Matched pairs of groups and bismash products of Hopf algebras. Communications in Algebra 1981; 9: 841-882.