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Axioms 2012, 1(3), 291-323; doi:10.3390/axioms1030291
The Hecke Bicategory
Received: 19 July 2012; in revised form: 4 September 2012 / Accepted: 5 September 2012 / Published: 9 October 2012
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations)
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract: We present an application of the program of groupoidification leading up to a sketch of a categorification of the Hecke algebroid—the category of permutation representations of a finite group. As an immediate consequence, we obtain a categorification of the Hecke algebra. We suggest an explicit connection to new higher isomorphisms arising from incidence geometries, which are solutions of the Zamolodchikov tetrahedron equation. This paper is expository in style and is meant as a companion to Higher Dimensional Algebra VII: Groupoidification and an exploration of structures arising in the work in progress, Higher Dimensional Algebra VIII: The Hecke Bicategory, which introduces the Hecke bicategory in detail.
Keywords: Hecke algebras; categorification; groupoidification; Yang–Baxter equations; Zamalodchikov tetrahedron equations; spans; enriched bicategories; buildings; incidence geometries