The Hecke bicategory
Summary: 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 [\textit{J. C. Baez} et al., Theory Appl. Categ. 24, 489--553 (2010; Zbl 1229.18003)] and an exploration of structures arising in the work in progress [\textit{J. C. Baez} and the author, ``Higher dimensional algebra. VIII: The Hecke bicategory'', \url{http://math.ucr.edu/home/baez/hecke.pdf}], which introduces the Hecke bicategory in detail.
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