Categorification and Groupoidification of the Heisenberg Algebra

These lectures, prepared for Higher Structures in China III, held in Changchun, Aug 2012, describe a relationship between two forms of categorification of algebras by giving a combinatorial model for Khovanov’s categorification of the Heisenberg algebra in a 2-category of spans of groupoids. This is joint work with Jamie Vicary. The goal here is to describe two notions of “categorifying an algebra”, and see how they are related in one special example that of the Heisenberg algebra. The most important difference is a higher-categorical analog of the difference between a presentation and a representation of an algebra. The model we find via Baez-DolanTrimble “groupoidification” is a concrete representation of the abstract categorification of Khovanov. There are two lectures, which will cover the following: Lecture 1 • Categorification and Decategorification • The Heisenberg Algebra and the Fock Space Representation • Groupoidification and the Heisenberg Algebra Lecture 2 • Diagrammatic Categorification of the Heisenberg Algebra • Biadjointness and “process movies” • The Combinatorial Representation 1 First Lecture (Aug 13, 2012) 1.1 Categorification and Decategorification A decategorification operation is one which takes a category with structure to a set with related structure. Example 1.1. The cardinality operation: takes S ∈ Sets to its isomorphism class, which is labelled by the number |S| ∈ N. So N is a decategorification of Sets. Sets also has structure respected by | · |: