Groupoidification and Khovanov's Categorification of the Heisenberg Algebra
(Joint work with Jamie Vicary) The aim of this talk is to describe the connection between two approaches to categorification of the Heisenberg algebra. The groupoidification program of Baez and Dolan has been used to give a representation of the quantum harmonic oscillator in the category Span(Gpd) where the Fock space is represented by the groupoid of finite sets and bijections. This naturally gives a combinatorial interpretation of the (one-variable) Heisenberg algebra in the endomorphisms of this groupoid. On the other hand, Khovanov has given a categorification in which the integral part of the (many variable) Heisenberg algebra is recovered as the Grothendieck ring of a certain monoidal category described in terms of a calculus of diagrams. I will describe how an extension of the groupoidification program to a 2-categorical form of Span(Gpd), known as "2-linearization", recovers the relations used by Khovanov's construction, and how to interpret them combinatorially in terms of the groupoid of finite sets.