The Temperley-Lieb Tower and the Weyl Algebra

We define a monoidal category $\operatorname{\mathbf{W}}$ and a closely related 2-category $\operatorname{\mathbf{2Weyl}}$ using diagrammatic methods. We show that $\operatorname{\mathbf{2Weyl}}$ acts on the category $\mathbf{TL} :=\bigoplus_n \operatorname{TL}_n\mathrm{-mod}$ of modules over Temperley-Lieb algebras, with its generating 1-morphisms acting by induction and restriction. The Grothendieck groups of $\operatorname{\mathbf{W}}$ and a third category we define $\operatorname{\mathbf W}^\infty$ are closely related to the Weyl algebra. We formulate a sense in which $K_0(\operatorname{\mathbf W}^\infty)$ acts asymptotically on $K_0(\mathbf{TL})$.