The Temperley-Lieb Tower and the Weyl Algebra

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