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Around 2010, Khovanov defined the Heisenberg category, which is a monoidal category presented using diagrammatic generators and relations. Conceptually, the generating objects should be thought of as  induction and restriction functors between symmetric groups, and morphisms are natural transformations between (compositions of) these functors. Khovanov (and Brundan-Savage-Webster) proved its Grothendieck group is the Heisenberg algebra, and Cautis-Lauda-Licata-S.-Sussan proved the Licata-Savage q-deformed Heisenberg category has the elliptic Hall algebra as its trace (or, Hochschild homology). In this talk we discuss joint work in progress with M. Harper where we define an analogous category using induction and restriction functors between Temperley-Lieb algebras. One surprising technical fact about this new category is that "its bubble subalgebra is spanned by idempotents." Its Grothendieck group has some similarities to the Weyl algebra, but the generators and relations that we have found so far still seem quite mysterious. (The talk will start with a brief description of the Temperley-Lieb algebras and not assume much knowledge beyond that, but will sweep some technical details under a rather large rug.)