Khovanov's Heisenberg category (and its later generalizations) acts on the category of modules over the tower of symmetric groups (or Hecke algebras). The generating objects act via compositions of induction and restriction functors, and generating morphisms map to certain natural transformations between these compositions. In joint work, Harper and I defined the "Weyl category,'' whose starting point is Temperley-Lieb algebras instead of Hecke algebras. In this talk, I explain the construction of this category, its action, and what we know and/or expect about its Grothendeick group and trace. One new feature is the existence of "idempotent bubbles."