Infinite-Dimensional Towers and a Categorification of Differential Operators on ${\mathbb{A}}_{\mathbb{Z}[v,v^{-1}]}^1$

We introduce axioms for towers of infinite-dimensional algebras such that the corresponding Grothendieck groups of projective and finite-dimensional modules are Hopf dual to each other. This duality gives rise to an action of the Hesienberg double – a generalization of the classical Hesienberg algebra – on the Grothendieck group. We categorify this action and, as an application, construct a categorical realization of quantum differential operators on ${\mathbb{A}}_{\mathbb{Z}[v,v^{-1}]}^1$.