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

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 $\mathbb{A}^1_{\mathbb{Z}[v,v^{-1}]}$.