On 2-Verma modules for quantum \({\mathfrak {sl}_{2}}\)

In this paper we study the superalgebra \(A_n\), introduced by the authors in previous work on categorification of Verma modules for quantum \({\mathfrak {sl}_{2}}\). The superalgebra \(A_n\) is akin to the nilHecke algebra, and shares similar properties. We also prove a uniqueness result about 2-Verma modules on \(\Bbbk \)-linear 2-categories.