Extension, deformation and categorification of $\text{AssDer}$ pairs

In this paper, we consider associative algebras equipped with derivations. A pair consisting of an associative algebra and a distinguished derivation is called an AssDer pair. We study central extensions and formal one-parameter deformations of AssDer pairs in terms of cohomology. Finally, we define $2$-derivations on associative $2$-algebras and show that the category of associative $2$-algebras with $2$-derivations is equivalent to the category of $2$-term $A_{\mathrm{\infty }}$-algebras with homotopy derivations.