Categorification of two-dimensional cohomological Hall algebras

In the present paper, we provide a full categorification, at the level of stable $\infty$-categories, of two-dimensional cohomological Hall algebras of curves and surfaces. This is achieved by producing a suitable derived enhancement of the relevant moduli stacks entering in the constructions of such algebras. This method categorifies the cohomological Hall algebra of Higgs sheaves on a curve and the cohomological Hall algebra of coherent sheaves on a surface. Furthermore, it applies also to several other moduli stacks, such as the moduli stack of vector bundles with flat connections on a curve $X$ and the moduli stack of finite-dimensional representations of the fundamental group of $X$