Let M denote the moduli stack of either coherent sheaves on a smooth projective surface or Higgs sheaves on a smooth projective curve X. The convolution algebra structure on the Borel-Moore homology of M is an instance of two-dimensional cohomological Hall algebras. These examples were defined by Kapranov-Vasserot and by Schiffmann and me, respectively.
During the first part of the talk, I will give a gentle introduction to cohomological Hall algebras and their relevance in algebraic geometry and representation theory. The second part of the talk is devoted to the description of the full categorification of the cohomological Hall algebra of M. I will explain the construction and some applications in the theory of moduli spaces of sheaves. This is based on a joint work with Mauro Porta.