The first part of the talk is devoted to a gentle introduction to the theory of cohomological Hall algebras (COHAs for brevity) of surfaces and their categorification. Moreover, their “quantum” nature will be discussed. In addition, the Dolbeaut cohomological Hall algebras of curves will be introduced. During the second part of the talk, I will address the problem of constructing pairs consisting of a COHA and a representation of it canonically associated to torsion pairs of the abelian category of coherent sheaves on a smooth projective complex surface S.