I am wondering whether there is a pedagogic exposition of the cohomology theory for vertex operator algebras (VOAs). As far as I am aware there are several theories.
One source here gives a theory for computing $H^1$ and $H^2$ for a given VOA and VOA module. There is also a small section in Kac's book on VOAs that described two cohomologies, one "reduced."
I would also like to know, in the same way that singular cohomology is a topological invariant, what kind of invariant would a cohomology theory for VOAs produce? In other words, what kind of changes to the VOA change the cohomology and which preserve it?
In terms of the viewpoint by which it is presented, as I learnt cohomology from an algebraic geometry background, that is how I understand it best, with minimal category theory.