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This text is an updated version of material used for a course at Universit\'ede Nantes, part of `Functor homology and applications', April 23-27, 2012. Theproof by Touz\'e of my conjecture on cohomological finite generation (CFG) hasbeen one of the successes of functor homology. We will not treat the originalproof in any detail. Instead we will focus on a formality conjecture ofCha{\l}upnik that leads to second generation proof of the existence of theuniversal classes of Touz\'e.

Lectures on bifunctors and finite generation of rational cohomology algebras