Homotopical algebra and relative Koszul duality theory

Given two homotopically equivalent chain complexes and an algebraic structure on one side, one can transfer it to the other side. However the transferred structure does not satisfy strictly the algebraic relations but only up to homotopy. The Koszul duality theory of operads gives the exact definition of what should be this homotopy algebraic structure. The first aim of my talk is to explain these results. In the end, I will show how to refine them, for instance introducing a new relative Koszul duality theory.