Free Monoid in Monoidal Abelian Categories

We give an explicit construction of the free monoid in monoidal abelian categories when the monoidal product does not necessarily preserve coproducts. Then we apply it to several new monoidal categories that appeared recently in the theory of Koszul duality for operads and props. This gives a conceptual explanation of the form of the free operad, free dioperad and free properad.      

A cohomological construction of quantization functors of Lie bialgebras

We propose a variant to the Etingof-Kazhdan construction of quantization functors. We construct the twistor J_Φ associated to an associator Φ using cohomological techniques. We then introduce a criterion ensuring that the “left Hopf algebra” of a quasitriangular QUE algebra is flat. We prove that this criterion is satisfied at the universal level. This gives a construction of quantization functors, equivalent to the Etingof-Kazhdan construction.