# Report from Villars (2 in a series)

In English, Homology and Homotopy, Mathematics, Operads and PROPs.

So we hit the pistes during monday morning, those of us who actually already are here. Me, Bruno Vallette (Hi Stockholm!), Arne Weiner, Miles Gould, Paul Eugene Parents and Jonathan Scott, Dev Sinha and Muriel Livernet. Skiing was MARVELOUS. Me, Arne and Miles shot off on our own, and damn did we have a good time.

As I'm writing this, they're still out there - I went back when the pain in my legs caused tears in my eyes for just turning on the skis. The techniques were solid as concrete. The muscles not so much. It took half an hour in the sauna to get to the point where I actually was able to walk again.

Among the more amusing things that happened was that when I was stopping to wait up for Arne and Miles in a narrow forest path, I ended up standing too close to the edge, which subsequently gave up and dropped me down into a few meters of powder just under a fern tree. Getting out of there was awkward - to begin with my legs were in the wrong angle to get out of the ski bindings; and once Miles helped me out, the only reason I wasn't buried in snow to my shoulders was that I packed it as I stood on it, and the carrying point ended up being roughly waistdeep.

Go on. You try it. Get up from waistdeep loose powder snow. Straight up, a meter or so, onto that hardened shell that you once were skiing on. It's an extremely amusing and rather hard exercise.

Now, for the mathematics. The (only) talk today was by Bruno Vallette, on

## Manin products and Koszul duality

For associative quadratic algebras, given as a quotient of the free tensor algebra on a space [tex]V[/tex] by [tex]A(V,R)=T(V)/(R)[/tex], Manin defines two different products, by [tex]A(V,R)\circ A(W,S)=A(V\otimes W,\tau(R\otimes W^2+V^2\otimes S))[/tex] for [tex]\tau a\otimes b\otimes c\otimes d=a\otimes c\otimes b\otimes d[/tex] the standard twisting homomorphism; and [tex]A(V,R)\bullet A(W,S)=A(V\otimes W,\tau(R\otimes S))[/teX]. The main and most relevant result here is that the two products are Koszul dual to each other, i.e. [tex](A\circ B)^!=A^!\bullet B^![/tex].

Using the concept of lax 2-monoidal categories, it is possible to generalize this to basically any possible interesting category - with the same particular construction viable for Algebras, Nonsymmetric and symmetric operads, dioperads, coloured operads, properads, PROPs (and probably ½PROPs as well).

Notes on this will be forthcoming, and I'll post here once I get a relevant URL.