Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

n-commutative cosimplicial monoids

(joint work with Alexey Davydov)
The goal of my talk is to introduce a new interesting algebraic concept of n-commutative cosimplicial monoids. For n=0 these are just cosimplicial objects in the category of monoids in a monoidal category. For n= \infty these are cosimplicial objects in the the category of commutative monoids in a symmetric monoidal category. The most famous example of such an object is cosimplicial cochain complex of a space with coefficient in a commutative ring. It was proved by several people that the total complex of cosimplicial cochain complex is an E_{\infty}-algebra. Michael Mandell proved that the homotopy type of this E_{\infty}-algebras over integers is a complete homtopy invariant of nilpotent spaces of finite type.
In my talk I will introduce an n-commutativity condition for cosimplicial monoids for all positive n and I will provide two examples of such objects. Namely, the Davydov-Yetter deformation complex of a tenzor functor is a 1-commutative cosimplicial monoid and the Davydov-Yetter deformation complex of a tenzor category is a 2-commutative cosimplicial monoid. The interest of study of n-commutative cosimplicial monoids comes from the following result of mine and Alexei Davydov: the total complex of an n-commutative monoid in chain complexes is naturally a E_{n+1}-algebra.
In the second part of my talk I will sketch a proof of this theorem and explain how to get very explicit Steenrod cup-product operations and Poisson bracket on total complex of an n-commutative monoid in terms of certain sums over paths on the p x q rectangular lattice. To the best of my knowledge these formulas are new even for classical cochain complex of a space.