Hopf algebras, tetramodules, and n-fold monoidal categories

This paper is an extended version of my talk given in Zürich during the Conference “Quantization and Geometry”, March 2-6, 2009. The main results are the following. 1. We construct a 2-fold monoidal structure [BFSV] on the category Tetra(A) of tetramodules (also known as Hopf bimodules) over an associative bialgebraA. According to an earlier result of R.Taillefer [Tai1,2], Ext q Tetra(A)(A,A) is equal to the GerstenhaberSchack cohomology [GS] of A, which governs the infinitesimal deformations of the bialgebra structure on A. 2. Given an n-fold monoidal category C with a common unit object A and some mild property (*) formulated in the paper, we consider the graded vector space W q = Ext q C(A,A). We prove that W q has a natural (n+ 1)-algebra structure whose product is the Yoneda product. As a conclusion, the Gerstenhaber-Schack cohomology of any Hopf algebra A is a 3-algebra. 3. We find an operad of Z-modules which acts on the Hochschild cohomology of any associative Z-algebra A flat over Z. The k-th component of this operad is the graded space ⊕iπ −i (D 2 k ) of stable homotopy groups of the space D k , the k-th component of the little disks operad. We establish as well an n-monoidal version of this result. 4. We define a contravariant functor from the homotopical category of topological spaces with values in graded vector spaces, depending on an n-fold monoidal category (“Hochschild cohomology depending on topological space”). In particular, such a functor is assigned to any associative algebra, any bialgebra, etc.