(Higher) categorical methods in deformation theory: the case of associative bialgebras

In some rather simple cases (like associative algebras case) all necessary structures on the deformation complex (the Hochschild cohomological complex) can be constructed ''by hand''. Here we mean the Gerstenhaber Lie bracket and the cup-product. On the other hand, HH(A)=Ext(A,A) in the category of A-bimodules. The crucial remark is: the category of A-bimodules is monoidal, and A is the unit object in it. It is interesting and instructive for further cases to prove that HH(A) is a 2-algebra only from the above property. We discuss this construction, due to S.Schwede. The new results concern to the case of associative bialgebras, where the Gerstenhaber-Schack cohomology of any Hopf algebra is a 3-algebra.
We do not know any "naive" way to prove, constructing some operations on the Gerstenhaber-Schack complex. On the other hand, the direct analogue of the category of bimodules to the Hopf algebra case, the category of tetramodules, is 2-fold monoidal. We use this property in the construction of the 3-algebra on the GS cohomology. As well, we can talk on general theorems concerning n-fold monoidal abelian categories and Ext's there, if the time will allow. The constructions are rather elementary, but the proofs seem unavoidably use one of the enchanced spectra categories (eg, symmetric spectra), as a tool of linearization of categories.