Application of (higher) operads in deformation theory

As well as to an associative algebra are assigned the modules over it, to an operad are assigned its algebras. The algebras of the operad Assoc are the same as associative algebras, the algebras over the operad Lie are Lie algebras, and so on. In other words, operads encode types of algebraic structures.
Among the operads there is a very important family $E_n$, $n\geqslant1$, the dimension $n$ little discs operads. The deformation theory of associative algebras is connected with the operad $E_2$, and this idea gave rise (in a work of D. Tamarkin) to a proof of famous Kontsevich formality theorem. In my talk, I'll start with these definitions and constructions. An interesting and partially open problem is deformation theory of associative bialgebras, and of the corresponding formality therein. This problem is 3-dimensional, that is, is connected with the operad $E_3$. Classically, deformations of associative bialgebras are controlled by the Gerstenhaber-Schack complex. However, any higher structure on it hasn't been known yet. Recently I've came to an understanding that a conceptually better one is the deformation theory of (dg) monoidal categories, and the corresponding complex. I'll define this complex and tell on an approach to constructing of higher structures over it. It is done through a concept of an $n$-operad, a deep generalisation of a conventional operad due to M. Batanin.