## Abstract

In algebra, one studies algebraic operations (compositions) on a given vector space, obeying some compatibility laws. The simplest example is an associative algebra structure on a vector space. The famous Eckmann-Hilton argument shows that, given two compatible unital associative algebra structures on the same vector space, both structures are equal and are commutative. It can be interpreted by saying that the world of vector spaces is very rigid, and no interesting higher structures can be derived from two compatible associative algebra structures. The higher category theory provides a more relaxed environment than the category of vector spaces. Thus, two compatible associative structures in such a relaxed environment produce a homotopy 2-algebra structure on a given space. The generalized Deligne conjecture aims to derive higher structures similar to a homotopy 2-algebra, starting with a (higher n-) monoidal abelian category. For n=1 it gives precisely a homotopy 2-algebra structure, as it was proven in two recent promoter's papers. In historically the first case, considered by P.Deligne, and referred to as the classical Deligne conjecture, one has a homotopy 2-algebra structure on the Hochschild cochain complex of an associative algebra. The main goal of this project is to generalize our previous results for an arbitrary n-monoidal abelian category, for n greater than 1. The case n=2 is of special importance for deformation theory of associative bialgebras.

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