Differential graded categories and Deligne conjecture

We prove a version of the Deligne conjecture for n-fold monoidal abelian categories A   over a field kk of characteristic 0, assuming some compatibility and non-degeneracy conditions for A  . The output of our construction is a weak Leinster (n,1)(n,1)-algebra over kk, a relaxed version of the concept of Leinster n  -algebra in Alg(k)Alg(k). The difference between the Leinster original definition and our relaxed one is apparent when n>1n>1, for n=1n=1 both concepts coincide.We believe that there exists a functor from weak Leinster (n,1)(n,1)-algebras over kk to C•(En+1,k)C•(En+1,k)-algebras, well-defined when k=Qk=Q, and preserving weak equivalences. For the case n=1n=1 such a functor is constructed in [31] by elementary simplicial methods, providing (together with this paper) a complete solution for 1-monoidal abelian categories.Our approach to Deligne conjecture is divided into two parts. The first part, completed in the present paper, provides a construction of a weak Leinster (n,1)(n,1)-algebra over kk, out of an n  -fold monoidal kk-linear abelian category (provided the compatibility and non-degeneracy condition are fulfilled). The second part (still open for n>1n>1) is a passage from weak Leinster (n,1)(n,1)-algebras to C•(En+1,k)C•(En+1,k)-algebras.As an application, we prove in Theorem 8.1 that the Gerstenhaber–Schack complex of a Hopf algebra over a field kk of characteristic 0 admits a structure of a weak Leinster (2,1)(2,1)-algebra over kk extending the Yoneda structure. It relies on our earlier construction [30] of a 2-fold monoidal structure on the abelian category of tetramodules over a bialgebra.