Generalised Joyal disks, ${\Theta }_d$-colored $(d+1)$-operads, and higher Deligne conjecture

Abstract

The classical Deligne conjecture (which by now has found several different proofs) says that the Hochschild cohomological complex of a small dg category over a field $k$ has a structure of a $C(E_2,k)$-algebra. A similar statement for any small dg weak $d$-category is called the generalised Deligne conjecture, it says the cohomological Hochschild complex of a dg weak $d$-category (appropriately defined) admits an action of the operad $C(E_{d+1},k)$. This paper is the first in a series of papers, in which our goal is to construct a $(d+1)$-algebra providing a solution to the generalised Deligne conjecture, on the level of complexes. This paper contains the combinatorial core of this $(d+1)$-algebra structure, for any $d\ge 1$. Namely, we construct a colored $(d+1)$-operad in sets (in the sense of M.Batanin), denoted by ${\mathbf{seq}}_d$. Its category of colors (= the category of unary operations) is the category ${\Theta }_d$ of A.Joyal [J], dual to the category of Joyal $d$-disks [J], [Be2,3]. For $d=1$, our $\Delta$-colored operad ${\mathbf{seq}}_1$ coincides with the operad $\mathbf{seq}$ of D.Tamarkin [T3]. We prove that the construction indeed gives rise to a $(d+1)$-operad, and that this ${\Theta }_d$-colored $(d+1)$-operad ${\mathbf{seq}}_d$ is contractible in the dg and in the topological condensations, for any $d\ge 1$. The contractibility of a $(d+1)$-operad $\mathcal{O}$ is a key property, which, due to the Batanin symmetrisation theorem [Ba1,2], endows any its ($d$-terminal) algebra with an $E_{d+1}$-algebra structure.

v2: essentially elaborated, Conjectures 1 and 2 of v1 are proven as Key-Lemmas 5.10 and 5.12 in v2, the results hold now for any d. 68 pages, 6 figures