Generalised Joyal disks and $\Theta_d$-colored $(d+1)$-operads

Abstract

In this paper, we propose a method for constructing a colored $(d+1)$-operad $\mathbf{seq}_d$ in $\mathrm{Sets}$, in the sense of Batanin [Ba1,2], whose category of colors (=the category of unary operations) is the category $\Theta_d$, dual to the Joyal category of $d$-disks [J], [Be2,3]. For $d=1$ it is the Tamarkin $\Delta$-colored 2-operad $\mathbf{seq}$, playing an important role in his paper [T3] and in the solution loc.cit. to the Deligne conjecture for Hochschild cochains. We expect that for higher $d$ these operads provide a key to solution to the the higher Deligne conjecture, in the (weak) $d$-categorical context. For general $d$ the construction is based on two combinatorial conjectures, which we prove to be true for $d=2,3$. We introduce a concept of a generalised Joyal disk, so that the category of generalised Joyal $d$-disks admits an analogue of the funny product of ordinary categories. (For $d=1$, a generalised Joyal disk is a category with a ``minimal'' and a ``maximal'' object). It makes us possible to define a higher analog $\mathcal{L}^d$ of the lattice path operad [BB] with $\Theta_d$ as the category of unary operations. The $\Theta_d$-colored $(d+1)$-operad $\mathbf{seq}_d$ is found ``inside'' the desymmetrisation of the symmetric operad $\mathcal{L}^d$. We construct ``blocks'' (subfunctors of $\mathcal{L}^d$) labelled by objects of the cartesian $d$-power of the Berger complete graph operad [Be1], and prove the contractibility of a single block in the topological and the dg condensations. In this way, we essentially upgrade the known proof given by McClure-Smith [MS3] for the case $d=1$, so that the refined argument is generalised to the case of $\Theta_d$. Then we prove that $\mathbf{seq}_d$ is contractible in topological and dg condensations (for $d=2,3$, and for general $d$ modulo the two combinatorial conjectures).