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

TL;DR

This work constructs explicit $Θ_d$-colored $(d+1)$-operads $\mathbf{seq}_d$ via generalized Joyal $d$-disks and a higher lattice path framework, aiming to address the higher Deligne conjecture in weak $d$-categorical contexts. Central to the approach is the higher lattice path operad $\mathcal{L}^d$ and its block-decomposition controlled by Berger posets; the authors prove contractibility of single blocks in both topological and dg condensations and show how to assemble these blocks into the contractible operad $\mathbf{seq}_d$ under two combinatorial conjectures (proved for $d=2,3$). The paper then establishes a refined generalization of Tamarkin’s 2-operad to the $d$-dimensional setting: the existence of a contractible $Θ_d$-colored $(d+1)$-operad that acts, after Batanin symmetrisation, as an $E_{d+1}$-algebra on derived modifications in $d$-categories. The results provide explicit higher braces and universal structures for deformation problems in weak $d$-categories, with potential to unify and extend Deligne-type phenomena beyond the classical $d=1$ case. While the full general $d$-case depends on combinatorial conjectures, complete verification for $d=2,3$ already yields concrete higher-operadic tools and a clear path to higher Deligne-type formality results.

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 $Θ_d$, dual to the Joyal category of $d$-disks [J], [Be2,3]. For $d=1$ it is the Tamarkin $Δ$-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 $Θ_d$ as the category of unary operations. The $Θ_d$-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 $Θ_d$. 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).

Figures (5)

• Figure 1: The (skew-)symmetrization of the weighted sum $L_1\pm L_2$ gives a closed element $D_1\{\{D_2\}\}$, where $L_i$ is the element corresponded to the $i$-th line of the l.h.s.
• Figure 2: A 2-level graph $T$ (left) and the disk $\overline{T}$ (right)
• Figure 3: A 2-level tree $T$ (left) and the corresponding 2-globular diagram $T^*$ (right)
• Figure 4: A lattice path $\omega$ (left) and the lattice path $c(\omega)$ (right)
• Figure 6: A quasi-bijection $(1,\operatorname{\rm id})\to (2,(12))$ in $\mathcal{M}^4_2$

Theorems & Definitions (29)

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