Theta_d-Colored (d+1)-Operad seq_d

• Colored (d+1)-operad seq_d is a higher categorical structure that replaces a simple color set with the category Theta_d, enabling refined operadic compositions.
• It employs the lattice path operad L^d and Berger’s posets to organize pruned (d+1)-trees and manage blockwise complexity in compositions.
• After condensation and symmetrization, seq_d provides an algebraic model for E_{d+1}-operads, supporting higher Hochschild-type actions fundamental to the higher Deligne conjecture.

A colored $(d+1)$‑operad $\mathbf{seq}_d$ is a higher categorical operadic structure where the set of “colors” is not simply a set but the objects of a category—the Joyal category $\Theta_d$, which is dual to the category of $d$-disks. This framework generalizes the Tamarkin $\Delta$-colored 2-operad $\mathbf{seq}$ ($d = 1$) and provides a concrete model for $(d+1)$‑fold algebraic structures, crucial for addressing conjectures in higher category theory such as the higher Deligne conjecture. The construction of $\mathbf{seq}_d$ centers on combinatorial and categorical data: the use of pruned $(d+1)$-level trees, the lattice path operad $\mathcal{L}^d$, and the handling of colorings and blockwise complexity via Berger’s posets $\mathcal{K}(2)^{\times d}$. Its contractibility (under appropriate combinatorial conditions) implies that, after condensation, $\mathbf{seq}_d$ serves as a topological or dg model for an $E_{d+1}$‑operad, enabling actions on higher Hochschild-type complexes and thereby interpreting core structures in weak $d$-categories (Shoikhet, 7 Oct 2025).

1. The Color Category $\Theta_d$ and Joyal $d$-Disks

The central innovation of $\mathbf{seq}_d$ is replacing the set of colors with the category $\Theta_d$, which is dual to the category of generalized Joyal $d$-disks. Objects in $\Theta_d$ are pruned $d$-level trees, equivalent to $d$-disks in topology via Joyal’s duality. This categorification of colors enables gluings and operadic compositions that respect intricate symmetry and duality present in higher categorical and geometric contexts. In the classical ($d=1$) case, the colors reduce to the simplex category $\Delta$.

$\Theta_d$-colored operads naturally support actions by operations whose compatibility is dictated by categorical morphisms and dualities. This feature allows direct modeling of field-theoretic and topological objects with boundaries and marked points carrying monodromy or symmetry data, as exemplified in operads for moduli spaces of $G$-covers (Petersen, 2012).

2. The Lattice Path Operad $\mathcal{L}^d$ and Block Structure

The lattice path operad $\mathcal{L}^d$ is defined as a symmetric operad in sets (or spaces/chain complexes), with $\mathcal{L}^d(T_1, \dots, T_k; T)$ encoding generalized lattice paths between pruned trees $T_1, \dots, T_k$ and $T$. Each $\omega \in \mathcal{L}^d$ is further organized by blockwise complexity, with blocks indexed by tuples in $\mathcal{K}(2)^{\times d}$—these are the Berger complete graph posets. For each pair of leaves, the block parameter constrains the direction-change complexity per dimension.

The blocks form subfunctors of $\mathcal{L}^d$ corresponding to operations with bounded complexity, and the union over suitable blocks gives rise to the colored operad $\mathbf{seq}_d$. The block structure ensures compatibility with the categorical input and output organization coming from $\Theta_d$.

3. Operadic Composition and Combinatorial Consistency

Composition in $\mathbf{seq}_d$ is inherited from $\mathcal{L}^d$, with the domain and codomain determined by the categorical structure of $\Theta_d$. The arity component for a pruned $(d+1)$-tree $T$ is defined as:

$$\mathbf{seq}_d(T) = \left\{ \omega \in \mathcal{L}^d(|T|) \;\middle|\; \forall i <_c j,\; (\cdot,\cdot)(\omega_{ij}) \le (q_c, q_c) \text{ for some } q \right\}$$

where $<_c$ is the induced categorical ordering on leaves, and $(q_c, q_c)$ are prescribed block parameters per categorical position.

Two critical conjectures—proven for $d=2,3$—ensure that:

1. Prescribed block labels are compatible under quasi-bijections (operadic substitutions).
2. The poset of allowed blocks $P_{\mathbf{seq}_d}(T)$ is contractible, implying that after taking the homotopy colimit over all blocks, each arity component $\mathbf{seq}_d(T)$ is contractible.

4. Contractibility, Symmetrization, and Higher Deligne Conjecture

The essential contractibility of each component of $\mathbf{seq}_d$ (in topological and dg condensations) ensures, via Batanin’s symmetrization theorem, that the induced symmetric operad after color condensation is weakly equivalent to the $E_{d+1}$ little disks operad. Thus, $\mathbf{seq}_d$ provides a direct algebraic model for $E_{d+1}$ actions.

This property is pivotal for the higher Deligne conjecture, which posits $E_{d+1}$-actions on Hochschild-type complexes associated to (weak) $d$-categories. For $d=1$, this recovers Tamarkin’s solution using $\mathbf{seq}$; for higher $d$, the colored operad $\mathbf{seq}_d$ allows canonical higher operations, supporting algebraic structures on deformation complexes and factorization homology (Shoikhet, 7 Oct 2025).

5. Comparison with Other Colored Operads and Applications

The $\mathbf{seq}_d$ construction generalizes previous colored operadic frameworks by allowing colors to denote categorical data with nontrivial automorphism or duality. Related works (e.g., operads colored by groupoids (Petersen, 2012), category-colored Markl operads (Trnka, 2023)) demonstrate the utility of categorical coloring for encoding field theories with symmetry, moduli of covers, or algebraic QFTs (Benini et al., 2017). The block structure and Berger poset indexing in $\mathbf{seq}_d$ furnish fine control over allowable compositions, analogous to bud generating systems for combinatorial grammars (Giraudo, 2016), and enable detailed enumeration of operadic components.

In topology and configuration space theory, similar colored operads underlie models for iterated loop spaces and configuration spaces, with the contractibility of indexing posets generating Koszul and self-dual properties in polytopal operads (Arkhipov et al., 2021). The framework adapts naturally to rational homotopy theory of operadic modules via the colored approach (Willwacher, 15 Dec 2024).

6. Mathematical Formulas and Summary

Key constructions are centered on:

• Assignment of colors via categorical objects: $\mathrm{colors} = \mathrm{Obj}(\Theta_d)$
• Arity components as colimits over block posets:

$$\mathbf{seq}_d(T)[D] = \operatorname{colim}_{(\cdot,\cdot) \in P_{\mathbf{seq}_d}(T)} \mathcal{L}^d_{(\cdot,\cdot)}[D]$$

• Compositional constraints:

$$\forall\, i <_c j,\quad (\cdot,\cdot)(\omega_{ij}) \le (q_c, q_c) \text{ for some } q$$

Contractibility results undergird the deduction that $\mathbf{seq}_d$ (after condensation and symmetrization) yields $E_{d+1}$-type operads, which provides topological actions on higher Hochschild complexes:

Construction	Colors	Contractibility Result
$\mathbf{seq}_d$	Objects of $\Theta_d$	Contractible after colimit
$\mathcal{L}^d$	Pruned $d$-trees	Block contractibility
Berger posets $\mathcal{K}(2)^{\times d}$	Tuples of block indices	Poset contractibility

7. Impact and Open Directions

The $\Theta_d$-colored $(d+1)$‑operad $\mathbf{seq}_d$ provides a categorical and combinatorial foundation for higher operadic structures in weak $d$-categories, field theory, moduli problems, and algebraic topology. The contractibility and symmetrization yields powerful tools for proving existence of $E_{d+1}$ structures, supporting the higher Deligne conjecture and actions on deformation complexes associated to higher categorical objects. Open directions include extending the proofs of the key combinatorial conjectures for all $d$, exploring full implementations in model and derived categorical contexts, and further formalizing connections to factorization homology and quantum field theory settings (Shoikhet, 7 Oct 2025).