Little Three-Disks Actions on Operadic Homotopy Centers

Definitions and mathematical statement

Let $\mathcal E_3$ denote the little three-disks operad. For $n\geq 1$, its space $\mathcal E_3(n)$ consists of ordered configurations of $n$ embeddings $x\mapsto r_i x+a_i$, with $r_i>0$, of the open unit ball in ${\mathbb{R}}^3$ into itself, whose images are pairwise disjoint. Operadic composition inserts configurations into disks. The nullary convention must be specified when interpreting the question.

A nonsymmetric operad $\mathcal O$ consists of objects $\mathcal O(n)$ in an ambient monoidal category, with composition maps \[ \mathcal O(k)\otimes\mathcal O(n_1)\otimes\cdots\otimes\mathcal O(n_k) \longrightarrow \mathcal O(n_1+\cdots+n_k) \] for the permitted arities, satisfying associativity and unit axioms, without requiring symmetric-group actions. The supplied record does not specify the ambient category, permitted arities, or precise unit conventions.

Write $Z^h(\mathcal O)$ for the intended homotopy center of $\mathcal O$. This is a placeholder for the source's construction: the supplied record does not define that construction or its homotopical framework.

Source question. Does every nonsymmetric operad $\mathcal O$ have a homotopy center $Z^h(\mathcal O)$ admitting an action of the little three-disks operad?

If the intended homotopy center is a space, the proposed structure is an operad morphism \[ \mathcal E_3\longrightarrow \operatorname{End}_{Z^h(\mathcal O)}, \qquad \operatorname{End}_{X}(n)=\operatorname{Map}(X^n,X), \] with the chosen arity and unit conventions. In another enrichment, the acting operad and endomorphism operad require corresponding interpretation. The record does not determine whether the action is strict, exists on a suitable replacement, or is homotopy coherent, nor whether it must be natural or compatible with pre-existing structure. Consequently, this expresses the supplied conjectural target but does not determine a fully specified mathematical assertion.

Short English statement

The question asks whether the homotopy center of every nonsymmetric operad carries a little three-disks action. The supplied context presents this as an operadic analogue of a little two-disks action on the homotopy center of a monoid. A precise formulation requires the intended definition of homotopy center, ambient category, and action conventions.

Sources

• Local UnsolvedMath registry: supplied record associated with data/unsolvedmath/problems.json, permanent ID $30006031$, exact source problem code catalogue entry, title "Little Three-Disks Actions on Operadic Homotopy Centers." No proposer is named; the supplied proposed year is 2024.
• Supplied report citation: Homotopical Algebra and Higher Structures (2025), Oberwolfach Reports, pp. 2251--2308, DOI https://doi.org/10.4171/owr/2024/39. The record describes these reports as documenting mathematical workshops and their open-problem discussions. No page pinpoint for this question is supplied.
• Supplied literature reference labelled "Batanin--Markl": https://doi.org/10.1016/j.aim.2012.04.011. The original-statement field instead attributes the analogous monoid result to "Batanin and Berger" and refers to "[2]"; the supplied record does not resolve this attribution or identify that numbered reference separately.
• Literature assessment dated exactly 2026-08-22: this is supplied dated curation, not independently verified. It labels the problem open, describes the proposed $E_3$-action as a conjecture with a proof plan, and says the identified literature proves the analogous monoid case rather than this operadic claim. The supplied checked-source links are the report and Batanin--Markl DOI links above. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry supplies the status labels open and corrected_verified. Its verification note reports that the official report was checked, the question isolated from surrounding exposition, and notation repaired. These are source assertions; no independent source consultation or mathematical verification is claimed here.