Definition of homotopy n-commutative cosimplicial monoids

Develop a general definition of a homotopy n-commutative cosimplicial monoid that captures, for arbitrary n, the homotopical commutativity structure exhibited by cosimplicial monoids such as those arising from higher-dimensional brace operations.

Background

In connecting their construction to prior work on cosimplicial monoids and deformation theory, the authors note that while a notion of ‘2-commutativity’ is present for certain cosimplicial monoids (e.g., in the Davydov–Yetter complex), an appropriate general notion of homotopy n-commutative cosimplicial monoid has not been formulated.

They suggest that higher-dimensional brace operations might provide the homotopies expressing such commutativity, but a general definition is currently lacking.

References

We do not know how do define a homotopy n-commutative cosimplicial monoid in general, but the idea (for n=2 case) was that the “two-dimensional” brace (see the first row in Figure \ref{figureintro} below) provides a homotopy up to which the 2-commutativity holds.

— Generalised Joyal disks and $Θ_d$-colored $(d+1)$-operads  (2510.05813 - Shoikhet, 7 Oct 2025) in Introduction, subsection discussing the link with [BD]