Operadic Calculus for Higher Colour-Kinematics Duality

The search for algebraic foundations of colour-kinematics duality and the double-copy construction has brought into focus a generalization of Batalin--Vilkovisky algebras, referred to here as coexact BV-algebras and as $\textrm{BV}^\square$-algebras in other sources. While these structures capture the cubic sector, they fail to encode higher-valence phenomena, for which a homotopy-theoretic extension becomes necessary. This work introduces a conceptual notion of homotopy coexact BV-algebra, defined through the homotopical interplay of commutative and BV structures, and provides a concrete model in terms of generators and relations, obtained through an extension the theory of Koszul duality for operads. The resulting framework enables the systematic use of homotopical tools -- $\infty$-morphisms, homotopy transfer, rectification, and deformation theory -- and naturally accommodates the quartic-level structures recently identified in Yang--Mills theory.