A combinatorial approach to Kontsevich's Swiss cheese conjecture

From a coloured operad $\mathcal{P}$ and a $\mathcal{P}$-algebra $A$, we construct a new operad $\mathrm{S}\mathrm{C}(\mathcal{P})$ and a Hochschild object $\mathrm{H}\mathrm{o}\mathrm{c}\mathrm{h}(A)$ together with an $\mathrm{S}\mathrm{C}(\mathcal{P})$-action on the pair $(\mathrm{H}\mathrm{o}\mathrm{c}\mathrm{h}(A),A)$. We prove that if $\mathcal{P}$ is the little $n$-disks operad, then $\mathrm{S}\mathrm{C}(\mathcal{P})$ is equivalent to the Swiss cheese operad ${\mathcal{S}\mathcal{C}}_n$. This gives us a weak version of Kontsevich’s Swiss cheese conjecture (without the universal property). We apply our result to prove the existence of an $E_{n+1}$-action on the Hochschild-Pirashvili cochain of order $n$ of a commutative algebra.