Let $M$ be a smooth manifold. Then, by the well known Hochschild-Kostant-Rosenberg theorem, the cochain complex $C^*(C^\infty(M))$ of Hochshild cochains on the algebra $C^\infty(M)$ of smooth functions on $M$ provides an algebraic model (up to homotopy) for the algebra of multivector fields on $M$. In particular, the Gerstenhaber bracket on Hochschild-Kostant-Rosenberg on Hochshild cochains corresponds to the Schouten–Nijenhuis bracket on multivector fields.
If one looks at the garded algebra of multivector fields on $M$ as at the algebra of smooth functions on the shifted cotangent space of $M$, then one can write the statement of the Hochschild-Kostant-Rosenberg theorem in the form
$$ C^\infty(T^*[1]M)\simeq C^*(C^\infty(M)) $$
and the Gerstenhaber bracket on the right hand therefore corresponds to the canonical degree -1 Poisson bracket on the shifted cotangent bundle $T^*[1]M$.
What I'd like to know if there is a similar statement also for arbitrary shifts of the cotangent bundle, i.e., if one can write something like
$$ C^\infty(T^*[k]M)\simeq C_{(k)}^*(C^\infty(M)) $$
where $C_{(k)}^*$ is some version of Hochshild-type cochains (which for $k=1$ would be the usual Hochschild cochains). In particular, for an algebra $A$, the cochain complex $C_{(k)}^*(A)$ should be endowed with a canonical degree $-k$ Lie bracket (at least up to homotopy). I suspect some little $k$-discs operad construction may do the job, but I've not been able to figure out exactly.
