symmetric monoidal (∞,1)-category of spectra
The formality theorem (Kontsevich 97) states that there is an L-∞-algebra quasi-isomorphism from the dg-Lie algebra of polyvector fields (with zero differential and Schouten-Nijenhuis bracket) to the dg-Lie algebra of the shifted Hochschild cochain complex (with Hochschild differential and Gerstenhaber bracket), whose first Taylor coefficient is the HKR quasi-isomorphism.
Tamarkin alternatively proves the formality of the little disks operad (see also Kontsevich 1999) and proves that it implies the Kontsevich formality.
The Kontsevich formality theorem implies that every Poisson manifold has a deformation quantization, unique up to an element in the freely-acting piece of the automorphism infinity-group of the E1-operad. This is the Grothendieck-Teichmüller group (see there for more).
More generally, a result of Kontsevich and Tamarkin (…) says that over a field of characteristic 0 the canonical functor
from E-n algebras to Poisson n-algebras is an equivalence (since $P_n$ is the homology of $E_n$, this says that The $E_n$“ operad is formal over a field of characteristic 0.
But of course the automorphism infinity-group of both $E_n$ and $P_n$ acts on both sides and makes the space of all possible such equivalences a torsor over this group. A $P_n$ algebra may be thought of as encoding a prequantum field theory of higher dimension, of sors, and so formality says that the deformation quantization of factorization algebras always exists and that the choices are being acted on by the corresponding higher analog of the Grothendieck-Teichmüller group.
In agreement with Tsygan’s philosophy of noncommutative differential calculus and its relations to braces algebra, Willwacher extends the Kontsevich formality to a homotopy braces morphism and to a $G_\infty$-morphism in