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Here is a sketch of topological description of a Tamarkin-Tsygan precalculus. Consider the compactified configuration spaces $C_n$ and $D_{1,n}$ of $n$ points on $\mathbb{R}^2$ and $\mathbb{R}^2-\{( …

Your question might be Why are infinitesimal deformations typically considered as structures over the ring of dual numbers? A first order (or infinitesimal) deformation of an algebraic stru …

The only way I know to construct a formality quasi-morphism for poy-vector fields out of an homotopy is via Tamarkins approach (i.e. $G_\infty$-formality). What Tamrakin does is prove that there is …

Welcome to mathoverflow! There is actually a whole paper (in French) about choices of signs for Kontsevich formality: https://arxiv.org/pdf/math/0003003.pdf For instance, they define the Hochschild …

In addition to the references pointed out by Stefan, I would like to add Déformation, quantification, et théorie de Lie, by Catteno, Keller, and Torossian (Part I and Part III are actually in Engl …

If, by deformation, you mean formal one-parameter deformation (like, say, in deformation quantization), then it is already known that the Koszul duality between the symmetric and the exterior algebra …

You might want to have a look at §2.2 of An $L_\infty$ algebra structure on polyvector fields by Boris Shoikhet, where Boris constructs an exotic $L_\infty$-structure on poly-vector fields on a (possi …

I think you should have a look at the various papers of Louis Boutet de Monvel. But there is actually a construction of star-products on a symplectic manifold which makes use of the index theorem, due …

The action of GT on deformation quantization has been developed in http://arxiv.org/abs/1009.1654 (Willwacher) and before in http://arxiv.org/abs/math/0202039 (Tamarkin). The fact that GT is Aut(Cha …

Let me try an answer. It seems to me that the appropriate language to use is the one of ringed spaces. For a given ring space $(X,\mathcal{O}_X)$ one can consider the category of sheaves of right $ …

I think this does only work for so-called homological vector fields, i.e. vector fields of degree 1 which self-commute. Then you have $[v,v]=0$, which is the Maurer-Cartan equation in $T_{poly}$, and …

The answer to your second question is "no", I think. Let's assume that sufficiently nice means that it is a smooth algebraic variety over a field of characteristic $0$. Then, as written in Severin Bar …

Fedosov's work seems to be also available with details in a book Deformation quantization and index theory. Are the two references overlapping ? Yes, indeed. The book contains strictly more than …

One possible answer is in Toën-Vezzozi paper From HAG to DAG, who were themselves inspired by Ciocan-Fontanine and Kapranov (Derived Quot schemes and Derived Hilbert schemes). This approach works well …

Deformation of relations Answer to question 2 is the following: a deformation of an algebra $A_0$ parametrized by a pointed affine scheme $*\to X=Spec(B\to k)$ is the data of a $B$-algebra $A$ such t …