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We know that in Tamarkin's proof of Kontsevich's formality theorem, he defined the $G_\infty$ structure on the Hochschild cochain complex $C^\cdot(A,A)$ and constructed a $G_\infty$ morphism from $HH^\cdot(A,A)$ to $C^\cdot(A,A)$, which makes Kontsevich's original $L_\infty$ morphism as the $L_\infty$ part of it. So what is the precise definition of "the $L_\infty$ part of a $G_\infty$ morphism"? I know it looks like we just forget the homotopies for the cup product but how to make it precise? |
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A $G_\infty$-morphism $\phi$ is determined by structure maps $\phi^{k_1,\dots,k_n}$, $n\geq1$, $k_1,\dots,k_n\geq1$. The $L_\infty$-part of $\phi$ is the $L_\infty$-morphism $\ell$ with structure maps $\ell^k=\phi^{\overbrace{1,\dots,1}^{k~times}}$. In order to make this precise you might have to put (de)suspensions at the appropriate places. |
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