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?



What is the definition of "the $L_\infty$ part of a $G_\infty$ morphism"?
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.


