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It is known that one can associate a Kontsevich-type formality morphism toevery Drinfeld associator. We show that this morphism may be extended to aKontsevich-Shoikhet formality morphism of cochains and chains, by describingthe action of the Grothendieck-Teichm\"uller group on such objects (up tohomotopy).

The Grothendieck-Teichm\"uller group action on differential forms and formality morphism of chains