In our recent paper Shoikhet , a version of the “generalized Deligne conjecture” for abelian [math]-fold monoidal categories is proven. For [math] this result says that, given an abelian monoidal [math]-linear category [math] with unit [math], [math], [math] is the first component of a Leinster 1-monoid in [math] (provided a rather mild condition on the monoidal and the abelian structures in [math], called homotopy compatibility, is fulfilled).
In this article, we introduce a new concept of a graded Leinster monoid. We show that the Leinster monoid in [math], constructed by a monoidal [math]-linear abelian category in Shoikhet , is graded. We construct a functor, assigning an algebra over the chain operad [math], to a graded Leinster 1-monoid in [math], which respects the weak equivalences. Consequently, this article together with Shoikhet  provides a complete proof of the “generalized Deligne conjecture” for 1-monoidal abelian categories, in the form most accessible for applications to deformation theory (such as Tamarkin’s proof of the Kontsevich formality).
Journal Article. 30942 words.
Subjects: Mathematics ; Pure Mathematics
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