Journal Article

Graded Leinster Monoids and Generalized Deligne Conjecture for 1-Monoidal Abelian Categories

Boris Shoikhet

in International Mathematics Research Notices

Volume 2018, issue 19, pages 5857-5937
Published in print October 2018 | ISSN: 1073-7928
Published online March 2017 | e-ISSN: 1687-0247 | DOI: https://dx.doi.org/10.1093/imrn/rnx012
Graded Leinster Monoids and Generalized Deligne Conjecture for 1-Monoidal Abelian Categories

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Abstract

In our recent paper Shoikhet [47], 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 [47], 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 [47] 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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