Graded Leinster monoids and generalized Deligne conjecture for 1-monoidal abelian categories

TL;DR

This paper introduces graded Leinster monoids in abelian categories and proves a generalized Deligne conjecture for 1-monoidal abelian categories, linking algebraic structures to chain operads and deformation theory.

Contribution

It defines graded Leinster monoids and demonstrates their role in establishing the generalized Deligne conjecture for 1-monoidal abelian categories.

Findings

Leinster monoids in this setting are graded.

A functor to $C(E_2,k)$-algebras is constructed.

Complete proof of the generalized Deligne conjecture for 1-monoidal categories.

Abstract

In our recent paper [Sh1] a version of the "generalized Deligne conjecture" for abelian $n$-fold monoidal categories is proven. For $n=1$ this result says that, given an abelian monoidal $k$-linear category $\mathscr{A}$ with unit $e$, $k$ a field of characteristic 0, the dg vector space $\mathrm{RHom}_{\mathscr{A}}(e,e)$ is the first component of a Leinster 1-monoid in $\mathscr{A}lg(k)$ (provided a rather mild condition on the monoidal and the abelian structures in $\mathscr{A}$, called homotopy compatibility, is fulfilled). In the present paper, we introduce a new concept of a ${\it graded}$ Leinster monoid. We show that the Leinster monoid in $\mathscr{A}lg(k)$, constructed by a monoidal $k$-linear abelian category in [Sh1], is graded. We construct a functor, assigning an algebra over the chain operad $C(E_2,k)$, to a graded Leinster 1-monoid in $\mathscr{A}lg(k)$, which respects the weak equivalences. Consequently, this paper together with loc.cit. 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).