分次Leinster幺半群与1-幺半群阿贝尔范畴的广义Deligne猜想
Graded Leinster monoids and generalized Deligne conjecture for 1-monoidal abelian categories
摘要：
在我们最近的文章[Sh 1]中，我们证明了交换$n$ -fold monoidal范畴的“广义Deligne猜想”的一个版本.对于$n=1$，这个结果说，给定一个交换幺半群$k$ -线性范畴$\mathscr{A}$，单位为$e$，$k$是一个特征为0的域，dg向量空间$\mathrm{RHom}_{\mathscr{A}}(e,e)$是$\mathscr{A}lg(k)$中Leinster 1-幺半群的第一个分支（假设$\mathscr{A}$中幺半群和交换结构满足一个相当温和的条件，称为同伦相容）。本文引入了${\it graded}$ Leinster幺半群的新概念。证明了由[Sh 1]中的monoidal $k$ -线性交换范畴构造的$\mathscr{A}lg(k)$中的Leinster幺半群是分次的.我们构造了一个函子，将链操作数$C(E_2,k)$上的一个代数赋给$\mathscr{A}lg(k)$中的分次Leinster 1-幺半群，它尊重弱等价.因此，本文与loc.cit.为1-monoidal阿贝尔范畴的“广义Deligne猜想”提供了一个完整的证明，其形式最容易应用于变形理论（如Tamarkin对Kontsevich形式的证明）。
英文摘要：
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).
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