1. Graded Leinster monoids and generalized Deligne conjecture for 1-monoidal abelian categories
- Author
-
Boris Shoikhet
- Subjects
Monoid ,Pure mathematics ,Functor ,Conjecture ,General Mathematics ,Homotopy ,010102 general mathematics ,Field (mathematics) ,Mathematics - Category Theory ,01 natural sciences ,Mathematics::K-Theory and Homology ,Mathematics::Quantum Algebra ,Mathematics::Category Theory ,0103 physical sciences ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Category Theory (math.CT) ,010307 mathematical physics ,Abelian category ,0101 mathematics ,Abelian group ,Unit (ring theory) ,Mathematics - 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)., v5 (68 pages) the final version (minor corrections and improvements to v4 are made)
- Published
- 2018