Protoperads II: Koszul duality

Abstract

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Protoperads i: Combinatorics and definitions

“…We have an analogous statement for double Poisson structures: as the properad As, which encodes associative algebras, is Koszul (see [LV12]), the properad DPois ∼ = As⊠ Val c DLie is Koszul if, and only if, the properad DLie is Koszul (see [Ler18,Sect. 4]).…”

“…Using the framework of protoperads is successful: we prove in [Ler18] that there is a Koszul duality theory for protoperads which is compatible with that for properads via Ind, and we use it to prove that the properads DLie and hence DPois are Koszul. Section 1 -Bricks and walls.…”

“…This fundamental observation allows us to reduce the problem to one in the category of S-modules, and to define the minimal framework, called protoperad, which encode double Lie algebras. We will see (in [Ler18]) that the homotopy theory of these new objects is more simpler than the homotopy of properads. We resume the most important results of this article in the following theorem.…”

“…This is the first of two papers in which the author develops the notion of protoperad, which is a kind of properad (see [Val03,Val07]), and the homotopy theory of these new objects (see [Ler18]). Properad is an algebraic notion which encodes types of bialgebras, i.e.…”

“…Double Poisson structure is properadic in nature. It is encoded by the properad DPois (see [Ler18]); it gives us a good framework to study what is its version up to homotopy (cf. [Val03, MV09a,MV09b]).…”