A Koszul duality for props

Koszul duality is a fundamental correspondence between algebras for an operad $\mathcal{O}$ and coalgebras for its dual cooperad $B\mathcal{O}$, built from $\mathcal{O}$ using the bar construction. Francis-Gaitsgory proposed a conjecture about the general behavior of this duality. The main result of this paper, roughly speaking, is that Koszul duality provides an equivalence between the subcategories of nilcomplete algebras and conilcomplete coalgebras and that these are the ...

We survey the topology which led to the original bar and cobar constructions, for both associative algebras and coalgebras and for Lie algebras and commutative coalgebras. These constructions are often viewed as part of the larger theory of Koszul duality of operads, so this survey is meant to offer an historical perspective on the most prominent cases of that theory. We also explain recent work which shows that Hopf/linking invariants for homotopy are at the heart of the dua...

We develop a self-dual, bivariant extension of the concept of an operadic category, its associated operads and their algebras. Our new theory covers, besides all classical subjects, also generalized traces and bivariant versions of Kapranov's charades. It is, moreover, combinatorially rich and aesthetically pleasing.

This is a survey on recent progress in algebraic deformation theory and the application of algebraic operads to its study. We review the classical homotopical tools in the theory of algebraic operads, namely Koszul duality. We give concrete examples of applications of such tools to various flavours of problems related to deformations of algebraic structures. We also study formal moduli problems and related notions from the operadic point of view.

A theorem of Pridham and Lurie provides an equivalence between formal moduli problems and Lie algebras in characteristic zero. We prove a generalization of this correspondence, relating formal moduli problems parametrized by algebras over a Koszul operad to algebras over its Koszul dual operad. In particular, when the Lie algebra associated to a deformation problem is induced from a pre-Lie structure it corresponds to a permutative formal moduli problem. As another example we...

We discuss the consequences of the Poincar\'e duality, versus AS- Gorenstein property, for Koszul algebras (homogeneous and non homogeneous). For homogeneous Koszul algebras, the Poincar\'e duality property implies the existence of twisted potentials which characterize the corresponding algebras while in the case of quadratic linear Koszul algebras, the Poincar\'e duality is needed to get a good generalization of universal enveloping algebras of Lie algebras. In the latter ca...

We describe the Koszul dual of two quadratic operads on planar forests introduced to study the infinitesimal Hopf algebra of planar rooted trees and prove that these operads are Koszul.

This is the central article of a series of three papers on cross product bialgebras. We present a universal theory of bialgebra factorizations (or cross product bialgebras) with cocycles and dual cocycles. We also provide an equivalent (co-)modular co-cyclic formulation. All known examples as for instance bi- or smash, doublecross and bicross product bialgebras as well as double biproduct bialgebras and bicrossed or cocycle bicross product bialgebras are now united within a s...

The purpose of this foundational paper is to introduce various notions and constructions in order to develop the homotopy theory for differential graded operads over any ring. The main new idea is to consider the action of the symmetric groups as part of the defining structure of an operad and not as the underlying category. We introduce a new dual category of higher cooperads, a new higher bar-cobar adjunction with the category of operads, and a new higher notion of homotopy...

This is a cornucopia of types of algebras with some of their properties from the operadic point of view.