Curved Koszul duality theory

TL;DR

This paper extends Koszul duality to operads and properads with curvature, enabling resolutions and dualities for algebraic structures with non-augmented relations, impacting homotopy and cohomology theories.

Contribution

It introduces a curved Koszul duality framework for operads and properads with quadratic, linear, and constant relations, including non-augmented cases.

Findings

Provides a curved Koszul duality theory for operads and properads.

Constructs cofibrant resolutions for properads like Frobenius algebras.

Applies the theory to homotopy and cohomology of unital associative algebras.

Abstract

We extend the bar-cobar adjunction to operads and properads, not necessarily augmented. Due to the default of augmentation, the objects of the dual category are endowed with a curvature. We handle the lack of augmentation by extending the category of coproperads to include objects endowed with a curvature. As usual, the bar-cobar construction gives a (large) cofibrant resolution for any properad, such as the properad encoding unital and counital Frobenius algebras, a notion which appears in 2d-TQFT. We also define a curved Koszul duality theory for operads or properads presented with quadratic, linear and constant relations, which provides the possibility for smaller relations. We apply this new theory to study the homotopy theory and the cohomology theory of unital associative algebras.