D-structures and derived Koszul duality for unital operad algebras

TL;DR

This paper introduces D-structures for unital operad algebras, establishing a derived category equivalence that generalizes Koszul duality to unital settings and unifies various algebraic frameworks.

Contribution

It defines D-structures on unital operad algebras, extending Koszul duality to a unital context and connecting different algebraic theories.

Findings

Established an equivalence of derived categories for unital operad algebras.

Generalized Koszul duality to include unital operads.

Unified multiple algebraic contexts of Koszul duality.

Abstract

Generalizing a concept of Lipshitz, Ozsv\'ath and Thurs-ton from Bordered Floer homology, we define $D$-structures on algebras of unital operads, which can also be interpreted as a generalization of a seemingly unrelated concept of Getzler and Jones. This construction gives rise to an equivalence of derived categories, which can be thought of as a unital version of Koszul duality using non-unital Quillen homology. We also discuss a multi-sorted version of the construction, which provides a framework for unifying the known algebraic contexts of Koszul duality.