Bar-cobar duality for operads in stable homotopy theory

TL;DR

This paper extends bar-cobar duality to operads of spectra in stable homotopy theory, establishing a Quillen equivalence and introducing a new model for cooperads, with implications for derived Koszul duality.

Contribution

It generalizes bar-cobar duality from chain complexes to spectra, providing a new homotopical model for cooperads and a derived Koszul duality framework.

Findings

Established a Quillen equivalence between operads and cooperads of spectra.

Constructed a weak equivalence between W-construction and cobar-bar construction for operads.

Developed a model structure on pre-cooperads that captures cooperad homotopy theory.

Abstract

We extend bar-cobar duality, defined for operads of chain complexes by Getzler and Jones, to operads of spectra in the sense of stable homotopy theory. Our main result is the existence of a Quillen equivalence between the category of reduced operads of spectra (with the projective model structure) and a new model for the homotopy theory of cooperads of spectra. The crucial construction is of a weak equivalence of operads between the Boardman-Vogt W-construction for an operad P, and the cobar-bar construction on P. This weak equivalence generalizes a theorem of Berger and Moerdijk that says the W- and cobar-bar constructions are isomorphic for operads of chain complexes. Our model for the homotopy theory of cooperads is based on `pre-cooperads'. These can be viewed as cooperads in which the structure maps are zigzags of maps of spectra that satisfy coherence conditions. Our model structure on pre-cooperads is such that every object is weakly equivalent to an actual cooperad, and weak equivalences between cooperads are detected in the underlying symmetric sequences. We also interpret our results in terms of a `derived Koszul dual' for operads of spectra, which is analogous to the Ginzburg-Kapranov dg-dual. We show that the `double derived Koszul dual' of an operad P is equivalent to P (under some finiteness hypotheses) and that the derived Koszul construction preserves homotopy colimits, finite homotopy limits and derived mapping spaces for operads.