Bar-cobar duality for operads in stable homotopy theory

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.

💡 Research Summary

The paper lifts the classical bar‑cobar duality, originally formulated by Getzler and Jones for operads in the category of chain complexes, to the realm of operads in spectra, i.e. within stable homotopy theory. The authors work with reduced operads of spectra equipped with the projective model structure, where weak equivalences are the underlying stable equivalences of symmetric sequences. Their main achievement is the construction of a Quillen equivalence between this operadic model category and a newly introduced model for cooperads of spectra, called “pre‑cooperads”.

A pre‑cooperad is a relaxed version of a cooperad: instead of strict structure maps, one allows zig‑zags of maps of spectra that satisfy a coherent system of homotopies. The authors endow the category of pre‑cooperads with a model structure in which every object is weakly equivalent to an honest cooperad, and a map between genuine cooperads is a weak equivalence precisely when it is a weak equivalence on the underlying symmetric sequences. This model category is designed so that the homotopy theory of cooperads can be accessed without having to resolve strict composition maps.

The technical heart of the work is the comparison between two constructions associated to a given operad (P). The Boardman‑Vogt (W)-construction (W(P)) provides a cofibrant resolution of (P) by freely inserting higher‑dimensional “trees”. On the other hand, the cobar‑bar construction (B,C(P)) first takes the bar construction (a cooperad) and then applies the cobar construction (returning an operad). In the chain‑complex setting Berger and Moerdijk proved that these two constructions are isomorphic. The authors prove a spectral analogue: there is a natural weak equivalence of operads \