Boardman-Vogt resolutions and bar/cobar constructions of (co)operadic (co)bimodules

Abstract

We develop the combinatorics of leveled trees in order to construct explicit resolutions of (co)operads and (co)operadic (co)bimodules. We build explicit cofibrant resolutions of operads and operadic bimodules in spectra analogous to the ordinary Boardman--Vogt resolutions and we express them as cobar constructions of indecomposable elements. Dually, in the context of CDGAs, we perform similar constructions, and we obtain fibrant resolutions of Hopf cooperads and Hopf cooperadic cobimodules. We also express them as bar constructions of primitive elements.