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

TL;DR

This paper constructs explicit resolutions of operads and bimodules using combinatorics of leveled trees, extending Boardman--Vogt and bar/cobar constructions to (co)operadic contexts in spectra and CDGAs.

Contribution

It introduces explicit cofibrant and fibrant resolutions of (co)operads and (co)operadic bimodules via leveled trees, generalizing classical constructions.

Findings

Explicit cofibrant resolutions of operads in spectra.

Fibrant resolutions of Hopf cooperads in CDGAs.

Resolutions expressed as cobar and bar constructions of primitive elements.

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.