The paper “Boardman–Vogt resolutions and bar/cobar constructions of (co)operadic (co)bimodules”, that Ricardo Campos, Julien Ducoulombier, and I wrote, has been accepted for publication in the journal Higher Structures! This technical part will serve as part of the foundation on which we are going to build in future works about knot-like structures. As a reminder, here is its abstract:
In this paper, we develop the combinatorics of leveled trees in order to construct explicit resolutions of (co)operads and (co)operadic (co)bimodules. We construct explicit cofibrant resolutions of operads and operadic bimodules in spectra analogous to the ordinary Boardman–Vogt resolutions and we express them as a cobar construction of indecomposable elements. Dually, in the context of CDGAs, we perform similar constructions to obtain fibrant resolutions of Hopf cooperads and Hopf cooperadic cobimodules and we express them as a bar construction of primitive elements.
I am of course very grateful to the anonymous referees, who provided very valuable input, and to the editors, for their time spent on our paper.