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

Ricardo Campos, Julien Ducoulombier, Najib Idrissi
In: High. Struct. 5.1, pp. 293–366 (2021), 74 pages.
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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.