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

Author(s)
Ricardo Campos, Julien Ducoulombier, Najib Idrissi.
Publication
In: High. Struct. 5.1, pp. 293–366 (2021), 74 pages.
Online on
Updated on
Accepted on
Full text
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DOI:10.21136/HS.2021.09
https://doi.org/10.21136/HS.2021.09
arXiv:1911.09474
https://arxiv.org/abs/1911.09474
hal-01786218
https://hal.science/hal-01786218
MR4367224
https://mathscinet.ams.org/mathscinet/article?mr=4367224
Zbl 7482172
https://zbMath.org/?q=an:7482172

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.