Higher Structures, Vol. 5, No. 1, pp. 310-383, 2021

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

Julien Ducoulombier, Najib Idrissi, Ricardo Campos

Received January 14th 2020. Published online December 16th 2021.

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.
Keywords:  operads; bimodules over operads; bar construction; model category; homotopy theory; cofibrant and fibrant resolutions
Classification MSC:  18M70, 18M75, 18N40

PDF available at:  Institute of Mathematics CAS

Affiliations:   Julien Ducoulombier, Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany, e-mail: julien@mpim-bonn.de; Najib Idrissi, Université de Paris and Sorbonne Université, IMJ-PRG, CNRS, F-75006 Paris, France, e-mail: najib.idrissi-kaitouni@imj-prg.fr; Ricardo Campos, IMAG, Université de Montpellier, CNRS, Montpellier, France, e-mail: ricardo.campos@umontpellier.fr

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