Operadic twisting as an adjunction
Abstract
For operads with a map from the curved homotopy Lie operad, we introduce a corresponding curved variant `cTw' of Willwacher's operadic twisting comonad `Tw'. We show that cTw-coalgebra structures on such an operad are in bijection with certain splittings (not respecting the differential) of the projection to its quotient by the curvature operation. We derive a similar classification of Tw-coalgebras. For the class of operads whose Koszul dual admits a unital extension, we give explicit formulas for the cTw-coalgebra structures on their curved homotopy resolutions, recovering the convolution Lie algebra's ``gauge group action'' of Dotsenko, Shadrin, and Vallette.
Cite
@article{arxiv.2510.02229,
title = {Operadic twisting as an adjunction},
author = {Guillaume Laplante-Anfossi and Adrian Petr and Vivek Shende},
journal= {arXiv preprint arXiv:2510.02229},
year = {2025}
}
Comments
48 pages, new section 5 on monadicity and uncurving, and new appendix D on gauge group action via Yoneda