Operadic twisting as an adjunction

Guillaume Laplante-Anfossi; Adrian Petr; Vivek Shende

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arXiv · 2510.02229

Operadic twisting as an adjunction

Guillaume Laplante-Anfossi·Adrian Petr·Vivek Shende

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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.

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Guillaume Laplante-Anfossi, Adrian Petr, Vivek Shende. 2025-10-02. Operadic twisting as an adjunction. https://arxiv.org/abs/2510.02229

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