Operadic twisting as an adjunction
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912759133765632 |
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| author | Laplante-Anfossi, Guillaume Petr, Adrian Shende, Vivek |
| author_facet | Laplante-Anfossi, Guillaume Petr, Adrian Shende, Vivek |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_02229 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Operadic twisting as an adjunction Laplante-Anfossi, Guillaume Petr, Adrian Shende, Vivek Algebraic Topology Category Theory 18M70 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. |
| title | Operadic twisting as an adjunction |
| topic | Algebraic Topology Category Theory 18M70 |
| url | https://arxiv.org/abs/2510.02229 |