arXiv subjects

Adrian Petr

Publications and source records attributed to Adrian Petr.

4 recordsLinked to original sources

arXiv2025

Operadic twisting as an adjunction

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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arXiv2024

A universal characterization of the curved homotopy Lie and associative operads

We study the category of nonsymmetric dg operads valued in strict graded-mixed complexes, equipped with a distinguished arity zero weight one element which generates the weight grading, and whose differential has weight one. We show that the initial object is the curved A-infinity operad, that the forgetful functor to the category of operads under it admits a right adjoint, and that the unit of the adjunction encodes the operation of twisting a curved A-infinity algebra by a Maurer-Cartan element. The corresponding notions for symmetric operads characterize the curved L-infinity operad and the corresponding twisting procedure.

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arXiv2024

On Fukaya categories and prequantization bundles

We show: the Floer homology over the Novikov ring of (nonexact!) rational Lagrangians in an (nonexact!) integral symplectic manifold can be computed in terms of exact Lagrangians in an exact filling of the prequantization bundle. As a consequence, we give a Fukaya-sheaf correspondence for rational (nonexact!) Lagrangians in Weinstein manifolds, as conjectured by Ike and the first-named author. We also show that bounding cochains for immersed rational Lagrangians transform naturally under Legendrian isotopy, as conjectured by Akaho and Joyce. As an illustration, we show that quantum cohomology of the complex projective line -- which requires the counting of one holomorphic sphere -- can be recovered from purely sheaf-theoretic calculations.

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arXiv2022

Mapping tori of $A_{\infty}$-autoequivalences and Legendrian lifts of exact Lagrangians in circular contactizations

We study mapping tori of quasi-autoequivalences $\tau : \mathcal{A} \to \mathcal{A}$ which induce a free action of $\mathbf{Z}$ on objects. More precisely, we compute the mapping torus of $\tau$ when it is strict and acts bijectively on hom-sets, or when the $A_{\infty}$-category $\mathcal{A}$ is directed and there is a bimodule map $\mathcal{A} (-, -) \to \mathcal{A} (-, \tau (-))$ satisfying some hypotheses. Then we apply these results in order to link together the Fukaya $A_{\infty}$-category of a family of exact Lagrangians, and the Chekanov-Eliashberg DG-category of Legendrian lifts in the circular contactization.

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