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Localization and
flexibilization in symplectic geometry
Oleg Lazarev, Zack Sylvan and Hiro Lee Tanaka |
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Geometry & Topology 30 (2026) 1829–1898
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DOI: 10.2140/gt.2026.30.1829
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Abstract |
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We introduce the critical Weinstein -category — the result of stabilizing the category of Weinstein sectors and inverting subcritical morphisms — and for every finite collection of integers, construct a -flexibilization endofunctor. Our main result is that -flexibilization is an idempotent localization functor of the critical Weinstein -category, allowing us to characterize the essential image of the endofunctor by a universal property. This localization has the effect of replacing every Weinstein sector with one in which is invertible in the wrapped Fukaya category and hence is a symplectic analog of topological localization of Bousfield and Sullivan, answering a question of Abouzaid and Seidel. When , our construction recovers Cieliebak and Eliashberg’s flexibilization procedure. Moreover, we show that -flexibilization is symmetric monoidal as a functor of higher categories, and hence gives rise to a new way of constructing -commutative algebra objects from symplectic geometry. |
Keywords
sectors, Liouville sectors, localization, Fukaya
categories, primes, flexible, symplectic
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Mathematical Subject Classification
Primary: 53D35, 57R17
Secondary: 18E35, 55P60
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References |
Publication
Received: 27 March 2024
Revised: 1 September 2025
Accepted: 19 November 2025
Published: 18 July 2026
Proposed: Mohammed Abouzaid
Seconded: Ciprian Manolescu, Yakov Eliashberg
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Authors |
| Oleg Lazarev | |
| Department of Mathematics University of Massachusetts Boston Boston, MA United States |
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| Zack Sylvan | |
| New York, NY United States |
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| Hiro Lee Tanaka | |
| Department of Mathematics Texas State University San Marcos, TX United States |
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| © 2026 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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