Localization and flexibilization in symplectic geometry

TL;DR

This paper introduces a new localization functor in symplectic geometry called P-flexibilization, which generalizes existing flexibilization procedures and provides a universal, symmetric monoidal framework for symplectic sectors and their Fukaya categories.

Contribution

It constructs the P-flexibilization endofunctor as an idempotent localization in the critical Weinstein infinity-category, extending previous flexibilization methods and connecting to algebraic structures.

Findings

P-flexibilization is an idempotent localization functor.

It generalizes Cieliebak and Eliashberg's flexibilization.

The functor is symmetric monoidal and constructs E-infinity-commutative algebra objects.

Abstract

We introduce the critical Weinstein infinity-category -- the result of stabilizing the category of Weinstein sectors and inverting subcritical morphisms -- and for every finite collection P of integers, construct a P-flexibilization endofunctor. Our main result is that P-flexibilization is an idempotent localization functor of the critical Weinstein infinity-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 P is invertible in the wrapped Fukaya category and hence is a symplectic analogue of topological localization of Bousfield and Sullivan, answering a question of Abouzaid and Seidel. When P = {0}, our construction recovers Cieliebak and Eliashberg's flexibilization procedure. Moreover, we show that P-flexibilization is symmetric monoidal as a functor of higher categories, and hence gives rise to a new way of constructing E-infinity-commutative algebra objects from symplectic geometry.