Localization and flexibilization in symplectic geometry

We introduce the critical Weinstein 1-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 1-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 analog of topological localization of Bousfield and Sullivan, answering a question of Abouzaid and Seidel.When P D f0g, 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 1 -commutative algebra objects from symplectic geometry.1. Introduction 1829 2. Liouville geometry background 1841 3. 1-categorical background 1863 4. Two models for P -flexibilization 1868 5. Comparison of P -flexibilization functors 1877 6. Idempotency of P -flexibilization 1886