Any ring gives us a rich linear category: its category of modules. More generally, any scheme, any stack, and (tautologically) any non-commutative space does, too. The incredible surprise is that symplectic manifolds also give rise to rich linear categories through the Fukaya category construction. Even better, mirror symmetry should allow us to recast and study all the examples in the first two sentences through symplectic techniques. But we're not there yet. In this talk, I'll talk about recent progress in proving that the flimsy, geometric setting of symplectic geometry actually displays some of the structured and homotopically convenient properties that homotopy theorists and algebraists know and love. In a sentence: The oo-category of Weinstein sectors is a localization of an ordinary category, and the oo-category admits localizations that behave very much like inverting prime numbers (and more generally, nullifying finite stable homotopy types like Moore spectra). This is joint work with Oleg Lazarev and Zachary Sylvan.