# Localizing the Fukaya category of a Stein manifold

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We introduce a new class of non-compact symplectic manifolds called

Liouville sectors and show they have well-behaved, covariantly functorial

Fukaya categories. Stein manifolds frequently admit coverings by Liouville

sectors, which can be used to understand the Fukaya category of the total

space (we will study this geometry in examples). Our first main result in

this setup is a local-to-global criterion for generating Fukaya categories.

Our eventual goal is to obtain a combinatorial presentation of the Fukaya

category of any Stein manifold. This is joint work (in progress) with John

Pardon and Vivek Shende.