The infinity-category of stabilized Liouville sectors

Abstract

We prove the surprising fact that the infinity-category of stabilized Liouville sectors is a localization of an ordinary category of stabilized Liouville sectors and strict sectorial embeddings.From the perspective of homotopy theory, this result continues a trend of realizing geometrically meaningful mapping spaces through the categorically formal process of localizing. From the symplectic viewpoint, these results allow us to reduce highly non-trivial coherence results to much simpler verifications. For example, we prove that the wrapped Fukaya category is coherently functorial on stabilized Liouville sectors: Not only does a wrapped category receive a coherent action from stabilized automorphism spaces of a Liouville sector, spaces of sectorial embeddings map to spaces of functors between wrapped categories in a way respecting composition actions. As we will explain, our methods immediately establish such coherence results for most known sectorial invariants, including Lagrangian cobordisms.As another application, we show that this infinity-category admits a symmetric monoidal structure, given by direct product of underlying sectors. The existence of this structure relies on a computation-familiar from the foundations of factorization homology-that localizations detect certain isotopies of smooth manifolds. Moreover, we characterize the symmetric monoidal structure using a universal property, again producing a simple-as-possible criterion for verifying whether invariants are both continuously and multiplicatively coherent in a compatible way.Much of this paper is devoted to the foundational work of rigorously constructing the infinitycategory of stabilized Liouville sectors, where any sector M is identified with its stabilization M × T * [0, 1]. We begin with a verification that our construction is indeed an infinity-category (which relies on new constructions in Liouville geometry), then show that our infinity-category computes the "correct" mapping spaces (employing new simplicial techniques). A significant ingredient is a demonstration that two notions of sectorial maps give rise to homotopy equivalent spaces of maps.