The infinity-category of stabilized Liouville sectors

TL;DR

This paper establishes that the infinity-category of stabilized Liouville sectors can be viewed as a localization of a simpler category, enabling simplified coherence proofs for symplectic invariants and introducing a symmetric monoidal structure.

Contribution

It proves the localization of the infinity-category of stabilized Liouville sectors and demonstrates its implications for functoriality and monoidal structures in symplectic geometry.

Findings

Wrapped Fukaya category is coherently functorial on stabilized Liouville sectors

Wrapped Floer theory works in families

The infinity-category admits a symmetric monoidal structure

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 a consequence, we observe that wrapped Floer theory for sectors works in families. 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.