The Fukaya category pairs with Lagrangian cobordisms

Abstract

Fix a suitably convex, exact symplectic manifold M. We consider the stable oo-category Lag(M) of non-compact Lagrangians whose (higher) morphisms are (higher) Lagrangian cobordisms between them. We show that this oo-category pairs with the Fukaya category Fuk(M) of compact branes. In fact, we also show that there is a subcategory of Lag(M) which pairs with the wrapped Fukaya category of M. This is a first step in a project to enrich wrapped Fukaya categories over cobordism spectra. As a corollary, we show that cobordant compact branes are equivalent in the Fukaya category. We will also mention several other applications (without proof) of the oo-categorical appraoch: One can realize Seidel's representation as a \pi_0-level consequence of a map of spaces; stable cobordism groups of non-compact branes map to Floer cohomology groups; some of Biran-Cornea's results can be recovered from the colored planar operad associated to the s-dot constructions of each category; and there is an Eoo map of spectra from exact Lagrangian cobordisms in Euclidean space to the integers.