Lagrangian Cobordism and Fukaya Categories

TL;DR

This paper introduces a new categorical framework linking Lagrangian cobordisms with the derived Fukaya category of a symplectic manifold, enhancing understanding of their algebraic and geometric relationships.

Contribution

It constructs a functor from a category of Lagrangian cobordisms to a derived Fukaya category, incorporating triangulated structures to unify geometric and algebraic perspectives.

Findings

Established a functorial relationship between cobordism categories and Fukaya categories.

Enhanced the algebraic structure of Lagrangian cobordisms with triangulated category techniques.

Provided a new perspective on symplectic topology through categorical constructions.

Abstract

Given a symplectic manifold M, we consider a category with objects finite ordered families of Lagrangian submanifolds of M (subject to certain additional constraints) and with morphisms Lagrangian cobordisms relating them. We construct a functor that maps this category to a variant of the derived Fukaya category of M in a way that takes into account the triangulated structure of the latter.