A stable infinity-category of Lagrangian cobordisms

TL;DR

This paper constructs a stable infinity-category of Lagrangian cobordisms in symplectic geometry, conjecturally linking it to the partially wrapped Fukaya category, and establishes its triangulated homotopy category structure.

Contribution

It introduces a new stable infinity-category of Lagrangian cobordisms and proves its stability and triangulated structure, connecting geometric cobordisms with algebraic categories.

Findings

Lag is a stable infinity-category.

Its homotopy category is triangulated.

The shift functor matches the grading shift for Lagrangian branes.

Abstract

Given an exact symplectic manifold M and a support Lagrangian \Lambda, we construct an infinity-category Lag, which we conjecture to be equivalent (after specialization of the coefficients) to the partially wrapped Fukaya category of M relative to \Lambda. Roughly speaking, the objects of Lag are Lagrangian branes inside of M x T*(R^n), for large n, and the morphisms are Lagrangian cobordisms that are non-characteristic with respect to \Lambda. The main theorem of this paper is that Lag is a stable infinity-category, so that its homotopy category is triangulated, with mapping cones given by an elementary construction. In particular, its shift functor is equivalent to the familiar shift of grading for Lagrangian branes.