Functoriality for Lagrangian correspondences in Floer theory
Katrin Wehrheim, Chris T. Woodward
TL;DR
This paper develops a functorial framework for Lagrangian correspondences in Floer theory using quilted Floer cohomology, establishing a categorification that respects composition under smooth, embedded conditions.
Contribution
It introduces a composition functor for Lagrangian correspondences in Floer theory and proves its compatibility with geometric composition in specific cases.
Findings
Defines a composition functor using quilted Floer cohomology
Shows functor agrees with geometric composition when smooth and embedded
Establishes categorification commutes with composition for Lagrangian correspondences
Abstract
Using quilted Floer cohomology and relative quilt invariants, we define a composition functor for categories of Lagrangian correspondences in monotone and exact symplectic Floer theory. We show that this functor agrees with geometric composition in the case that the composition is smooth and embedded. As a consequence we obtain 'categorification commutes with composition' for Lagrangian correspondences.
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