Some speculations on pairs-of-pants decompositions and Fukaya categories

TL;DR

This paper explores speculative ideas on how Fukaya categories can be constructed from simpler local pieces, inspired by tropical geometry and mirror symmetry, aiming to understand their sheaf-like properties.

Contribution

It proposes a speculative framework for viewing Fukaya categories as local objects assembled from basic components, connecting to tropical geometry and mirror symmetry.

Findings

Conceptual links between Fukaya categories and tropical geometry

Potential sheaf-theoretic structures for Fukaya categories

Foundational ideas for future rigorous formulations

Abstract

These are notes from a 2010 talk. They concern possible ways in which Fukaya categories might be considered as "local", which means glued together from simpler pieces in a loosely sheaf-theoretic sense. As the title suggests, this is purely speculative. Much of the motivation comes from tropical geometry and mirror symmetry.