Associahedral categories, particles and Morse functor

TL;DR

This paper introduces associahedral categories derived from particles in smooth manifolds, establishing a topological field theory framework and connecting Morse-Witten theory to chain complexes via a functor.

Contribution

It defines associahedral categories and interprets Morse-Witten theory as a functor to chain complexes, linking topology, algebra, and Morse theory in a novel categorical setting.

Findings

Introduction of associahedral categories related to particles

Establishment of a topological field theory structure

Interpretation of Morse-Witten theory as a functor to chain complexes

Abstract

Every smooth manifold contains particles which propagate. These form objects and morphisms of a category equipped with a functor to the category of Abelian groups, turning this into a 0+1 topological field theory. We investigate the algebraic structure of this category, intimately related to the structure of Stasheff's polytops, introducing the notion of associahedral categories. An associahedral category is preadditive and close to being strict monoidal. Finally, we interpret Morse-Witten theory as a contravariant functor, the Morse functor, to the homotopy category of bounded chain complexes of particles.