Associahedral categories, particles and Morse functor

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.