  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.
