Higher Morse moduli spaces and n-categories

We generalize Cohen & Jones & Segal’s flow category whose objects are the critical points of a Morse function and whose morphisms are the Morse moduli spaces between the critical points to an n-category. The n-category construction involves repeatedly doing Morse theory on Morse moduli spaces for which we have to construct a class of suitable Morse functions. It turns out to be an almost strict' n-category, i.e. it is a strict n-category up to canonical isomorphisms’.