A Weak ∞-Functor in Morse Theory

In the spirit of Morse homology initiated by Witten and Floer, we construct two ∞-categories \({\cal A}\) and \({\cal B}\). The weak one \({\cal A}\) comes out of the Morse–Smale pairs and their higher homotopies, and the strict one \({\cal B}\) concerns the chain complexes of the Morse functions. Based on the boundary structures of the compactified moduli space of gradient flow lines of Morse functions with parameters, we build up a weak ∞-functor \({\cal F}:{\cal A} \rightarrow {\cal B}\). Higher algebraic structures behind Morse homology are revealed with the perspective of defects in topological quantum field theory.