  In the spirit of Morse homology initiated by Witten and Floer, we construct
two $\infty$-categories $\mathcal{A}$ and $\mathcal{B}$. The weak one
$\mathcal{A}$ comes out of the Morse-Samle pairs and their higher homotopies,
and the strict one $\mathcal{B}$ 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
$\infty$-functor $\mathcal{F}: \mathcal{A}\rightarrow \mathcal{B}$. Higher
algebraic structures behind Morse homology are revealed with the perspective of
defects in topological quantum field theory.
