  In this paper, we define a relative Morse complex for manifold with boundary
using the handlebody decomposition of the manifold. We prove that the homology
of the relative Morse complex is isomorphic to the relative singular homology.
Furthermore, we construct $A_\infty$-category structure on the relative Morse
complex by counting the trajectory trees among the critical points of different
Morse functions. Our result generalizes Fukaya's construction on closed
manifold and Manabu's construction of absolute homology on manifold with
boundary.
