  This paper introduces the notion of $n$-morphisms between two
$A_\infty$-algebras, such that 0-morphisms correspond to standard
$A_\infty$-morphisms and 1-morphisms correspond to $A_\infty$-homotopies
between $A_\infty$-morphisms. The set of higher morphisms between two
$A_\infty$-algebras then defines a simplicial set which has the property of
being an algebraic $\infty$-category. The operadic structure of
$n-A_\infty$-morphisms is also encoded by new families of polytopes, which we
call the $n$-multiplihedra and which generalize the standard multiplihedra.
These are constructed from the standard simplices and multiplihedra by lifting
the Alexander-Whitney map to the level of simplices. Rich combinatorics arise
in this context, as conveniently described in terms of overlapping partitions.
Shifting from the $A_\infty$ to the $\Omega B As$ framework, we define the
analogous notion of $n$-morphisms between $\Omega B As$-algebras, which are
again encoded by the $n$-multiplihedra, endowed with a thinner cell
decomposition by stable gauged ribbon tree type. We then realize this higher
algebra of $A_\infty$ and $\Omega B As$-algebras in Morse theory. Given two
Morse functions $f$ and $g$, we construct $n-\Omega B As$-morphisms between
their respective Morse cochain complexes endowed with their $\Omega B
As$-algebra structures, by counting perturbed Morse gradient trees associated
to an admissible simplex of perturbation data. We moreover show that any inner
horn of higher morphisms arising from a count of perturbed Morse gradient trees
can always be filled, not only algebraically but also geometrically.
