  The main objective of this paper is to extend Morse-Forman theory to
vector-valued functions. This is mostly motivated by the need to develop new
tools and methods to compute multiparameter persistence. To generalize the
theory, in addition to adapting the main definitions and results of Forman to
this vectorial setting, we use concepts of combinatorial topological dynamics
studied in recent years. This approach proves to be successful in the following
ways. First, we establish a result which is more general than that of Forman
regarding the sublevel sets of a multidimensional discrete Morse function.
Second, we find a way to induce a Morse decomposition in critical components
from the critical points of such a function. Finally, we deduce a set of Morse
equation and inequalities specific to the multiparameter setting.
