  Inspired by the works of Forman on discrete Morse theory, which is a
combinatorial adaptation to cell complexes of classical Morse theory on
manifolds, we introduce a discrete analogue of the stratified Morse theory of
Goresky and MacPherson. We describe the basics of this theory and prove
fundamental theorems relating the topology of a general simplicial complex with
the critical simplices of a discrete stratified Morse function on the complex.
We also provide an algorithm that constructs a discrete stratified Morse
function out of an arbitrary function defined on a finite simplicial complex;
this is different from simply constructing a discrete Morse function on such a
complex. We then give simple examples to convey the utility of our theory.
Finally, we relate our theory with the classical stratified Morse theory in
terms of triangulated Whitney stratified spaces.
