  Here, we introduce a new definition of regular point for piecewise-linear
(PL) functions on combinatorial (PL triangulated) manifolds. This definition is
given in terms of the restriction of the function to the link of the point. We
show that our definition of regularity is distinct from other definitions that
exist in the combinatorial topology literature. Next, we stratify the Jacobi
set/critical locus of such a map as a poset stratified space. As an
application, we consider the Reeb space of a PL function, stratify the Reeb
space as well as the target of the function, and show that the Stein
factorization is a map of stratified spaces.
