  We apply the theory of directed topology developed by Grandis [9, 10] to the
study of stratified spaces by describing several ways in which a stratification
or a stratification with orientations on the strata can be used to produce a
related directed space structure. This description provides a setting for the
constructions of state-sum TQFTs with defects of [5, 8], which we extend to a
similar construction of a Dijkgraaf- Witten type TQFT in the case where the
defects (lower dimensional strata) are not sources or targets, but sources on
one side and targets on the other, according to an orientation convention.
