  In this paper we prove a duality for constructible sheaves on conically
smooth stratified spaces. Here we consider sheaves with values in a stable and
bicomplete $\infty$-category equipped with a closed symmetric monoidal
structure, and in this setting constructible means locally constant along
strata and with dualizable stalks. The crucial point where we need to employ
the geometry of conically smooth structures is in showing that Lurie's version
of Verdier duality restricts to an equivalence between constructible sheaves
and cosheaves: this requires a computation of the exit path $\infty$-category
of a compact stratified space, that we obtain via resolution of singularities.
