  In a series of papers the authors introduced the so-called blown-up
intersection cochains. These cochains are suitable to study products and
cohomology operations of intersection cohomology of stratified spaces. The aim
of this paper is to prove that the sheaf versions of the functors of blown-up
intersection cochains are realizations of Deligne's sheaves. This proves that
Deligne's sheaves can be incarnated at the level of complexes of sheaves by
soft sheaves of perverse differential graded algebras. We also study Poincar\'e
and Verdier dualities of blown-up intersections sheaves with the use of
Borel-Moore chains of intersection.

Algebraic Topology