  We develop a generalization to non-Witt spaces of the intersection homology
theory of Goresky-MacPherson. The second author has described the self-dual
sheaves compatible with intersection homology, and the other authors have
described a generalization of Cheeger's L2 de Rham cohomology. In this paper we
extend both of these cohomologies by describing all sheaf complexes in the
derived category of constructible sheaves that are compatible with middle
perversity intersection cohomology, though not necessarily self-dual. On
Thom-Mather stratified spaces this refined intersection cohomology theory
coincides with the analytic de Rham theory.
