Refined intersection homology on non-Witt spaces

We investigate a generalization to non-Witt stratified spaces of the intersection homology theory of Goresky-MacPherson. The second-named 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 first extend both of these cohomology theories 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. Our main result is that this refined intersection cohomology theory coincides with the analytic de Rham theory on Thom-Mather stratified spaces. The word "refined" is motivated by the fact that the definition of this cohomology theory depends on the choice of an additional structure (mezzo-perversity) which is automatically zero in the case of a Witt space.