  Intersection homology is defined for simplicial, singular and PL chains. In
the case of a filtered simplicial complex, it is well known that the three
versions are isomorphic. This isomorphism is established by using the PL case
as an intermediate between the singular and the simplicial situations.
  Here, we give a proof similar to the classical proof for ordinary simplicial
complexes. We also study the intersection blown-up cohomology that we have
previously introduced.
  In the case of a pseudomanifold, this cohomology owns a Poincar\'e
isomorphism with the intersection homology, for any coefficient ring, thanks to
a cap product with a fundamental class. We prove that the simplicial and the
singular blown-up cohomologies of a filtered simplicial complex are isomorphic.
From this result, we can now compute the blown-up intersection cohomology of a
pseudomanifold from a triangulation.
  Finally, we introduce a blown-up intersection cohomology for PL-spaces and
prove that it is isomorphic to the singular ones. We also show that the cup
product in perversity $ \overline 0$ of a CS-set coincides with the cup product
of the singular cohomology of the underlying topological space.
