  Let X be a pseudomanifold. In this text, we use a simplicial blow-up to
define a cochain complex whose cohomology with coefficients in a field, is
isomorphic to the intersection cohomology of X, introduced by M. Goresky and R.
MacPherson.
  We do it simplicially in the setting of a filtered version of face sets, also
called simplicial sets without degeneracies, in the sense of C.P. Rourke and
B.J. Sanderson. We define perverse local systems over filtered face sets and
intersection cohomology with coefficients in a perverse local system. In
particular, as announced above when X is a pseudomanifold, we get a perverse
local system of cochains quasi-isomorphic to the intersection cochains of
Goresky and MacPherson, over a field. We show also that these two complexes of
cochains are quasi-isomorphic to a filtered version of Sullivan's differential
forms over the field Q. In a second step, we use these forms to extend
Sullivan's presentation of rational homotopy type to intersection cohomology.
  For that, we construct a functor from the category of filtered face sets to a
category of perverse commutative differential graded Q-algebras (cdga's) due to
Hovey. We establish also the existence and unicity of a positively graded,
minimal model of some perverse cdga's, including the perverse forms over a
filtered face set and their intersection cohomology. Finally, we prove the
topological invariance of the minimal model of a PL-pseudomanifold whose
regular part is connected, and this theory creates new topological invariants.
This point of view brings a definition of formality in the intersection setting
and examples are given. In particular, we show that any nodal hypersurface in
CP(4), is intersection-formal.
