  We complete the details of a theory outlined by Kontsevich and Soibelman that
associates to a semi-algebraic set a certain graded commutative differential
algebra of "semi-algebraic differential forms" in a functorial way. This
algebra encodes the real homotopy type of the semi-algebraic set in the spirit
of the DeRham algebra of differential forms on a smooth manifold. Its
development is needed for Kontsevich's proof of the formality of the little
cubes operad.
