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 de Rham algebra of differential forms on a smooth manifold. Its development is needed for Kontsevich’s proof of the formality of the little cubes operad.
"Real homotopy theory of semi-algebraic sets." Algebr. Geom. Topol. 11 (5) 2477 - 2545, 2011. https://doi.org/10.2140/agt.2011.11.2477