A cdga for compactly supported cohomology (à la Sullivan's algebra of polynomial forms)

Let $M$ be a smooth manifold, and let $\Omega^\bullet(M)$ be the commutative dg-algebra of differential forms on $M$. It is quasi-isomorphic to the dg-algebra of singular cochains on $M$. If $M$ is no longer a manifold one can still (under mild hypotheses) find a commutative dga quasi-isomorphic to the singular cochains, namely Sullivan's polynomial de Rham complex $A^\bullet(M)$. This is the basis of Sullivan's approach to rational homotopy theory.

Now consider instead the cdga $\Omega_c^\bullet(M)$ of compactly supported differential forms. It is quasi-isomorphic to the dga of singular cochains with compact support. Now what happens if $M$ is not necessarily a manifold -- can one construct a commutative version of the dga of singular cochains with compact support, à la Sullivan?

• There are commutative models for abstract reasons since the singular cochain algebra with compact support is strongly homotopy commutative, i.e. $E_\infty$. I don't know if you can construct it in the Sullivan way, i.e. as the cochain algebra of simplicial maps $M\rightarrow A_{PL}$ to a simplicial commutative cochain algebra. I'd rather take the usual thing in the target and a subset of these maps satisfying some local compacity or properness assumption. – Fernando Muro Feb 11 '15 at 23:05