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?