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?

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    $\begingroup$ 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. $\endgroup$ Commented Feb 11, 2015 at 23:05

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I think something close to what you are looking for is done on page 8 of http://arxiv.org/pdf/1404.1127.pdf for diffeological spaces. It's not "a la Sullivan", in that it's not done with polynomial forms, but it certainly covers a large class of non-manifold cases. The main obstacle in the paper is proving the existence of partitions of unity for covers of diffeological spaces, which is required for the exactness of the Mayer Vietoris sequence.

Would be interesting though to try and do this for PL-spaces (I'm imagining locally finite simplicial sets) using polynomial forms.


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