I was wondering if it was possible to extend the De Rham theorem to differentiable spaces, as defined for instance in [this paper](http://www.jstor.org/stable/10.2307/1970846) of Chen. If it has been done, could anyone come up with a good reference?


Alternatively, I could use a reference proving the "classical" De Rham theorem, by using explicitly the chain map
$$
\alpha \mapsto \left[ \sigma \mapsto \int\_\Delta \sigma^\*\alpha \right]
$$
so that I can extend the proof to differentiable spaces.


[edit] Seeing many of the answers, there seems to be either another "de Rham theorem" or another way of seeing differential forms, so let me clarify: What I mean by "de Rham theorem" is the fact that the cohomology of differential forms is isomorphic to e.g. the singular cohomology, when describing smooth manifolds. Now perhaps the answers given treated that, but it's not apparent to me, so perhaps one could elaborate on the link?



De Rham theorem for differentiable spaces
You can also try Chapters 1 and 2 of



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> Johan L. Dupont: Curvature and Characteristic Classes, Lecture Notes in Mathematics, vol. 640, Springer Verlag, 1978.
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