I was wondering if it was possible to extend the De Rham theorem to differetiable spaces, as defined for instance in this paper 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 explicitely 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 aswers, there seem 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 describibg smooth manifolds. Now perhaps the answers given treated that, but it's not apparent to me, so perhaps one could elaborate on the link ?