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To every simplicial manifold is associated its simplicial deRham complex.

Is there any literature that discusses explicitly to which extent this classical construction, regarded as a (contravariant) functor from simplicial manifolds to dg-algebras, is homotopical?

For instance: simplicial manifolds naturally embed into the category of simplicial presheaves on the category of manifolds, on which we have the standard local model structure on simplicial presheaves. On dg-algebras there is the standard model structure on dg-algebras.

Is there any literature that discusses explicitly the respect of the simplicial deRham complex operation of the respective weak equivalences?

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Some of this is handled in the nLab, I believe, and there are some references dotted around in the relevant entries. –  Tim Porter Apr 13 '11 at 6:13
    
This paper: front.math.ucdavis.edu/0810.1747 contains a covariant version of simplicial de Rham theory, with good categorical properties. I wrote it with applications to model structures in mind, although no such applications are actually in the paper. –  Neil Strickland Apr 13 '11 at 6:43
    
Could someone retag this? (I'm not really competent to do it myself.) –  Yemon Choi Apr 13 '11 at 7:53
    
I retagged it, but did not take flags off, maybe I should have. –  Sean Tilson Apr 13 '11 at 12:14

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