I have two related questions. Here $M$ is a real smooth manifold, $TM$ is its tangent bundle, $T^n M := T ... TM$ is the $n$-th iterated tangent bundle.


1. Fiberwise linear smooth functions $TM \to \mathbf R$ are the same as smooth one-forms on $M$. Is there a handy generalization of this to $n$-forms and some functions $T^n M \to \mathbf R$?
2. Can de Rham cohomology be expressed in terms of (the simplicial abelian group of) functions $T^\bullet M \to \mathbf R$?


Thank you. 



de Rham cohomology vs. iterated tangent bundles?
**EDIT**: I think an answer to your **first** question is explained in the papers:


* P.-A. Meyer, *Qu'est ce qu'une différentielle d'ordre $n$*, Exposition. Math. 7 (1989), 249–264.
* Laksov, Dan; Thorup, Anders, *These are the differentials of order $n$*. Trans. Amer. Math. Soc. 351 (1999), no. 4, 1293–1353. Freely available [online](http://www.ams.org/journals/tran/1999-351-04/S0002-9947-99-02120-0/S0002-9947-99-02120-0.pdf).


Quote from the second:



> 
> The higher order differentials were part of the *folklore* of mathematics up to the end of the previous century, and formulas like $d^2f=f'\_xd^2x+f'\_yd^2y+{f''\_{x^2}}dx^2+2{f''\_{xy}}dxdy+{f''\_{y^2}}dy^2$ can be found in most classical calculus books. (...) the higher order differentials vanished (...) because the extensive user of exterior differentials led mathematicians to believe that $d^2$ should always be zero.
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> (...)
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> Let $C\_n:=\mathcal{C}^\infty(T^nX)$. The differential of $\mathcal{C}^\infty$- functions on $T^nX$ can be viewed as a k-linear map $d:C\_n\rightarrow C\_{n+1}$, and we obtain a sequence of linear maps...
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Then $\Omega^n$ is defined as a suitable submodule of $C\_n$, and there is a product $\Omega^p \otimes \Omega^n \rightarrow \Omega^{p+n}$ such that $d(\omega\cdot\pi)=d\omega\cdot\pi + \omega\cdot d\pi$.


