TITLE: Two fancy ways of defining differential forms: How does one show that they are equivalent?
QUESTION [13 upvotes]: Given a smooth manifold M, the following procedures yield the differential graded algebra (Ω*(M),ddR) of differential forms:

Procedure 1 (synthetic geometry).
For each n, consider the object of infinitesimal n-simplices in M. This is a ringed space whose underlying space is the diagonal M ⊂ Mn+1, and whose structure sheaf is $C^\infty (M^{n+1}) / I^2$, where $I$ is the ideal defining the diagonal.
The objects of infinitesimal n-simplices assemble into a simplicial ringed space, and by taking global function one gets a cosimplicial algebra. The normalized Moore complex of that cosimplicial algebra is then canonically isomorphic to the deRham complex of M.
Procedure 2 (supergeometry).
Letting ℝ0|1 denote the odd line, we may consider the internal hom
$Map(\mathbb R^{0|1} , M)$ in the category of supermanifolds. The global functions $C^\infty( Map(\mathbb R^{0|1} , M))$ then form a ℤ/2-graded algebra. Noting that the supergroup of automorphisms of ℝ0|1 acts on $Map(\mathbb R^{0|1} , M)$, one can then upgrade that ℤ/2-graded algebra to a ℤ-graded algebra with differential d. Once again, that procedure recovers the deRham complex of M.


Why do these two recipes produce the same outcome?

More generally, one can imagine applying Procedures 1 and 2 to objects M that are more general than manifolds (supermanifolds, singular spaces, schemes, derived manifolds, differential stacks, infinity stacks, ...). Are those two procedures always going to agree?
If not, when do they agree, when do they disagree, and why?

REPLY [2 votes]: The first construction is an explicit realization of the oo-categorical derived de Rham space functor, which identifies all infinitesimal neighbour points, up to higher equivalence. So it's an explicit realization of the derived dR functor in Simpson-Teleman, I think. http://ncatlab.org/schreiber/show/path+%E2%88%9E-groupoid#Infinitesimal