Étale cohomology of schemes $X$ is constructed as follows: one associates to $X$ the so-called étale topos of $X$, and then one just takes the sheaf cohomology of that topos.


Is it possible to associate to each smooth manifold $M$ a "de Rham topos of $M$", whose sheaf cohomology yields the de Rham cohomology of $M$?



De Rham via topoi
One can define an analogue of the crystalline topos for smooth manifolds.


This is known as the [de Rham stack](https://ncatlab.org/nlab/show/de+Rham+space) of $M$.


One of the easiest constructions of the de Rham stack
embeds smooth manifolds fully faithfully (using the Yoneda embedding)
into the category of ∞-sheaves on affine smooth loci, the latter being defined as the opposite category of [$\def\Ci{{\rm C}^∞} \Ci$-rings](https://ncatlab.org/nlab/show/C%5E%E2%88%9E-ring) satisfying certain properties.


In this language, the de Rham stack of an ∞-sheaf $F$
is the ∞-sheaf $\def\dR{{\rm dR}} \dR(F)$ defined by $\def\Spec{\mathop{\rm Spec}} \def\red{{\rm red}} \dR(F)(\Spec A)=\dR(F)(\Spec(\red(A))$,
where $\Spec A$ denotes the spectrum of a $\Ci$-ring (defined purely formally in this context) and $\red(A)$ denotes the quotient of $A$ by its ideal of nilpotent elements.


One can then prove that the commutative differential graded algebra of smooth functions on $\dR(M)$ is precisely the de Rham algebra of $M$.


The de Rham stack has other exciting properties: vector bundles (and, more generally, sheaves) on $\dR(M)$ can be identified with D-modules, etc.


The cited nLab article has the relevant pointers to the literature.


The de Rham stack is also closely related to the definition
of the differential graded algebra of differential forms in [synthetic differential geometry](https://ncatlab.org/nlab/show/synthetic%20differential%20geometry)
as the differential graded algebra of infinitesimal smooth singular cochains equipped with the cup product.
See the nLab article [differential forms in synthetic differential geometry](http://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/differential+forms+in+synthetic+differential+geometry) for further pointers to the literature.


