Question is the following:



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> Is the functor $H^n\_{dR}:\text{Man}\rightarrow \text{Set}$ a sheaf with respect to open cover topology on $\text{Man}$?
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More generally, are cohomology functors sheaves in general (in any reasonably non trivial Grothendieck topology)?


I am also interested in cohomology functors that arise in Algebriac geometry/topology.


Is there a way of sheafification in this setup?


I have nothing much to support this question, this is completely out of curiosity.


Edit : I am also interested in answers/references related to the comment of Piotr Achinger; that reads



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> "in what way is cohomology a sheaf" leads one to notions like $\infty$-topoi etc.
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Are cohomology functors sheaves?

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> Is the functor H^n\_dR:Man→Set a sheaf with respect to open cover topology on Man?
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As already pointed out in the comments, the answer is no for n>0, yes for n=0.



> 
> "in what way is cohomology a sheaf" leads one to notions like ∞-topoi etc.
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> 


In the context of this question,
the assignment M↦Ω(M) yields a contravariant
functor from smooth manifolds to cochain complexes,
and this functor satisfies the homotopy descent condition.


This was first proved by Weil in Sur les théorèmes de de Rham ([DigiZeitschriften](http://www.digizeitschriften.de/download/PPN358147735_0026/PPN358147735_0026___log13.pdf), [DOI](https://doi.org/10.1007/bf02564296), [EuDML](https://eudml.org/doc/139040)).
An accessible exposition is given by Bott and Tu in §8 of [Differential Forms in
Algebraic Topology](http://gen.lib.rus.ec/book/index.php?md5=F8BA716DD25F37E6D3085BCB201D08E3).


