Let $M$ be a manifold and $f \in {\cal C}^{\infty}(M)$ a smooth function. I define a differential over the exterior forms:

$$d_f(g)(X)=f X(g)$$

$$d_f \alpha (X,Y)= fX\alpha (Y) -f Y\alpha (X) - f \alpha ([X,Y])-X(f)\alpha (Y) + Y(f)\alpha(X)$$

I define $d_f$ over the higher degree by the Leibniz rule:

$$d_f (\alpha \wedge \beta)= d_f(\alpha)\wedge \beta +(-1)^{deg(\alpha)} \alpha \wedge d_f(\beta)$$

We have:

$$ d_f \circ d_f=0$$

So the cohomology is:

$$H^*_f(M,{\bf R})=Ker(d_f)/Im(d_f)$$

Can we have a functor of categories from the category of manifolds with function to the category of algebras?