One can define the de Rham space / stack of (say) a scheme by identifying infinitesimally closed points. Modules over this space, also called crystals, may be regarded as an algebraic analogue of flat connections on a smooth vector bundle over a manifold.
My question is why we call this space the de Rham space. My guess is that this space is related to algebraic de Rham cohomology (or crystalline cohomology, whatever that is) of a scheme, but references on this topic say nothing about this. Differential-geometric intuition would be more than welcome.