geometry of triangulated category and D-modules theory

To me it seems as if the most conceptual way to think about this is as follows:

So the category of D-modules on a space is something like its category of flat (generalized) vector bundles.

As currently discussed for instance here on the nCafe, the way to think about this in terms of notions of space is the following:

in the oo-context, the plain overcategory of our ambient oo-tops over a space X is just this space X regarded dually in terms of the oo-topos of oo-stacks over it. This is a change of perspective (space to things on the space) that essentially does not lose information.

But then we can instead first form the overcategory and then stabilize it. This does lose information. And in fact, this turns out to form the category not of oo-stacks but of quasicoherent sheaves over $X$. These are to be thought of as a "linearization of all oo-stacks". I try to talk about that here.

So this gives a description of the original space which is a little more indirect. Now, with D-modules, it becomes yet a bit more indirect: instead of all stabilized oo-stacks, we just retain those that have a flat connection, in a way.

The original underlying space need not be fully reconstructible from this. But then one take the perspective that we are just interested in the linearized and flat situation, and take a stable oo-ocategory to be a formal dual of a possibly fictitious space.