[contravariant]

Yeah you'll want hypergraphs (or even better simplicial complexes) in general but you can make sense of manifolds as discrete objects. In fact I'm fairly sure you can triangulate any (smooth?) manifold.

The tangent space can be defined in terms of derivations, as soon as you define what a smooth function on the 'discrete' manifold should look like (you may have to define the derivative at a face, rather than a vertex, or have the function take values on faces rather than vertices).

[woopwoop]

Sure. I think the notion of a manifold wouldn't be very interesting if it didn't have some kind of discrete analogue. But, for me at least, smooth manifolds are much easier to think about than simplicial complexes or PL manifolds.