Hope my question is not too vague.
I've studied John M. Lee "Introduction to smooth manifolds" and now want to move to study differential geometry on simplicial complexes, the goal eventually is to study Laplace-Beltrami operators on graphs. But I want to go slowly and understand first the relationship between differential geometry on simplicial complexes and the smooth one, and then finally relate differential geometry on simplicial complexes to that on graphs.
I found a lot of material on differential geometry focusing on discrete domain separately (here, or here). With definition where the discrete case is given as analogy of a smooth case, but not as extension of one from the another.
I was wondering if there is a unified theory that combines smooth and non-smooth manifolds, such that it includes also discrete manifolds like graphs as a special case?
I could imagine a treatment where one obtains differential geometry on simplexes as a "limit" of the smooth theory where the functions are averaged over simplices (integrated out in some sense) and various differential operators are defined to include not just smooth domains but domains with corners.
Any comments are welcome.