[enriquto]

the tangent space is the set of edges

vector fields (sections of the tangent bundle) correspond to real-valued functions defined on edges

dimension is always 2 ;)

if you want higher dimension you have to consider higher-dimensional cliques beyond edges (that are 2-cliques): triangles, tetrahedra, and so on. But very often this is not necessary, even when discretizing 3d stuff. For example, for Poisson equation, and the associated classical pde, you only need the laplacian, which acts on functions, regardless of the dimension.

[contravariant]

That seems a weird choice of tangent space, surely a circle and a disk should have different tangent spaces?

[enriquto]

And they do. When you discretize a circle as a graph, all the edges are along the boundary. When you discretize the disk, you fill its whole interior with edges in all directions.

[woopwoop]

Is a triangle a circle or a disk?