This is called "discrete exterior calculus" and you can find it easily.

The idea is that k-forms are functions defined on the k-cliques of the graph. TFA is just the 1-dimensional case of this.

The 2-dimensional case would be:

0-forms: functions defined on vertices

1-forms: functions defined on edges

2-forms: functions defined on triangles

The exterior derivative is defined in a natural way, by taking differences along signed boundaries.

In the case of a triangulated surface, the Hodge dual has a nice interpretation via the dual triangulation.