Consider a triangulated orientable surface with the following data: on each edge a vector with integer coordinates is written so that for each triangle the sum of the vectors corresponding to three edges of its boundary is $0$.
May such an object be interpret as a discrete version of some topological construction, say, flat connection?
Since for each triangle sum of the vectors corresponding to its boundary is $0$, it is possible to assign to each triangle its "area"- the determinant of the matrix formed by any two vectors of the boundary of the triangle. I also wonder, if there exists some interpretation of determinants of these triangles. Maybe, the 2-cochain formed by them is representing some characteristic class?