Discrete version of some topological object. 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?
 A: This should be a comment.  
Since the image is Abelian $\phi$ factors through the universal Abelian cover (much smaller than the universal cover) and your objects are equivalent to a map $\psi$ from the integer first homology of your surface to $Z^n$ along with the data of an equivariant map from the one skeleton of this cover to $R^n$ linear on edges and with vertices going to integer points.
Since this is equivariant this descends to a map $\tau$ from the surface to the torus $R^n$ mod $Z^n$.  
In terms of flat connections $\psi$ is the monodromy action of the (Abelianization of) the fundamental group on the fiber ($R^n$).  
In the case n=2 one characteristic class interpretation of your associated 2-cocycle class is as the pullback of the fundamental class of the torus $R^2$ mod $Z^2$ (classifying space of $Z$ x $Z$) via $\tau$.  
A: Let $\Sigma$ be your surface.
I'll construct a new surface $\tilde \Sigma$ as follows.
Take $\Sigma$ apart into individual triangles, and reglue them with a twist at each edge (the adjacency graphs for the faces of $\Sigma$, and for the faces of $\tilde\Sigma$ are the same). Note that the surfaces $\Sigma$ and $\tilde\Sigma$ could have different genus, and that $\tilde\Sigma$ could be non-orientable.
Your data is equivalent to having a piecewise linear map (not necessarily an embedding) of the universal cover of the new surface $\tilde\Sigma$ into $\mathbb R^n$, so that the vertices of the triangulation map to $\mathbb Z^n$. That map is well defined up to an overall translation.
