Flows in graphs and the homology of free categories

Abstract

We study the \(R\)-module of generalized flows in a graph with coefficients in the \(R\)-representation of the graph over a ring \(R\) with 1 and show that this \(R\)-module is isomorphic to the first derived functor of the colimit. We generalize Kirchhoff’s laws and build an exact sequence for calculating the R-module of flows in the union of graphs.