  This paper generalizes the Max-Flow Min-Cut (MFMC) theorem from the setting
of numerical capacities to sheaves of partial semimodules over semirings on
directed graphs. Motivating examples of partial semimodules include probability
distributions, multicommodity capacity constraints, and logical propositions.
Directed (co)homology theories for such sheaves describes familar constructs on
networks. First homology classifies locally decomposable flows, an orientation
sheaf over a semiring generalizes directions, connecting maps for homology
assign values to flows, connecting maps for cohomology assign values to cuts,
and a Poincare Duality describes a decomposition of flows as local flows over
cuts. A consequent interpretation of feasible flow-values as a homotopy limit
generalizes MFMC for edge weights in certain ordered monoids [Frieze] and hence
also classical MFMC. Certain duality gaps are explained as a failure for
directed sheaf (co)homology to satisfy a natural generalization of exactness.
