Abstract: Sheaves of semimodules on locally ordered spaces model real-world local constraints on dynamical systems indescribable in the language of modules. This talk generalizes a homotopical construction of singular sheaf (co)homology for semimodule-valued sheaves over locally ordered spaces; terminal compactifications of ordered Euclidean space play the role of spheres. Examples of singular (co)homology semimodules include flows on a network subject to capacity constraints and causal singularities on spacetimes. Various calculational tools, incorporating previous cubical approximation theorems, will be presented. The main result presented is a Poincare Duality for sheaves over spacetimes and generalizations admitting top-dimensional topological singularities. Calculations on spacetime surfaces and directed graphs follow.

This talk assumes no familiarity with directed topology or semimodules.