Local coefficient system

Summary

A local coefficient system on a space $X$ gives an abelian group for each point of $X$. These groups may be transported along paths. The transport depends only on the homotopy class of the path. Homology and cohomology with local coefficients generalise ordinary homology and cohomology with a fixed coefficient group. They are needed for Poincaré duality on non-orientable manifolds, for obstruction theory and for the Serre spectral sequence of a fibration with non-simply-connected base.