Given a topological space , a local system is a functor from the fundamental groupoid of to a category.Spanier, chapter 1; Exercises F. As an important special case, a bundle of (abelian) groups on is a local system valued in the category of (abelian) groups. This is to say that a bundle of groups on assigns a group to each element of , and assigns a group homomorphism to each continuous path from to . In order to be a functor, these group homomorphisms are required to be compatible with the topological structure, so that homotopic paths with fixed endpoints define the same homomorphism; furthermore the group homomorphisms must compose in accordance with the concatenation and inversion of paths.Whitehead, section 6.1; page 257. One can define homology with coefficients in a bundle of abelian groups.Whitehead, section 6.2.
