A local coefficient system on a topological space 
assigns an Abelian group to each point
and an isomorphism between the assigned groups
to each homotopy class of paths
with fixed endpoints. These assignments respect path
composition and identity paths (Hatcher 2002, pp. 330-332).
For a connected CW-complex
with a basepoint, this is equivalently an Abelian
group with an action of the fundamental
group by group automorphisms. Transport
around loops can therefore give nontrivial group
automorphisms of the assigned group. For example, on
a manifold, the orientation character assigns multiplication
by 1 or 
on 
to loops and gives the orientation coefficient system used
in Poincaré duality.
Local Coefficient System
See also
Abelian Group, Cohomology Group, Fundamental Group, Group Action, Group Automorphism, Homology Group, Local System, Poincaré Complex, Poincaré Duality, Topological SpaceExplore with Wolfram|Alpha
References
Hatcher, A. Algebraic Topology. Cambridge, England: Cambridge University Press, 2002. https://pi.math.cornell.edu/~hatcher/AT/AT.pdf.Cite this as:
Weisstein, Eric W. "Local Coefficient System." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LocalCoefficientSystem.html