A local coefficient system on a 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). For a connected CW-complex
with a basepoint, this is equivalently an Abelian
group with an action of the fundamental group.
Transport around loops can therefore give nontrivial automorphisms of the assigned group.
For example, an orientation character assigns multiplication by 1 or 
to loops and gives the orientation
coefficient system used in Poincaré duality.
Local Coefficient System
See also
Cohomology Group, Fundamental Group, Homology Group, Poincaré Complex, Poincaré DualityExplore 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