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  "body": "\n## Summary\n\nA local coefficient system on a space $X$ gives an abelian group for each point of $X$.\nThese groups may be transported along paths.\nThe transport depends only on the homotopy class of the path.\nHomology and cohomology with local coefficients generalise ordinary homology and cohomology with a fixed coefficient group.\nThey 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.\n\n## Explanations\n\n### L1 human\n\nOrdinary cohomology attaches the same group, say $\\Z$, to every point of a space.\nA local system attaches a copy of a group to each point, but going around a loop may come back with a twist.\nFor example, on a circle, attach $\\Z$ to each point and let one trip around the circle act by $n \\mapsto -n$.\nLocally nothing has changed, but globally the coefficients are twisted, as the Möbius band is a twisted version of a cylinder.\n",
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