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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]] with [[fundamental groupoid|ZBHGXJF24E]] $\\Pi_1(X)$.\nA *local coefficient system* (or *local system*) of [[abelian groups|082GHNPJ53]] on $X$ is a functor $L \\colon \\Pi_1(X) \\to \\mathrm{Ab}$.\nIn detail, it consists of:\n\n1. an abelian group $L_x$ for each point $x \\in X$;\n2. an isomorphism $L_\\gamma \\colon L_x \\to L_y$ for each path-homotopy class $\\gamma$ of paths from $x$ to $y$;\n\nsuch that $L_{\\gamma \\cdot \\gamma'} = L_{\\gamma'} \\circ L_\\gamma$ for composable classes and $L_\\gamma = \\id$ for the class of a constant path.\n\n## Remarks\n\nLet $X$ be path-connected and $x_0 \\in X$.\nThen a local system is determined up to isomorphism by the group $L_{x_0}$ with its action of the [[fundamental group|S1XK3HXHFX]] $\\pi_1(X, x_0)$.\nThis is the same as a [[module|0NZ8JA25WK]] over the group ring $\\Z[\\pi_1(X, x_0)]$.\nIf $X$ is locally path-connected and semilocally simply connected, local systems are the same as locally constant sheaves of abelian groups on $X$.\n\nExample: on the circle $S^1$, let $L_x = \\Z$ for every $x$, and let transport along a path that winds $k$ times multiply by $(-1)^k$.\nThis respects composition, since $(-1)^{k + l} = (-1)^k (-1)^l$, so it is a local system.\nIt is not isomorphic to the constant system $\\Z$, since the generator of $\\pi_1(S^1)$ acts by $-1$ rather than $1$.\n\nNon-example: on $S^1$, assigning to every path the same isomorphism $n \\mapsto -n$ fails the clause on constant paths, which must act by the identity.\n\nThe orientation system of an $n$-manifold $M$, with $L_x = H_n(M, M \\setminus \\set{x}; \\Z)$, is the main example; it is constant exactly when $M$ is orientable.\n",
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