Local coefficient system

Remarks

Let $X$ be path-connected and $x_0 \in X$. Then a local system is determined up to isomorphism by the group $L_{x_0}$ with its action of the fundamental group $\pi_1(X, x_0)$. This is the same as a module over the group ring $\Z[\pi_1(X, x_0)]$. If $X$ is locally path-connected and semilocally simply connected, local systems are the same as locally constant sheaves of abelian groups on $X$.

Example: on the circle $S^1$, let $L_x = \Z$ for every $x$, and let transport along a path that winds $k$ times multiply by $(-1)^k$. This respects composition, since $(-1)^{k + l} = (-1)^k (-1)^l$, so it is a local system. It is not isomorphic to the constant system $\Z$, since the generator of $\pi_1(S^1)$ acts by $-1$ rather than $1$.

Non-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.

The 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.