(Constant sheaf)
Let \(A\) be an abelian group with the discrete topology. The constant sheaf \(\ul{A}\) on a topological space \(X\) has \(\ul{A}(U)\) the group of continuous maps \(U \to A\), with restriction of functions. If \(U\) is open and connected, then \(\ul{A}(U) = A\); more generally, if every connected component of the open set \(U\) is open, then \(\ul{A}(U) = \prod_{\pi_0(U)} A\), one factor for each connected component. [@Har10a, §II.1]
(Locally constant sheaf and local system)
A sheaf \(\mathcal{L}\) of abelian groups on a topological space \(X\) is locally constant if every point has an open neighbourhood \(U\) with \(\mathcal{L}|_U\) isomorphic to a constant sheaf. A local system is a locally constant sheaf of abelian groups, or of \(k\)-vector spaces, whose stalks are finitely generated, respectively finite-dimensional.
(Monodromy)
Let \(X\) be connected, locally path-connected and semilocally simply connected, with base point \(x\). Taking the stalk at \(x\) with its action by parallel transport along loops gives an equivalence between local systems of \(k\)-vector spaces on \(X\) and finite-dimensional representations \(\pi_1(X, x) \to \GL(\mathcal{L}_x)\). The constant local systems correspond to the trivial representations, and \(\Gamma(X, \mathcal{L}) = \mathcal{L}_x^{\pi_1(X,x)}\).
For the double cover \(f \colon S^1 \to S^1\), \(z \mapsto z^2\), the pushforward \(f_* \ul{\QQ}\) is the local system of rank \(2\) whose monodromy sends the generator of \(\pi_1(S^1) = \ZZ\) to the swap of the two coordinates of \(\QQ^2\). It decomposes as the constant local system \(\ul{\QQ}\) plus the rank-one local system with monodromy \(-1\), which has no nonzero global sections.
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