Locally constant sheaves of finite rank

Let $S$ be a topological space (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison), let $k$ be a field and let $\mathcal{F}$ be a sheaf of $k$-vector spaces on $S$ (A sheaf on a topological space, Presheaves and sheaves of groups, rings, and modules). Then $\mathcal{F}$ is locally constant of finite rank if every point $x\in S$ has an open neighbourhood $U\subseteq S$ for which there are an integer $n\ge 0$ and an isomorphism of sheaves on $U$ $\mathcal{F}{\mathrm{\mid }}_U\cong k_U^n,$ where $k_U$ is the constant sheaf on $U$ with value $k$ and $\mathcal{F}{\mathrm{\mid }}_U$ is the restriction of $\mathcal{F}$ to the open subspace $U$ (Restriction of a sheaf to an open subspace, Morphisms of presheaves, The constant sheaf is the sheaf of locally constant functions). A sheaf with this property is also called a finite-rank local system of $k$-vector spaces on $S$. This is a property of the sheaf on the space; no identification of such sheaves with functors on the fundamental groupoid is part of the definition.

The sheaf $k_U^n$ is the $n$-fold direct sum of $k_U$ in the category of sheaves of $k$-vector spaces, with $k_U^0$ the zero sheaf; its sections over an open $W\subseteq U$ are the locally constant functions $W\to k^n$. The integer $n$ is quantified locally, so no global rank is asserted and the rank may differ from point to point; the definition asserts no existence of a global trivialization. The final sentence records only that a functorial groupoid presentation of local systems is a different, uncompared object here.