My question: Let X be a sufficiently 'nice' topological space. Then there is an equivalence between representations of the fundamental group of X and local systems on X, i.e. sheaves on X locally isomorphic to a constant sheaf. Does anyone know of a self contained, detailed treatment of this suitable for my background? I've looked at the first few pages of Delignes "Équations différentielles à points singuliers réguliers" (which my advisor suggested I take a look at) but here it just says that the equivalence is "well known", giving no reference. Neither googling ("local systems representations fundamental group") (nothing usable comes up), wiki nor the nLab entry (not detailed anough and more interested in generalisation) on local systems were of much help to me. I apologise in case the equivalence should obvious once one knows about universal covering spaces/deck transformations. I haven't learned those yet. If so, please let me know.

Why I care: I am trying to read Simpsons "Higgs bundles and local systems",. Publ. Math. I. H. E. S. 75 (1992) 5–95". Simpson assumes this equivalence but gives no references.

My background: I am a Diplom (roughly equivalent to MSc) student at a german university. For my Diplom thesis I am aiming to understand Narasimhan and Seshadris Theorem (from "Stable and unitary vector bundles on a compact Riemann surface") which I think roughly states that a holomorphic vector bundle of degree zero on a compact Riemann surface X of genus g ≥ 2 is stable if and only if it arises from an irreducible unitary representation of the fundamental group of X. I also hope to read and understand parts of Hitchins paper "The Self-Duality Equations on a Riemann Surface" (Proc. London Math. Soc. 1987 s3-55: 59-126) and as mentioned above parts of Simpsons article "Higgs bundles and local systems".

If there is any way this question could be improved upon, please let me know.

Feel free to retag.