I am working on a topic related to the Riemann-Hilbert Correspondence, and I was wondering if there is a reference that explicitly goes over the following equivalence of categories (when $M$ is a smooth connected manifold for example) : $$\text{Fun}(\Pi_1, \text{Vect}_{\mathbb{R}}^{\text{f.d.}})\simeq \text{Sh}_{\text{Loc-Cst}}(M, \text{Vect}_\mathbb{R}^{\text{f.d}}) \simeq \text{Cat of Vector Bun. Over M with Flat Connections}$$

where the morphisms in the third category are connection preserving.

If possible, I would also like to know about any references that thoroughly go over the connection (no pun intended) between the notion of parallel transport and the notion of a (Ehresmann, Koszul) connection, as well as how parallel transport is homotopy invariant for a flat connection. Here is what I've been looking at:

  1. Geometry, Topology, and Physics; 2nd edition Nakahara: This has been very helpful, and I think this is about the closest to what I would like.

  2. Principal Bundles, the Classical Case; Sontz: This got me started, and has a lot of good material. However, this and the book above are quite physics based and it's been a bit since I've had to look at it that way.

  3. Differential Geometry; Bundles, Connections, Metrics and Curvature ; Taubes: This has been less useful to me, apart from helping me with the construction of an associated bundle.

I've been using the universal covering space of $M$ as a principal bundle so I can get associated vector bundles corresponding to functors out of the fundamental groupoid, but it would be nice to see it elsewhere.


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