Representation of fundamental group and flat connections I read Differential Geometry Of Complex Vector Bundles by Kobayashi, and he says there that a vector bundle $E$ has flat connection is equivalent to $E$ being defined by a representation of $\pi_1$. But he doesn't prove this. There are any suggestion how to start to prove this, or anyone has reference with a proof of this statement (I couldn't find any such reference).
 A: Complete and elementary proofs of this fact  may be found in detail in the following books:

*

*Morita's "Geometry of Characteristic Classes" Section 2.1.4 pg. 55 or

*Taubes' "Differential Geometry" Section 13.9.1 pg. 162.

A: The category of representations of the fundamental group $π_1$ can be replaced by the equivalent category of representations of the fundamental groupoid, i.e., functors $\def\Vect{{\sf Vect}} π_{≤1}(M)→\Vect$, where $\Vect$ is the category of complex vector spaces and $π_{≤1}(M)$ is the fundamental groupoid of $M$.
Now one can define a functor from the category of vector bundles to the category of functors $π_{≤1}(M)→\Vect$: send a vector bundle $E→M$
to the functor $π_{≤1}(M)→\Vect$ that maps $m∈M$ to the fiber $E_m∈\Vect$ and a homotopy class of paths $m→m'$ to the parallel transport map $E_m→E_{m'}$,
which is well defined because the connection on $E$ is flat.
Observe that the functor defined above is natural in $M$
and defines a morphism of stacks of categories over the site of smooth manifolds.
To show that a morphism between two such stacks is a weak equivalence of stacks, it suffices to show that for $\def\R{{\bf R}} M=\R^n$ (respectively ${\bf C}^n$ if we are working over complex manifolds) we get an equivalence of categories.
Indeed, the category of complex vector bundles with a flat connection on $\R^n$ is equivalent to the category $\Vect$ of vector spaces via the functor that takes the fiber over $0∈\R^n$.
Likewise, the category of functors $π_{≤1}(\R^n)→\Vect$
is equivalent to the category $\Vect$ of vector spaces
via the functor that evaluates at $0∈π_{≤1}(\R^n)$.
