Is there a notion of "flat vector bundle over a topological space"? I am reading this paper and at the top of page 5 the author makes reference to categories consisting of flat complex vector bundles over $X$ where $X$ is an arbitrary topological space. However, the only notion of flat vector bundle which I have seen uses a connection on the vector bundle, which in turn requires that $X$ have a cotangent bundle, hence $X$ must be a smooth manifold.
My questions are therefore as follows:


*

*What is the definition of a flat vector bundle over a topological space?

*What is the action of $\pi_1(X)$ on such a vector bundle? (I need to understand $H^\bullet(X;\text{Hom}_X(V,W))$ for flat vector bundles $V,W$ over $X$.)

 A: A flat vector bundle over a topological space is a bundle whose transition functions can be taken to be locally constant; equivalently, over a path-connected space, it's the same data as a principal $G$-bundle ($G = GL_n(\mathbb{R})$ or $GL_n(\mathbb{C})$ as appropriate) where $G$ is given the discrete topology. Over a reasonable space $X$ this is the same thing as a functor from the fundamental groupoid $\Pi_1(X)$ to vector spaces. 
A: According to Kamber-Tondeur (1967), a principal $G$-bundle over a space $X$

is flat if it is induced from the universal covering bundle of $X$ by a homomorphism $\pi_1X\to G$. In the differentiable case this is equivalent to the existence of a connection with curvature zero [15, Lemma 1].
(...)
A vector bundle is called flat, if its associated principal bundle is flat.

A: A way to rephrase QY's answer is to say that a vector bundle E is flat if there is a local system L such that $L \otimes_{\mathbb{C}} C_X \simeq E$ where $C_X$ is the trivial vector bundle (or, said differently, $C_X$ is the bundle whose sections on an open U are given by $C_X(U)$ - the continuous functions $f\colon U \to \mathbb C$).
