I am reading [this paper](http://arxiv.org/pdf/0712.0160v3.pdf) 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:


1. What is the definition of a flat vector bundle over a topological space?
2. 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 $X$.)



Is there a notion of "flat vector bundle over a topological space"?
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. 


