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 $X$.)