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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:

  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 $X$.)
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    $\begingroup$ For sensible spaces, vector bundles with discrete structure group are a good candidate $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Sep 28, 2015 at 2:36

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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.

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    $\begingroup$ I disagree with Qiaochu: it's not "a bundle whose transition functions can be taken to be locally constant". That would be a bundle with a property, whereas flat vector bundles are bundles equipped with extra structure. One way to describe that structure is that there's a certain distinguished class of local sections which are singled out, which one calls the "flat sections". The flat sections should form a locally constant sheaf, and the stalks of that sheaf should be the fibers of the vector bundle. $\endgroup$
    – André Henriques
    Commented Oct 1, 2015 at 20:26
  • $\begingroup$ @Andre: my apologies for being imprecise. Of course the issue is that I haven't said what a morphism of flat vector bundles is, and of course the correct notion is not the same as a morphism of underlying vector bundles; I think you also need a local constancy condition here and then things are fine? The discrete structure group definition is maybe clearer (in particular, it's also clear in this formulation that flatness is a structure rather than a property) but I wanted to avoid a formulation which doesn't allow vector bundles to jump in rank across different components. $\endgroup$
    – Qiaochu Yuan
    Commented Oct 2, 2015 at 2:05
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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.

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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 $f\colon U \to \mathbb C$).

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    $\begingroup$ I don't understand this argument. Doesn't this just amount to the (true) remark that "Another name for a flat vector bundle is a local system"? $\endgroup$
    – Tom Church
    Commented Sep 29, 2015 at 0:08
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    $\begingroup$ @Tom: I think pro wants to identify a local system with its sheaf of flat sections but wants to identify a flat vector bundle with its sheaf of continuous sections. $\endgroup$
    – Qiaochu Yuan
    Commented Sep 29, 2015 at 3:35
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    $\begingroup$ @TomChurch I am not saying anything deep. I like sheaves: while the local system L is a Q-sheaf of finite rank, the trivial vector bundle C_X is infinite dimensional. When you don't have Kahler differentials this is the only way I know to say that a vector bundle is flat (this and by using representations of $\pi_1$ as QY said). $\endgroup$
    – pro
    Commented Sep 29, 2015 at 4:50

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