Fix a topological space $X$. Now consider a functor from the fundamental groupoid of $X$ to the category $Vect$. In other words, we assign a vector space to each point of $X$, we allow ourselves to flow our vectors around our space, and this flow is consistent with respect to homotopy.

This seems to give us a vector bundle. Is this true? Also, can we get all vector bundles this way?

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    $\begingroup$ You cannot get all vector bundles in this way. $\endgroup$ – Alex Degtyarev Jun 1 '15 at 7:18

This does give you a vector bundle, and it comes with a flat (i. e. path homotopy invariant) parallel transport. You cannot get all vector bundles, but you can get exactly the ones with such a parallel transport. (On a manifold, those are exactly the ones that admit a flat connection)

An easy way to justify this is that those vector bundles have structure group $O(n)$ with the discrete topology, so their classifying space is $K(O(n), 1)$. But maps into a K(G,1) factor through the 1-type, i. e. the fundamental groupoid.

  • $\begingroup$ An explicit obstruction to admitting a flat connection is that, by Chern-Weil theory, all of the real Pontryagin classes must vanish. $\endgroup$ – Qiaochu Yuan Jun 1 '15 at 3:23

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