Representations of the Fundamental Group

olliemath

This is not important, but it's been bugging me for a while.
I'm struggling to see how the locally constant sheaves of vector spaces on X give rise to representations of the fundamental group of X.

The approach I've been thinking of is the following. Given a locally constant sheaf F on X, the associated topological space |F| is a covering space for X. Thus, given a loop [tex]\gamma[/tex] in X with base point x and a point y in [tex]F_x[/tex] we can lift to a uniqe curve [tex]\gamma'[/tex] in |F| with initial point y. Setting [tex]\gamma\cdot y=\gamma'(1)[/tex] we obtain an action of [tex]\pi(X,x)[/tex] on [tex]F_x[/tex] which is called the monodromy action. [tex]F_x[/tex] is a vector space, but I don't see how we know that the map [tex]y\mapsto\gamma\cdot y[/tex] is linear.
Or possibly this is not the right approach?

Any help is greatly appreciated - O

Hurkyl

You have to use the fact that you're working in a sheaf of vector spaces, not just a sheaf of sets where you've put a vector space structure on one of the fibers.