Local system of groups

Category-theoretic definition
A local system of groups is a functor to the category of groups from the category whose:


 * Objects are points of the topological space
 * Morphisms are homotopy classes of paths in the topological space (a path is viewed as a morphism from the start point to the end point)

Long-hand definition
A local system of groups $$T$$ on a topological space $$X$$ is the following data:


 * For every point $$x \in X$$, a group $$T(x)$$
 * For every path $$\phi$$, a map $$T(\phi): T(\phi(0) \to T(\phi(1))$$

such that:


 * $$T$$ of the constant path is the identity map
 * $$T$$ of homotopic paths is the same
 * $$T$$ of the composite of paths is the composite of their $$T$$s

Fundamental groups form a local system
The fundamental groups at the points form a local system of groups.

Fundamental groups act on any local system
Observe that any loop at a point $$x$$ defines an automorphism on $$T(x)$$. Further, homotopy-equivalent loops give the same automorphism, and the composite of two loops gives the composite automorphism. Combining all these facts, we get a map:

$$\pi_1(X) \to Aut(T(X))$$

In other words, the local system of fundamental groups acts on any local system of groups, in a canonical way.

Note that the action of the local system of fundamental groups on itself is simply the action by conjugation.