Local system of groups

Definition

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

Facts

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.