nLab Chevalley fundamental group

Idea

The term Chevalley fundamental group is sometimes used for the fundamental group of a space defined via the automorphisms of a universal cover, hence its étale fundamental group (see there for more). In algebraic geometry/arithmetic geometry this is essentially the absolute Galois group

This terminology is used by Borceux and Janeldize in their book on Galois Theories. To quote from that source:

‘’ There are two classical definitions of the fundamental group of a topological space which give isomorphic groups for certain ‘’good’‘ spaces.“

The Chevalley fundamental group $Aut(p)= Aut(E,p)$ is defined only for connected spaces $B$ which admit a universal covering map $p: E\to B$ with connected $E$, and, of course, depends on it, but again, different $p$ produce isomorphic groups.’‘

The term algebraic fundamental group is also sometimes used for this, although more usually that term is reserved for Grothendieck’s fundamental group of a scheme.

References

• Francis Borceux, George Janelidze, Galois Theories, Cambridge Studies in Advanced Mathematics 72, Cambridge University Press, 2001.

• C. Chevalley, Theory of Lie groups, Princeton University Press, 1946.