Fundamental group of a topos nLab
From review of Brief an G. Faltings, in Geometric Galois actions, LMS Lecture notes 242. Outline of Groth philosophy of anabelian geometry. For anabelian schemes, the geometry of X should be determined by the profinite fundamental group, together with a homomorphism from it to the absolute Galois group of the base field. The kernel of this homomorphism is the geometric fundamental group of X, which is the completion of the topological fundamental group, in case K is a subfield of the complex numbers. The property of being anabelian should be geometric, i.e. depend only on the base change to alg closure. A connected nonsingular curve in char zero is anabelian iff its geometric fundamental group is not abelian. In char p, look at its maximal prime to p quotient instead.
Donu Arapura - Purdue University Title: Hodge structure on the fundamental group revisited. Abstract: Given a CW complex X with a map from its fundamental group to a group G, one gets a map X to K(G,1), which induces a map from the cohomology of G to X. When X is a variety, I would like to discuss a Hodge theoretic/motivic analogue of this, where G would be the group associated to the Tannakian category of variations of mixed Hodge structure/motivic sheaves on X. While I’m at it, I’d like to compare this “Hodge structure” on the fundamental group with those of Hain, Morgan, and Simpson.
nLab page on Fundamental group