etale homotopy type of is entirely determined by its etale homotopy group. Note where is a geometric point. Further topics From a category-theoretic point of view, the fundamental group is a functor {Pointed algebraic varieties} → {Profinite groups}. The inverse Galois problem asks what groups can arise as fundamental groups (or Galois groups of field extensions). Anabelian geometry, for example Grothendieck's section conjecture, seeks to identify classes of varieties which are determined by their fundamental groups. studies higher étale homotopy groups by means of the étale homotopy type of a scheme. The pro-étale fundamental group have introduced a variant of the étale fundamental group called the pro-étale fundamental group. It is constructed by considering, instead of finite étale covers, maps which are both étale and satisfy the valuative criterion of properness. For geometrically unibranch schemes (e.g., normal schemes), the two approaches agree, but in general the pro-étale fundamental group is a finer invariant: its profinite completion is the étale fundamental group. See also étale morphism Fundamental group Fundamental group scheme Notes References Scheme theory Topological methods of algebraic geometry