Let be a connected scheme. Recall that a finite étale cover of is a finite flat surjection such that each fibre at a point is the spectrum of a finite étale algebra? over the local ring at . Fix a geometric point .
For a finite étale cover, , we consider the geometric fibre, , over , and denote by its underlying set. This gives a set-valued functor on the category of finite étale covers of .
The algebraic fundamental group, is defined to be the automorphism group of this functor.
For more on this area, see at étale homotopy.
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or in a lengthier form:
and earlier version is to be found here.