This is a question based on the heuristics that most things in algebraic/differential topology has an analogue in algebraic geometry.
The fundamental group classifies the covering spaces of a (pointed, connected, path connected, semilocally simply connected topological) space. First we construct the universal cover as a space of paths and equip it with compact-open topology. And then we use the action of the fundamental group on the cover to define the space as a quotient. And by actions of the subgroups, we construct all the covering spaces, which are in one-one correspondence with the conjugacy classes of subgroups of fundamental group of the pointed space.
The question is, up to what extent can we carry over this setup to algebraic geometry? There are the appropriate notions of etale coverings and etale fundamental groups already, and by GAGA for smooth algebraic varieties one can already see some hope.
So, are there algebraic geometric analogues of the theorem on existence of universal cover and the theorem classification of covering spaces in one-one correspondence with conjugacy classes of subgroups of the fundamental groups?
Best Answer
Yes, (and of course): The very definition of the étale fundamental group is that it classifies finite étale covers.
Precisely: Let $X$ be a connected scheme and $x$ be a geometric point of $X$. There is by construction an equivalence of categories between finite $\pi_1^{\mathrm{ét}}(X,x)$--sets and finite étale coverings of $X$, connected coverings corresponding to transitive sets. This property characterises $\pi_1^{\mathrm{ét}}(X,x)$ up to unique isomorphism.
A universal cover as in topology also exists, although only as a profinite cover.
A fine source for this is T. Szamuely's book "Fundamental groups and Galois groups" (Cambridge 2009).