A Universe of Sorts

Given equations in $A$ , can find solutions in any $B$ such that we have $\phi: A \to B$
Can translate topological ideas to geometry.
Fundamental theorem of riemann: fundamental group with finitely many covering becomes algebraic (?!)
So we can look at finite quotients of the fundamental group.
As variety, we take line minus one point. This can be made by considering $xy - 1 = 0$ in $R[x, y]$ and then projecting solutions to $R[x]$ .
If we look at complex solutions, then we get $\mathbb C - \{0 \} = C^\times$ .
The largest covering space is $\mathbb C \xrightarrow{\exp} \mathbb C^\times$ . The fiber above $1 \in C^\times$ (which is the basepont) is $2 \pi i$ .
Finite coverings are $C^\times \xrightarrow{z \mapsto z^n} C^\times$ . The subsitute for the fundamental group is the projective (inverse) limit of these groups.
The symmetry of $Gal(\overline{\mathbb Q} / \mathbb Q)$ acts on this fundamental group.
One can get not just fundamental group, but any finite coefficients!
Category of coverings is equivalent to category of sets with action of fundamental group.
Abel Prize: Pierre Delinge

§ Hidden Symmetries of Alg Varieties