Standard conjectures in model theory, and categoricity of comparison isomorphisms: A model theory perspective

We formulate two conjectures about étale cohomology and fundamental groups motivated by categoricity conjectures in model theory. One conjecture says that there is a unique (Formula presented.) -form of the étale cohomology of complex algebraic varieties, up to (Formula presented.) -action on the source category; put differently, each comparison isomorphism between Betti and étale cohomology comes from a choice of a topology on (Formula presented.) Another conjecture says that each functor to groupoids from the category of complex algebraic varieties which is similar to the topological fundamental groupoid functor (Formula presented.) in fact factors through (Formula presented.) up to a field automorphism of the complex numbers acting on the category of complex algebraic varieties. We also try to present some evidence towards these conjectures, and show that some special cases seem related to Grothendieck standard conjectures and conjectures about motivic Galois group.