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Working paper

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

Les Publications mathématiques de l’IHES. IHES/M/19/03. IHES, 2019
We formulate two conjectures about ´etale cohomology and fundamental groups motivated by categoricity conjectures in model theory. One conjecture says that there is a unique Z-form of the ´etale cohomology of complex algebraic varieties, up to AutˆC-action on the source category; put differently, each comparison isomorphism between Betti and ´etale cohomology comes from a choice of a topology on C. Another conjecture says that each functor to groupoids from the category of complex algebraic varieties which is similar to the topological fundamental groupoid functor π top 1 , in fact factors through π top 1 , 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 indicate their relation to Grothendieck standard conjectures and conjectures about the motivic Galois group.