Suppose I have an allergy to non-Hausdorff spaces but I really want to do, say, arithmetic geometry. I wonder if there is some perverse way I could develop scheme theory that would accomodate for my needs.

I am not sure what is the best way to formalize this question but here is an attempt. Does there exist a functor from schemes to the category of Hausdorff topological spaces/continuous maps that is one of the following

- full and faithful
- faithful and admits a left adjoint
- full and admits a left adjoint
- faithful and admits a right adjoint
- full and admits a right adjoint?

P.S. I believe that there is a faithful functor from schemes to sets (take the union of the underlying set and all the stalks and do some power-set acrobatics) and there is a faithful functor from sets to Hausdorff spaces (discrete topology) so at least a faithful functor should exist.