A computable functor from graphs to fields

Abstract

Fried and Kollár constructed a fully faithful functor from the category of graphs to the category of fields. We give a new construction of such a functor and use it to resolve a longstanding open problem in computable model theory, by showing that for every nontrivial countable structure S, there exists a countable field F of arbitrary characteristic with the same essential computable-model-theoretic properties as. Along the way, we develop a new computable category theory, and prove that our functor and its partially defined inverse (restricted to the categories of countable graphs and countable fields) are computable functors.