A Computable Functor From Graphs to Fields

Abstract

We construct a fully faithful functor from the category of graphs to the category of fields. Using this functor, we resolve a longstanding open problem in computable model theory, by showing that for every nontrivial countable structure S, there exists a countable field F with the same essential computable-model-theoretic properties as S. 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.