A Computable Functor From Graphs to Fields
Logic
2015-10-27 v1 Category Theory
Number Theory
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.
Cite
@article{arxiv.1510.07322,
title = {A Computable Functor From Graphs to Fields},
author = {Russell Miller and Bjorn Poonen and Hans Schoutens and Alexandra Shlapentokh},
journal= {arXiv preprint arXiv:1510.07322},
year = {2015}
}