Bicategories, Biequivalence, and Bi-Interpretability
Abstract
We make explicit the correspondence between syntax and syntactic categories for coherent first-order logic, providing a categorical characterization of bi-interpretability. This is done by creating a biequivalence between a bicategory of coherent theories and the (strict) bicategory of coherent categories. While the biequivalence concerns the stronger equality-preserving bi-interpretability, we use it to obtain a necessary and sufficient condition for two theories to be bi-interpretable in general, by relating the exact completions of their syntactic categories. These results extend analogously to familiar fragments of first-order logic, thereby clarifying the long-intuited relation between logical syntax and syntactic categories.
Keywords
Cite
@article{arxiv.2011.14056,
title = {Bicategories, Biequivalence, and Bi-Interpretability},
author = {Anthony D'Arienzo and Vinny Pagano and Ian M. J. McInnis},
journal= {arXiv preprint arXiv:2011.14056},
year = {2023}
}
Comments
60 pages, 4 figures; Major rewrite of article reformulating theorems in bicategorical framework, removing nonessential theorems, further contextualizing results, and making more explicit constructions and proofs