When Lawvere meets Peirce: an equational presentation of boolean hyperdoctrines

TL;DR

This paper establishes a correspondence between fo-bicategories and Lawvere's hyperdoctrines, providing an equational and complete proof system for first-order logic through categorification.

Contribution

It introduces peircean bicategories to simplify the characterization of fo-bicategories and connects them to hyperdoctrines, advancing categorical logic.

Findings

Fo-bicategories form a proof system for first-order logic

Peircean bicategories offer a more succinct characterization

A correspondence between fo-bicategories and hyperdoctrines is established

Abstract

Fo-bicategories are a categorification of Peirce's calculus of relations. Notably, their laws provide a proof system for first-order logic that is both purely equational and complete. This paper illustrates a correspondence between fo-bicategories and Lawvere's hyperdoctrines. To streamline our proof, we introduce peircean bicategories, which offer a more succinct characterization of fo-bicategories.