Sheaf Representations and Duality in Logic

TL;DR

This paper explores the application of sheaf representations and duality theories to logic through algebraization and categorification, revealing new completeness results and connections between logic and geometry via topos theory.

Contribution

It introduces a novel perspective on logic by applying sheaf representations and duality, extending classical theories to categorical logic and topos theory.

Findings

New completeness theorems in logic derived from duality theories

Establishment of a connection between logic and geometry via categorification

Application of sheaf representations to algebraic structures in logic

Abstract

The fundamental duality theories relating algebra and geometry that were discovered in the mid-20th century can also be applied to logic via its algebraization under categorical logic. They thereby result in known and new completeness theorems. This idea can be taken even further via what is sometimes called ``categorification'' to establish a new connection between logic and geometry, a glimpse of which can also be had in topos theory.