Duality theory for categorical theories

TL;DR

This paper extends duality theory to categorical theories in model theory, establishing a correspondence between certain pretopoi and profinite monoids, and exploring their geometric properties via topoi.

Contribution

It generalizes categorical theory notions to coherent theories and proves a duality between categorical pretopoi and profinite monoids, linking topos theory and geometry.

Findings

Established a duality between categorical pretopoi and profinite monoids.

Identified profinite monoids as a full sub-2-category of topoi.

Explored the geometry of profinite monoids through classifying topoi.

Abstract

We have generalised the notion of categorical theory in model theory to the context of coherent theories. We prove a duality result between the full sub-2-category of pretopoi which are categorical, and the 2-category of profinite monoids. We also study the geometry of profinite monoids via the classifying topos construction, and show it identifies them as a full sub-2-category of the 2-category of topoi.