Formal Model Theory & Higher Topology

TL;DR

This paper develops a categorical framework connecting bounded ionads, accessible categories, and topoi to advance formal model theory, providing reconstruction, completeness results, and relations to geometric theories.

Contribution

It introduces a unified categorical approach to formal model theory using 2-categories and relates it to topoi and geometric sketches, offering new reconstruction and completeness results.

Findings

Relation between bounded ionads and accessible categories.

Reconstruction and completeness results in categorical framework.

Connection of abstract elementary classes to locally decidable topoi.

Abstract

We study the $2$-categories BIon, of (generalized) bounded ionads, and $\text{Acc}_\omega$, of accessible categories with directed colimits, as an abstract framework to approach formal model theory. We relate them to topoi and (lex) geometric sketches, which serve as categorical specifications of geometric theories. We provide reconstruction and completeness-like results. We relate abstract elementary classes to locally decidable topoi. We introduce the notion of categories of saturated objects and relate it to atomic topoi.