A categorical approach to internality

TL;DR

This paper develops a categorical framework for internality, extending model theoretic concepts to category theory and establishing analogous results about automorphism groups in this broader context.

Contribution

It introduces a categorical analogue of internality and proves a corresponding theorem, generalizing model theoretic results to category theory.

Findings

Categorical internality conditions lead to definable automorphism groups.

The model theoretic result is recovered via the category of definable sets.

Provides a new perspective linking model theory and category theory.

Abstract

Model theoretic internality provides conditions under which the group of automorphisms of a model over a reduct is itself a definable group. In this paper we formulate a categorical analogue of the condition of internality, and prove an analogous result on the categorical level. The model theoretic statement is recovered by considering the category of definable sets.