Classifying spaces and the Lascar group

TL;DR

This paper establishes a natural isomorphism between the Lascar group of a first order theory and the fundamental group of the classifying space of its models, linking model theory with algebraic topology.

Contribution

It introduces a novel connection between the Lascar group in model theory and the fundamental group of a topological space derived from models.

Findings

Lascar group is isomorphic to the fundamental group of the classifying space

Provides a topological interpretation of the Lascar group

Bridges model theory and algebraic topology

Abstract

We show that the Lascar group $\operatorname{Gal}_L(T)$ of a first order theory $T$ is naturally isomorphic to the fundamental group $\pi_1(|\operatorname{Mod}(T)|)$ of the classifying space of the category of models of $T$ and elementary embeddings.