Homotopy Types of Abstract Elementary Classes

TL;DR

This paper demonstrates that for any given homotopy type, there exists an abstract elementary class whose classifying space has that homotopy type, illustrating a deep connection between homotopy theory and model theory.

Contribution

It constructs specific abstract elementary classes that realize any homotopy type as their classifying space, establishing a novel link between these mathematical areas.

Findings

Any homotopy type can be realized by an AEC's classifying space.

Constructs explicit examples of such AECs.

Shows the richness of AECs in modeling topological spaces.

Abstract

We prove that for any homotopy type $X$, there is an abstract elementary class $\mathcal{C}$, with joint embedding, almagamation and no maximal models such that the classifying space realizes the homotopy type $X$. We provide a few explicit examples.