$\mu$-Abstract Elementary Classes and other generalizations

TL;DR

This paper introduces $0$-Abstract Elementary Classes ($0$-AECs), a broad framework in model theory that encompasses structures like boolean algebras and Dirichlet series, and explores their classification theory.

Contribution

It defines $0$-AECs, connects them to accessible categories with monomorphisms, and initiates their classification-theoretic analysis.

Findings

$0$-AECs include various mathematical structures.

Classification results for $0$-AECs transfer to accessible categories.

Preliminary classification theory for $0$-AECs is developed.

Abstract

We introduce $\mu$-Abstract Elementary Classes ($\mu$-AECs) as a broad framework for model theory that includes complete boolean algebras and Dirichlet series, and begin to develop their classification theory. Moreover, we note that $\mu$-AECs correspond precisely to accessible categories in which all morphisms are monomorphisms, and begin the process of reconciling these divergent perspectives: not least, the preliminary classification-theoretic results for {\mu}-AECs transfer directly to accessible categories with monomorphisms.