\(\mu\)-Abstract Elementary Classes and other generalizations

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.