μ-Abstract Elementary Classes and other generalizations

We introduce µ-Abstract Elementary Classes (µ-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 µ-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 µ-AECs transfer directly to accessible categories with monomorphisms.