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 metric spaces, 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: for example, the preliminary classification-theoretic results for μ-AECs transfer directly to accessible categories with monomorphisms.