A viewpoint on amalgamation classes

TL;DR

This paper introduces a comprehensive perspective on amalgamation classes by emphasizing morphisms as primitive and allowing structures of arbitrary size, expanding the classical theory of universal-homogeneous models beyond finite structures.

Contribution

It presents a self-contained approach to universal-homogeneous models that generalizes existing theories by considering arbitrary morphisms and structures of any cardinality.

Findings

Provides a foundational framework for amalgamation classes with arbitrary morphisms.

Extends classical theory to infinite structures.

Facilitates applications requiring non-embedding morphisms.

Abstract

We provide a self-contained introduction to the classical theory of universal-homogeneous models (also known as generic structures, rich models, or Fra\"iss\'e limits). In the literature, most treatments restrict consideration to embeddings among finite structures. This is not suitable for some applications. We take the notion of morphisms as primitive and we allow structures to have arbitrary cardinality.