A global approach to AECs

TL;DR

This paper introduces categorical and topological frameworks for Abstract Elementary Classes (AECs), defining morphisms, constructions, and properties to deepen the understanding of AECs in model theory.

Contribution

It presents new categorical notions, constructions, and axioms for AECs, including morphisms, gluings, and properties like TRP and GRP, linking them to model-theoretic concepts.

Findings

Defined a natural notion of morphism between AECs

Introduced the category of Grothendieck's gluings of AECs

Proved the equivalence of TRP and GRP, and their relation to Craig interpolation

Abstract

In this work we present some general categorial ideas on Abstract Elementary Classes (AECs) %\cite{She}, inspired by the totality of AECs of the form $(Mod(T), \preceq)$, for a first-order theory T: (i) we define a natural notion of (funtorial) morphism between AECs; (ii) explore the following constructions of AECs: "generalized" theories, pullbacks of AECs, (Galois) types as AECs; (iii) apply categorial and topological ideas to encode model-theoretic notions on spaces of types %(see Michael Lieberman Phd thesis) ; (iv) present the "local" axiom for AECs here called "local Robinson's property" and an application (Robinson's diagram method); (v) introduce the category $AEC$ of Grothendieck's gluings of all AECs (with change of basis); (vi) introduce the "global" axioms of "tranversal Robinson's property" (TRP) and "global Robinson's property" (GRP) and prove that TRP is equivalent to GRP and GRP entails a natural version of Craig interpolation property.