Universal abstract elementary classes and locally multipresentable categories

TL;DR

This paper establishes categorical equivalences linking universal classes and locally multipresentable categories, as well as AECs with intersections and locally polypresentable categories, enhancing understanding of Shelah's presentation theorem.

Contribution

It introduces categorical equivalences that connect model-theoretic and category-theoretic frameworks for abstract elementary classes and related structures.

Findings

Equivalence between universal classes and locally multipresentable categories.

Equivalence between AECs with intersections and locally polypresentable categories.

Insights into Shelah's presentation theorem for AECs.

Abstract

We exhibit an equivalence between the model-theoretic framework of universal classes and the category-theoretic framework of locally multipresentable categories. We similarly give an equivalence between abstract elementary classes (AECs) admitting intersections and locally polypresentable categories. We use these results to shed light on Shelah's presentation theorem for AECs.