Universal categories

TL;DR

The paper explores the categorical properties of models of first-order theories, revealing that elementarily equivalent small categories embed into model categories, and introduces a logic ensuring categorical equivalence implies elementary equivalence.

Contribution

It demonstrates a universal embedding property for model categories of any theory and proposes a first-order logic aligning categorical and elementary equivalences.

Findings

Any small category elementarily equivalent to a model category embeds into it.

Ultrapower techniques are used to prove the embedding property.

A new first-order logic is proposed where categorical equivalence implies elementary equivalence.

Abstract

The category of models of any theory $T$ in any first-order language $L$ has the surprising property that any small category that is elementarily equivalent with it, already embeds in it. The proof uses an abstract argument via ultrapowers, leaving one wonder which concrete categorical axioms, depending on $T$ and $L$, are responsible for this embedding result. We also propose a first-order logic for which equivalent categories are always elementarily equivalent.