Elementary equivalences and accessible functors

TL;DR

This paper introduces $oldsymbol{ ext{λ-}} ext{equivalence}$ and $oldsymbol{ ext{λ-}} ext{embeddings}$ in categories, extending classical model-theoretic notions to categorical contexts and applications in algebraic geometry.

Contribution

It defines new categorical notions of $ ext{λ-}$equivalence and $ ext{λ-}$embeddings, generalizing classical model theory and connecting to functorial properties in algebraic geometry.

Findings

Introduces $ ext{λ-}$equivalence and $ ext{λ-}$embeddings in categories.

Extends results of Feferman and Eklof on local functors.

Facilitates formalization of Lefschetz's principle in algebraic geometry.

Abstract

We introduce the notion of $\lambda$-equivalence and $\lambda$-embeddings of objects in suitable categories. This notion specializes to $L_{\infty\lambda}$-equivalence and $L_{\infty\lambda}$-elementary embedding for categories of structures in a language of arity less than $\lambda$, and interacts well with functors and $\lambda$-directed colimits. We recover and extend results of Feferman and Eklof on "local functors" without fixing a language in advance. This is convenient for formalizing Lefschetz's principle in algebraic geometry, which was one of the main applications of the work of Eklof.