On colimits and elementary embeddings

TL;DR

This paper improves existing theorems on colimit preservation between categories of structures by providing sharper results and new proofs utilizing set-theoretic methods involving large cardinals, rather than traditional category-theoretic techniques.

Contribution

It offers a sharper version of a known theorem and a novel proof using elementary embeddings and large cardinal assumptions, advancing the understanding of colimit preservation.

Findings

Sharper version of Rosicky, Trnkova, and Adamek's theorem

New proof using set-theoretic arguments with large cardinals

Enhanced understanding of colimit preservation in categories of structures

Abstract

We give a sharper version of a theorem of Rosicky, Trnkova and Adamek, and a new proof of a theorem of Rosicky, both about colimit preservation between categories of structures. Unlike the original proofs, which use category-theoretic methods, we use set-theoretic arguments involving elementary embeddings given by large cardinals such as alpha-strongly compact and C^(n)-extendible cardinals.