Accessible images revisited

TL;DR

This paper improves the understanding of the conditions under which the powerful image of accessible functors is accessible, reducing large cardinal assumptions needed and linking to tameness in Abstract Elementary Classes.

Contribution

It lowers the large cardinal assumptions required for the accessibility of powerful images of accessible functors and connects this to tameness in Abstract Elementary Classes.

Findings

Reduced large cardinal assumptions to $L_{\mu,\omega}$-compact cardinals.

Proved the accessibility of the {\\lambda}-pure powerful image of F.

Linked tameness of Abstract Elementary Classes to weaker set-theoretic assumptions.

Abstract

We extend and improve the result of Makkai and Par\'e that the powerful image of any accessible functor F is accessible, assuming there exists a sufficiently large strongly compact cardinal. We reduce the required large cardinal assumption to the existence of $L_{\mu,\omega}$-compact cardinals for sufficiently large {\mu}, and also show that under this assumption the {\lambda}-pure powerful image of F is accessible. From the first of these statements, we obtain that the tameness of every Abstract Elementary Class follows from a weaker large cardinal assumption than was previously known. We provide two ways of employing the large cardinal assumption to prove each result - one by a direct ultraproduct construction and one using the machinery of elementary embeddings of the set-theoretic universe.