Accessible images revisited
arXiv:1506.01986
Abstract
We extend and improve the result of Makkai and Paré 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 -compact cardinals for sufficiently large μ, and also show that under this assumption the λ-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.
14 pages; amended after referee's comments, mostly to improve exposition. We have changed terminology for the large cardinal axiom - this is discussed in some depth after Definition 2.3