Supercompact cardinals, elementary embeddings and fixed points

Abstract

Supercompactness is usually defined in terms of the existence of certain ultrafilters. By the well-known procedure of taking ultrapowers of V (the universe of sets) and transitive collapses, one obtains transitive inner models of V and corresponding elementary embeddings from V into these inner models. These embeddings have been studied extensively (see, e.g. [3] or [4]). We investigate the action of these embeddings on cardinals. In particular, we establish a characterization, based upon cofinality, of which cardinals are fixed by these embeddings.