On Mar 7, 2008, at 5:18 AM, Michael Shulman wrote:
One categorical analogue of replacement comes from categories of classes in algebraic set theory. That is, we move from a categorical analogue of ZF to an analogue of Godel-Bernays set theory. But it seems natural to wonder whether there could be a categorical analogue of replacement expressible solely as a property of the category Set, without reference to how it sits in a category of classes. Has anyone studied this question?
yes: Carsten Butz, Thomas Streicher, Alex Simpson and I did. See the first two items under 2007 on the AST site: http://www.phil.cmu.edu/projects/ast/ The short answer is, it depends on how "Sets" sits in the category of classes. In fact, *any* topos can occur as a category of "Sets" satisfying replacement in a suitable category of classes constructed from the topos. Steve Awodey