On Sat, Mar 8, 2008 at 2:45 PM, Colin McLarty <colin.mclarty@case.edu> wrote:
What I would really like to know is, can one formulate an elementary property of a topos which *does* allow one to reproduce the standard arguments of Replacement?
Yes, What you do is start with ETCS, and adjoin an axiom scheme of replacement. [...]
Thank you! This is exactly what I was looking for.
This has been known from the earliest days of categorical set theory.
But it doesn't seem to be *well* known any more, or at least well-disseminated and exposited. Several people have told me that they didn't think it was possible to express replacement category-theoretically without using a category of classes. And even now knowing what I'm looking for, I am unable to find more than a sentence or two about it in any book on topos theory, none of which actually gives any version of the axiom.
AUTHOR = "McLarty, Colin", TITLE = "Exploring Categorical Structuralism",
This raises another question. You mention at the end of this paper that large-cardinal axioms are "routinely pursued in isomorphism-invariant terms". This is clear to me for many types of large cardinals, but not for the stronger ones that involve elementary embeddings of the universe of sets. Ultrapowers have a categorical analogue, of course (filterquotient) but then there is a transitive collapse of the entire universe, from which I don't see immediately how to eliminate the global membership predicate. Can you give a clue or a reference? Thanks again, Mike