If one assumes that epics split and that the Boolean algebra of subsets of 1 reduces to two elements, then (to a first approximation) the main effect of axiom schemes is to provide larger cardinals. That is explicitly exemplified in the document that has been available in the University of Chicago Math Library since 1965, and that has been available in recent years as a TAC Reprint. Using the definable classes of sets smaller than any given set, the postulate that there are arbitrarily large such classes closed under arbitrary definable operations phi is proposed. I am not aware of any further studies of that postulate. (This may be the first time that a geometer has shown persistent interest in replacement) Of course above I use the term "class" in the intuitive sense of a natural subset of every model of the theory ("meta-" in Mac lane's terminology), that objective meaning corresponding subjectively to a formula of the theory. That is, not in the sense of a half- hearted attempt to represent classes by elements V of the model. (Less half-hearted is the proposal that seems implicit in the 1963 reaction of Goedel & Bernays themselves when they heard, presumably from Kreisel, that work was underway on a categorical set theory. Namely try a category of classes of classes etc satisfying at least the intuitive property of cartesian closure.) The usual replacement scheme does seem at first to yield only cardinals smaller than given ones. But in the hierarchical view, that quotient set consists of elements which themselves have elements, thus the actual mathematical content is that of a family of sets, a concept whose geometrical expression is that of a fibration, hence Colin's formulation of the axiom. Indeed the formulation goes back many years, but I don't have a reference. Concerning an elementary self-embeddings of the universe, it is in any case an additional functor added to the basic structure, and since the basic structure of category is first-order, a scheme could be considered to the effect that such a functor commute with quantifiers. Bill On Wed Mar 12 23:37 , "Michael Shulman" sent:
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