Wednesday, March 12, 2008 11:37 pm Subject: Re: categories: Re: replacing set theory Writes of the replacement scheme in 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.
There are two issues here, because there are two things to mean by "replacing set theory." On one hand it can mean replacing the membership-theoretic approach of ZF by the categorical approach of ETCS. This has long been routine and full details are found in Osius "Categorical Set theory: A Characterization of the Category of Sets", (Journal of Pure and Applied Algebra, 1974, pages 79--119). People do not talk about it a lot because it was a well-focussed problem which got a perfectly good answer and at the time there more pressing and more open-ended research issues. Perhaps also because Saunders Mac~Lane tended to stress that lots of higher set theory (like lots of logic in general) is very weakly linked to most of mathematics -- which is true, but is not to say that higher set theory (or logic in general) need be abandoned. He hoped they could be brought back into better touch. This is the issue raised for example by Feferman and Rao in Giandomenico Sica ed. _What is category theory?_ {Polimetrica, 2006) when they claim it is "unclear" whether certain ZF constructions can be given at all in categorical terms. Yes, it is clear they can, proven in detail by Osius in 1976 (not the first proof but my favorite reference on it). On the other hand, for some purposes we want to replace the category Set (no matter how it is axiomatized) by something more general, often by any elementary topos, or it could be any Boolean topos, or category with a category of classes. The categorical replacement scheme I mentioned generalizes very well to any well-pointed topos, but that is little more general than Set. It makes much use of fibers over global elements. To put it in the terms I like, it does nothing to say a family of sets S-->I defined by replacement should be "smooth" or "continuous over the index set I" in anyway. In Set the index *set* is not smooth or continuous to begin with, it is a discrete set. The idea of categories of classes is to get some sense of large collections that *do* vary "smoothly" or "continuously" when the base has some smooth or continuous character (in a very general sense, so for example effective computability is the relevant "smoothness" for those types in the effective topos close to the natural numbers). The next question brings us back to the first perspective. It is about the category Set, but wants to approach that category by categorical tools.
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?
Within ZF itself the global membership predicate X \in S is just one guise of the relation "the membership-tree of X is (isomorphic to) the restriction of the membership-tree of S to some node directly below the top." Transitive collapse can be re-cast in these terms as dealing with all well-founded extensional relations and not only dealing with restricted membership (i.e. restricted to some inner model of ZF) and actual membership of sets. In isomorphism invariant terms, a transitive collapse of the universe means a uniform method of taking any well-founded extensional relation (not just ZF membership on each transitive closure) and restricting it to a sub-relation which is still well-founded and extensional (which we do not bother collapsing to membership on some transitive closure) while preserving some properties of the first relation. And then we ask if a given collapse has any fixed points: are there any well-founded extensional relation so big that this collapse leaves an isomorphic relation? I have no idea what the practical effect would be of recasting collapse in these terms. I have never worked with large cardinals and transitive collapse. But it certainly *can* be recast this way. Some one should try it. best, Colin