Mike Barr quoted a "thought of Chairman Pratt" that was attributed to him,
Monotone functions respect order, group homomorphisms respect the group operation, linear transformations respect linear combinations, and gangsters respect membership in the Cosa Nostra, but what morphism has ever respected membership in a set? It is sheer hubris for a relation that can't get no respect to claim to support all of mathematics. (Old argument of category theorist Mike Barr, new polemics.)
and commented on it,
That is not a bad rendition (save for the reference to Cosa Nostra) of what I actually said which was that we create these elaborate structures of well-founded trees subject to the rule that two chidren of the same leaf cannot be isomorphic. But then, unlike all other structures that we build, we make no hypothesis that functions preserve the structure. Indeed, I think a structure-preserving map must be the inclusion of a subset. And there are no non-identity endomorphisms.
Of course, I entirely agree with the sentiment that epsilon-structures are completely inappropriate as a basis of most ordinary mathematics. However, mathematics and mathematicians are contrAry beasts, who treat any statement of the form "there is no such thing as ..." as a challenge, whether it be about membership-preserving functions or the square root of -1. Indeed, it's quite interesting to look at "carriers equipped with membership relations" in the same way as "carries equipped with group multiplications". This is what I did in my paper "Intuitionistic Sets and Ordinals", JSL 61 (1996). For a more categorical treatment, we may regard the relation as a coalgebra structure for the full covariant powerset functor. This is what Gerhard Osius did in his "Categorical Set Theory" in JPAA 4 (1974) and what I did for other functors in my unpublished paper "Towards a Uniform Treatment of Induction - the General Recusion Theorem" in 1995-6. (This was presented at "Category Theory 1995" in Cambridge and part of it appeared in Section 6.3 of my book.) As Mike Barr says, and Gerhard Osius proved in his paper, for "extensional" structures ("well-founded trees subject to the rule that two > chidren of the same leaf cannot be isomorphic"), the structure-preserving map must be a subset inclusion. However, without extensionality, it is a COALGEBRA HOMOMORPHISM. Well founded coalgebras behave like fragments of the initial algebra (the von Neumann hierarchy, in the case of the powerset functor), so their rigidity (lack of endomorphisms) is related to the uniqueness of homorphisms out of the initial algebra. The sense in which coalgebra homomorphisms are like partial algebra homomorphisms is explored in the early sections of my unpublished paper. Osius's recursion scheme has attracted some attention in recent years amongst functional programmers as a way of describing recursive programs. See "Recursive Coalgebras from Comonads" by Venanzio Capretta, Tarmo Uustalu and Varmo Vene, in "Information and Computation" 2006. In fact, the description also works for imperative programs - see Section 2.5 of my book. I have a longer survey of this subject that I intend to publish on "categories" later this month. MY WEB PAGES AT www.cs.man.ac.uk/~pt While I'm here, I'd like to draw your attention to some new things that I have put on my web pages recently. * The slides that I have used at recent conferences and seminars. * The unpublished paper and scanned transparencies of 1995-6 talks on well founded coalgebras. * A new web page for my book, "Practical Foundations of Mathematics", including where to buy it, errata, who has used or cited it, etc. If you have used it in a lecture or seminar series, please send me a URL and your experiences. * The full text of Jean-Yves Girard's "Proofs and Types". * A new version of my TeX package for "commutative diagrams", together with an explanation of the "PostScript" mode, why the "pure DVI" mode is strongly deprecated (NB those long-standing users who have spoiled their otherwise excellent books, papers and online journals by using it), and how to overcome its uglier features if you really insist on using it. * A collection of other TeX macros. * Scanned manuscripts of my 1983 Cambridge Part III Essay (= MSc thesis) (see "domain theory") and undergraduate algebra lecture notes. Paul Taylor