On 5/27/2012 11:00 PM, FEJ Linton wrote:
Virtually no one ever wants to restrict attention to functions that respect (preserve or reflect) membership (other than "preserve" between ordinals).
Wouldn't that depend on the context in which membership arises? Certainly group homomorphisms aren't expected to respect membership of group elements in groups, but neither are they expected to respect the composition of group homomorphisms equipping the category Grp. The latter kind of respect is accorded categories by functors between them. By the same token the former kind is accorded elementary models of set theory by elementary homomorphisms between them, the appropriate counterpart of functors in that context. Elementary homomorphisms preserve elementary structure, which in the case of models of set theory has membership as a basic part. ZFC structure is very different (at the bottom few layers) from the algebraic-in-Grph structure of the category Set, which has function composition as a basic part. For those who prefer algebra to logic, Joyal, Moerdijk and Awodey offer Algebraic Set Theory, AST, as a middle ground here. This replaces membership by (set-sized) unions and singleton a |--> {a}. Homomorphisms then have their usual algebraic meaning, which is arguably less fiddly than for elementary maps. Union allows the subset relation to be defined as X <= Y iff X U Y = Y, from which one can then define membership X e Y as {X} <= Y. Both relations are preserved by the homomorphisms of AST. For a crash course see Awodey's http://www.andrew.cmu.edu/user/awodey/preprints/astIntroFinal.pdf ZFC, Set, and AST differ only at the bottom few layers, above which foundational variations are tied to more fundamental issues involving Choice vs. Determinacy etc. One might compare the differences at the bottom with the wave-particle dichotomy in quantum mechanics or the event-state and time-information dichotomies in concurrency that I spoke on at Physics & Computation 1992 and 1994, see http://boole.stanford.edu/pub/ph94.pdf and the earlier (1992) http://boole.stanford.edu/pub/ql.pdf Or at least that's how it all looks to this outsider. Happy to be corrected on details I've got wrong. Vaughan Pratt [For admin and other information see: http://www.mta.ca/~cat-dist/ ]