I am as skilled a category therist as I am a dentist. That having been said, I shall walk out on a limb here. Category Theory attempts to capture the notion of thingies and functions that map thingies to thingies, in the greatest possible generality. (Yes, I know that the thingies in the range might be of a different sort than the thingies in the domain.) If Category Theory could not capture the category of "sets and functions", it would fail at the outset. So a category can't be restricted to anything as paltry as a set. Any approach that starts out by saying that a category is a kind of set, is pretty much doomed. On Mon, Jul 13, 2009 at 11:22 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
On 7/14/09, sctfen@gmail.com <sctfen@gmail.com> wrote:
My understanding is that it depends deeply on the version of set theory you're using. If you're using something like ZFC without large cardinal axioms, categories and morphisms are proper classes. If you're using something like Tarski-Grothendeick set theory, then most (but not all) categories and morphisms can be put into sets. The exception seems to be the category Set, which would appear to be a class in any set theory you're working with. The workaround most people use is "the category of all sets smaller than some strongly inaccessable cardinal," which gives you a model of ZFC, but not of all the sets possible in your particular set theory.
Ouch! --- when am I ever going to learn _not_ to go looking beneath stones ... Perhaps the approach outlined below might circumvent these complications? WFL
<< An alternative, suggested by Lawvere in the early sixties, is to develop an adequate language and background framework for a category of categories ... the basic idea is to define what are called weak n-categories (and weak ω-categories), and what had been called categories would then be called weak 1-categories (and sets would be weak 0-categories) >> [from the Stanford article, by Jean-Pierre Marquis].
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun