Thanks to everyone for their replies. I provide some quick clarifications to some of Patrik's comments below: ``You now introduce the word "core", and I would need to understand what that mathematically and logically really means, at least intuitively.'' I mean ``core'' in the trivial sense. The axioms defining what it is to be a category (of some kind) are just first-order. The same is true, of course, for ZFC. However, in the set theory context it seems at least sensible to try and provide a second-order (quasi) categoricity proof, despite the diversity of models for first-order ZFC (and indeed we have one by the work of Zermelo, Shepherdson, etc.). To attempt the same for category theory would seem to be an absurd strategy---the point of categories is that they can be instantiated in many non-isomorphic models. ``Yes, indeed, across diverse contexts, with "context" somehow related to "system" in Hidekazu Iwaki's reply involving "mathematical system".'' I cannot find Hidekazu Iwaki's reply (this is rather strange, I do not know why). However it seems like similar usage to me: I don't have a formal definition of `mathematical context', but the point is just that there's diverse non-isomorphic models where the same categorial properties pop up. e.g. (somewhat obviously for this list, but I found it striking learning some category theory) the notion of topos appears all over the place in diverse structures, and allows us to have a reasonable notion of internal logic. Your comments on the different notions of hierarchy are interesting, but I will have to think about this some more before I have anything reasonable to add. Best Wishes, Neil -- Dr. Neil Barton Postdoctoral Research Fellow Kurt Gödel Research Center for Mathematical Logic University of Vienna [For admin and other information see: http://www.mta.ca/~cat-dist/ ]