Dear Ross, I'm sure you know what a locally small fibration P: X --> S is. You also know that this is a very strong notion from which many many results can be derived, especially if you make very mild assumptions on S, e.g. that S has pullbacks. In your mail you give a the following definition:
When a class C of small objects has been distinguished in a finitely complete 2-category K, an object x of K is locally small when each cospan a : u --> x <-- v : b with u and v small has a small comma object a/b. It is only seldom that R has an object w such that this gives a logically equivalent definition by restricting to u = v = w.
Best wishes, Ross
Could you please tell me: (i) Given a category S how does one chose the finitely complete 2- category K and the class C of small objects so that the locally small objects of K in your sense, are the locally small fibrations over S ? (ii) What can you prove about the locally small objects of K, especially since you assume nothing on C ? (iii) What significant mathematical examples can you give of your notion ? Best wishes, Jean [For admin and other information see: http://www.mta.ca/~cat-dist/ ]