It seems to me that there is an important distinction here that is not being emphasized. Isomorphisms of categories can be *technically* quite useful. Knowing that a given equivalence of categories is an isomorphism, rather than merely an equivalence, can certainly make things much simpler, or even make things possible that we didn't know how to do before. Many examples of this sort have been mentioned. Another that should be added to the list is the theory of strict 2-categories, 2-limits, 2-adjoints, and so on, all of which is defined using ordinary enriched category theory over Cat, and hence involves many isomorphisms of hom-categories. However, I find that in most or all of these examples, one is not actually interested in the fact of an isomorphism of categories for its own sake. There is no "real meaning" in the fact that two categories are isomorphic, rather than equivalent; generally it's a technical accident of how we chose to define them. It may be a very *convenient* technical accident, but it is an accident nonetheless. If we chose our definitions differently, or worked in a different foundational system (such as one where the notion of "isomorphism of categories" cannot even be defined), some or all of our isomorphisms of categories might change to become only equivalences, but I don't believe that any of the real content of category theory would go away; it would just become harder to prove. Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

On Mon, May 31, 2010 23:26, Joyal, André wrote:

Dear category theorists,

I have not been able to define the notion of Grothendieck fibration without using the equality relation between the objects of the base category. Can you?

In a dependent language (eg ML, FOLDS, etc), this can be done. The twist is that it is _not_ just some extra algebraic structure on a functor p: \E --> \C; for an arbitrary such p, as you say, the lifting conditions can't be specified without referring to equality of objects. Rather, it first of all requires the objects and arrows of \E to be dependent types over the objects of \C; then, the "equalities" needed become part of the _typing specification_ of a lifting. This refinement of the definition complicates the theory a little, but is a quite natural requirement, I think. (Just as in this setting, a fibration of types is no longer an arbitrary map f:A --> B, but rather a dependent type A over B.) ===== Precisely, you can define it as follows: Say \C is a category (i.e. a type C_0, and a dependent type C_1(a,b), for a,b: C_0, an "equality" type on C_1, and appropriate composition/identity operations). The a (cloven) Grothendieck fibration \E over \C is: a dependent type E_0(a), for a: C_0; a dependent type E_1(a,b;i,j), for a,b: C_0, i:E_0(a), j:E_0(b) (we can write just E_1(i,j), since a,b are implicit in the types of i,j); operations making this a category over E_0; and an operation which, given an arrow a,b:C_0, f:C_1(a,b), and a lift j:E_0(b) of the target, returns liftings i:E_0(a) and \bar{f}:E_1(i,j), with p(\bar{f}) = f, and appropriately cartesian. (The definition of cartesian similarly doesn't need to refer to equality of objects, since they're included in the typings.) ===== If we were dealing with higher categories, and hence didn't have equality on C_1, then we would require E_1 to be a dependent type not just over a,b:C_0 and i:E_0(a), j:E_0(b), but also over f:C_1(a,b), so the condition that \bar{f} is a lift of f would again be part of its typing. This is reminiscent of latching/matching objects etc... Cheers, -Peter. -- Peter LeFanu Lumsdaine Carnegie Mellon University [For admin and other information see: http://www.mta.ca/~cat-dist/ ]