Re: Equality and fibration

["Peter LeFanu Lumsdaine" <plumsdai@andrew.cmu.edu>]

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





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