dom fibration

dom fibration

[Adam Eppendahl]

Can anyone explain why the codomain fibration cod: C^\rightarrow -> C,
which requires pull-backs, gets loads of attention, while the domain
fibration dom: C^rightarrow -> C, which works for all C, hardly gets a
look in?  Is the dom fibration really such a poor relation?

Adam Eppendahl

Re: dom fibration

[Ross Street]

>Can anyone explain why the codomain fibration cod: C^\rightarrow -> C,
>which requires pull-backs, gets loads of attention, while the domain
>fibration dom: C^rightarrow -> C, which works for all C, hardly gets a
>look in?  Is the dom fibration really such a poor relation?

I have a couple of points to make about this.

	1) By duality, the "poor relation" fibration dom can be 
viewed as the opfibration cod.  So it is not so poor after all, just 
part of an even richer structure of cod.

	2) Fibrations over  C  "amount" to pseudofunctors  C^op --> 
Cat.  Let  S    be the pseudofunctor  S(u) = C/u  on objects, using 
pullback on arrows; this is the "rich guy".  The "poor guy"  T  is a 
covariant functor (not just pseudo)  T(u) = C/u ,  using composition 
on arrows.  To get a contravariant Cat-valued functor on  C,  follow 
T  by the presheaf construction  P : Cat^coop --> Cat.  It turns out 
then that the Yoneda embedding gives a fully faithful pseudonatural 
transformation  y :  S --> PT.  Now  PT  is a very important 
character; every internal full subcategory of the topos  E  of 
presheaves on  C  is a full subobject of  PT.  (A good example is 
where  E  is globular sets and  PT  is the globular category of 
higher spans.)  It is true that  S  can be thought of as an internal 
model of  E  in  Cat(E)  leading to indexed (or parametrized) 
category theory.  But the reason this works well is that S is a full 
subobject of  PT. Again  T  wins out!

Regards,
Ross