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categories: Re: dom fibration
>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!