Re: Discrete fibrations vs. functors into Set

[<streicher@mathematik.tu-darmstadt.de>]

See pp.16-17 of my notes on fibered cats available from the arxiv.
There is an obvious functor Set^(_) : cat^op -> Cat to which one can apply
the Grothendieck construction.

Moreover, a cartesian functor is a fibered equivalence iff all its fibers
are ordinary equivalences. All this is folklore and just documented in my
notes.

Thomas

> We consider two categories. The first category with objects given by a
> small category B and a functor F:B->Set and morphisms
> (H,alpha):(B,F)->(C,G) given by a functor H:B->C and a natural
> transformation alpha:F=>H;G. The second category has as objects discrete
> fibrations p:E->B and morphisms (H,phi):(E,p)->(D,q:D->C) are given by
> functors H:B->C and phi:E->D such that phi;q=p;H.
>
> 1. Are there any "standard" terms and notations for these categories?
> 2. For both categories we do have projection functors into Cat! Are
> these functors kind of (op)fibrations?
> 3. We know that the Grothendieck construction establishes equivalences
> between corresponding fibers of the two projection functors into Cat. Do
> these fiber-wise equivalences extend to an equivalence between the two
> categories?
>
> Thanks
>
> Uwe
>




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