CAUTION: The Sender of this email is not from within Dalhousie. 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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