CAUTION: The Sender of this email is not from within Dalhousie. In a non-discrete setting these categories and questions have been considered in part in our recent paper on ???Diagrams, fibrations, and the decomposition of colimits??? with George Peschke: arXiv:2006.10890v1[math.CT] Among other things, the paper extends results obtained by Rene??? Guitart who considered categories of diagrams in some of his papers in the Cahiers of the early 1970s. Regards, Walter From: Uwe Egbert Wolter <Uwe.Wolter@uib.no> Date: December 2, 2020 at 9:37:06 PM EST To: categories list <categories@mta.ca> Subject: categories: Discrete fibrations vs. functors into Set Reply-To: Uwe Egbert Wolter <Uwe.Wolter@uib.no> Dear all, 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 ---1530486996-297531643-1607096310=:1789-- [For admin and other information see: http://www.mta.ca/~cat-dist/ ]