From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10329 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: Uwe Egbert Wolter Newsgroups: gmane.science.mathematics.categories Subject: Discrete fibrations vs. functors into Set Date: Wed, 02 Dec 2020 14:53:30 +0100 Message-ID: Reply-To: Uwe Egbert Wolter Mime-Version: 1.0 Content-Type: text/plain; charset=US-ASCII; format=flowed Content-Transfer-Encoding: 7bit Injection-Info: ciao.gmane.io; posting-host="blaine.gmane.org:116.202.254.214"; logging-data="11735"; mail-complaints-to="usenet@ciao.gmane.io" To: categories list Original-X-From: majordomo@rr.mta.ca Thu Dec 03 03:36:09 2020 Return-path: Envelope-to: gsmc-categories@m.gmane-mx.org Original-Received: from smtp2.mta.ca ([198.164.44.74]) by ciao.gmane.io with esmtps (TLS1.2:ECDHE_RSA_AES_256_GCM_SHA384:256) (Exim 4.92) (envelope-from ) id 1kkeTY-0002tG-MZ for gsmc-categories@m.gmane-mx.org; Thu, 03 Dec 2020 03:36:08 +0100 Original-Received: from rr.mta.ca ([198.164.44.159]:44340) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1kkeOU-0002F3-FM; Wed, 02 Dec 2020 22:30:54 -0400 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1kkeRg-0000Mp-6S for categories-list@rr.mta.ca; Wed, 02 Dec 2020 22:34:12 -0400 Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10329 Archived-At: 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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]