From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10330 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: Newsgroups: gmane.science.mathematics.categories Subject: Re: Discrete fibrations vs. functors into Set Date: Thu, 3 Dec 2020 09:53:23 +0100 Message-ID: References: Reply-To: Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 8bit Injection-Info: ciao.gmane.io; posting-host="blaine.gmane.org:116.202.254.214"; logging-data="14459"; mail-complaints-to="usenet@ciao.gmane.io" Cc: To: Uwe Egbert Wolter Original-X-From: majordomo@rr.mta.ca Fri Dec 04 03:54:14 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 1kl1Eb-0003YY-NA for gsmc-categories@m.gmane-mx.org; Fri, 04 Dec 2020 03:54:13 +0100 Original-Received: from rr.mta.ca ([198.164.44.159]:44502) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1kl18q-0002CN-3c; Thu, 03 Dec 2020 22:48:16 -0400 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1kl1C6-00039o-Sp for categories-list@rr.mta.ca; Thu, 03 Dec 2020 22:51:38 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10330 Archived-At: 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 > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]