From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9249 Path: news.gmane.org!.POSTED!not-for-mail From: Thomas Streicher Newsgroups: gmane.science.mathematics.categories Subject: Re: Re: Functors arising from a relational Grothendieck construction Date: Sat, 24 Jun 2017 10:37:59 +0200 Message-ID: References: Reply-To: Thomas Streicher NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: blaine.gmane.org 1498594316 1223 195.159.176.226 (27 Jun 2017 20:11:56 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Tue, 27 Jun 2017 20:11:56 +0000 (UTC) Cc: categories@mta.ca To: Luc Pellissier Original-X-From: majordomo@mlist.mta.ca Tue Jun 27 22:11:52 2017 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.40]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1dPwq0-0008WD-Dg for gsmc-categories@m.gmane.org; Tue, 27 Jun 2017 22:11:52 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:48539) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1dPwqu-0002Eb-G6; Tue, 27 Jun 2017 17:12:48 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1dPwpY-0006k6-Jl for categories-list@mlist.mta.ca; Tue, 27 Jun 2017 17:11:24 -0300 Content-Disposition: inline In-Reply-To: <5B931A70-3299-433D-89AC-7DFA8627CC2B@lipn.univ-paris13.fr> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9249 Archived-At: > a variant of the one in (Nielsen 2004, TAC 12(7), pp 248???261), but > consider You mean Niefield and not Nielsen. This paper makes clear the relation to Giraud-Conduch'e functors. But they just study the Weak Factorization Lifting Property (WFLP) which in terms of distributors means that all components of the natural transformation corresponding to lax preservation of composition are surjective. Faithful functors to B reflecting identities correspond to "relational variable sets" on B as described in the Niefield paper. But they are NOT Conduch'e fibrations since they just validate WFLP and not FLP. p : E -> B is a Conduch'e fibration (i.e. validatates FLP) iff it is exponentiable in Cat/B but p validates WLFP iff it is exponentiable in Cat_f/B (Cor.4.2 in Niefield paper). What Niefield calls Grothendieck construction is an instance of the transition from a lax normalised functor from B^op to Dist to a functor to B (due to Benabou). But this has nothing to do with what you describe as Grothendieck construction which rather is chanke of base along a functor B* -> B. Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]