From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8276 Path: news.gmane.org!not-for-mail From: Paul Levy Newsgroups: gmane.science.mathematics.categories Subject: Re: cleavages and choice Date: Sun, 3 Aug 2014 16:17:32 +0100 Message-ID: References: <20140730150643.GC19613@mathematik.tu-darmstadt.de> <89048344-26F3-448F-8B41-9FF89AE1C892@wanadoo.fr> Reply-To: Paul Levy NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (Apple Message framework v936) Content-Type: text/plain; charset=US-ASCII; format=flowed; delsp=yes Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1407156508 31309 80.91.229.3 (4 Aug 2014 12:48:28 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 4 Aug 2014 12:48:28 +0000 (UTC) To: Categories list Original-X-From: majordomo@mlist.mta.ca Mon Aug 04 14:48:23 2014 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1XEHgc-0004iK-DH for gsmc-categories@m.gmane.org; Mon, 04 Aug 2014 14:48:22 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:50061) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XEHfu-0007SY-Lm; Mon, 04 Aug 2014 09:47:38 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XEHft-0002nz-IG for categories-list@mlist.mta.ca; Mon, 04 Aug 2014 09:47:37 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8276 Archived-At: Related to Marco's point: In order to define the category Set, we have to define Set(X,Y) for all sets X and Y. There are many isomorphic options, but to get a category we have to choose one. Now replace "category" by "category with distinguished binary products and exponentials.". We shall then have to choose a particular implementation of X x Y and X -> Y. Why do some people find this philosophically objectionable? How is it worse than choosing Set(X,Y)? (Personally, I'd be inclined to make the same choice for X - > Y as for Hom(X,Y).) Paul On 2 Aug 2014, at 11:58, Marco Grandis wrote: > Dear Eduardo, > > I agree with many things in your message, but I think you are taking > your argument too far. > Talking of pullbacks you say: > >> We precisely teach in category theory courses that you should not >> work with any >> particular choice between the choices. > > I agree that it is better to avoid such a choice when possible. Yet > you cannot define a bicategory of spans without assuming that such a > choice has been made; in the same way as you cannot define the > (good) monoidal structure of Ab without recurring to a choice of > tensor products. > Such a situation, we all know, generally arises in non-strict > bicategories (and monoidal categories, in particular). > > Unless you want to redefine bicategories replacing the composition > of arrows with an existence property. I still prefer working with a > choice (eg of pullbacks) to such a complicated structure. > > Best regards > > Marco > -- Paul Blain Levy School of Computer Science, University of Birmingham +44 121 414 4792 http://www.cs.bham.ac.uk/~pbl [For admin and other information see: http://www.mta.ca/~cat-dist/ ]