From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5514 Path: news.gmane.org!not-for-mail From: David Roberts Newsgroups: gmane.science.mathematics.categories Subject: Re: A challenge to all Date: Wed, 13 Jan 2010 10:33:04 +1030 Message-ID: References: Reply-To: David Roberts NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1263397854 18220 80.91.229.12 (13 Jan 2010 15:50:54 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 13 Jan 2010 15:50:54 +0000 (UTC) To: =?ISO-8859-1?B?Sm95YWwsIEFuZHLp?= , categories@mta.ca Original-X-From: categories@mta.ca Wed Jan 13 16:50:46 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1NV5U0-0004aI-CA for gsmc-categories@m.gmane.org; Wed, 13 Jan 2010 16:50:08 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NV5B0-0002ac-3p for categories-list@mta.ca; Wed, 13 Jan 2010 11:30:30 -0400 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5514 Archived-At: Dear Andre, > The test: Can we define a notion of category internal to > a symmetric monoidal category without using a coalgebra structure > on the object of objects? the quickest reference I can find is the very incomplete page http://ncatlab.org/nlab/show/internal+category+in+a+monoidal+category but more importantly, the thesis referenced there. As I remark at the above page, Ross Street would know something about this. As to the other idea, someone (Toby Bartels perhaps) once said that explicitly equipping 'sets' with a notion of when elements are equal goes back to Cantor. I find thinking of the undergraduate (or high school) definition of equality of functions |R --> |R as a pale shadow of the idea of preset. Consider the collection of expressions denoting functions (even restricting to polynomials), this is a preset with the equality relation given by when one algebraic expression gives the same function as another algebraic expression. Best, David Roberts 2010/1/13 Joyal, Andr=E9 : > Dear All, > > I cannot imagine a category without an equality relation between the obje= cts of this category. > Ok, I may have been brainwashed by my training in mathematics at an early= age. > But more seriously, I think that the equality relation is inseparable > from the idea of a set. I do not understand what a preset is: > > http://ncatlab.org/nlab/show/preset > > Two things are equal if they are the same, if they coincide (whatever tha= t mean!). > Without this notion, an element of a set has no identity, no individualit= y. > Of course, a set is often constructed from other sets, > as in arithmetic with congruence classes. > I am fully aware that the equality relation between the objects of a > category is not preserved by equivalences in general. > But the art of category theory consists partly in knowing > which construction on the objects and arrows of > a category is invariant under equivalences. > > I would like to propose a test for verifying if the > notion of category can be freed from the equality relation > on its set of objects. The equality relation on an ordinary > set S is defined by the diagonal S-->S times S. > The objects of a symmetric monoidal category have no diagonal in general, > ie no coalgebra structure. > > The test: Can we define a notion of category internal to > a symmetric monoidal category without using a coalgebra structure > on the object of objects? > > > Best, Andr=E9 > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]