From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5513 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: A challenge to all Date: Tue, 12 Jan 2010 20:28:54 -0600 Message-ID: References: Reply-To: Michael Shulman 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 1263397846 18177 80.91.229.12 (13 Jan 2010 15:50:46 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 13 Jan 2010 15:50:46 +0000 (UTC) To: joyal.andre@uqam.ca, categories@mta.ca Original-X-From: categories@mta.ca Wed Jan 13 16:50:38 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 1NV5U3-0004cn-SC for gsmc-categories@m.gmane.org; Wed, 13 Jan 2010 16:50:12 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NV5Di-0002us-Ls for categories-list@mta.ca; Wed, 13 Jan 2010 11:33:18 -0400 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5513 Archived-At: Dear Andre, You are absolutely right that the equality relation is inseparable from the idea of a set. What is being proposed, however, is that a category doesn't need to have a *set* of objects. In fact, the objects of a category don't need to form an object of any category at all, so I think your proposed test is misguided. The formulation of category theory in dependent type theory which Richard, Toby, I, and others are proposing makes perfect sense without any equality predicate for the objects. Best, Mike Joyal wrote: > Dear All, >=20 > I cannot imagine a category without an equality relation between the ob= jects of this category. > Ok, I may have been brainwashed by my training in mathematics at an ear= ly age. > But more seriously, I think that the equality relation is inseparable=20 > from the idea of a set. I do not understand what a preset is: >=20 > http://ncatlab.org/nlab/show/preset >=20 > Two things are equal if they are the same, if they coincide (whatever t= hat mean!). > Without this notion, an element of a set has no identity, no individual= ity. > Of course, a set is often constructed from other sets,=20 > as in arithmetic with congruence classes.=20 > I am fully aware that the equality relation between the objects of a=20 > 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.=20 >=20 > I would like to propose a test for verifying if the=20 > notion of category can be freed from the equality relation > on its set of objects. The equality relation on an ordinary=20 > set S is defined by the diagonal S-->S times S. > The objects of a symmetric monoidal category have no diagonal in genera= l, > ie no coalgebra structure. >=20 > 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? >=20 >=20 > Best, Andr=E9=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]