From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5515 Path: news.gmane.org!not-for-mail From: Jeff Egger Newsgroups: gmane.science.mathematics.categories Subject: Re: A challenge to all Date: Tue, 12 Jan 2010 17:02:35 -0800 (PST) Message-ID: References: Reply-To: Jeff Egger 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 1263397891 18419 80.91.229.12 (13 Jan 2010 15:51:31 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 13 Jan 2010 15:51:31 +0000 (UTC) To: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= , categories@mta.ca Original-X-From: categories@mta.ca Wed Jan 13 16:51:24 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 1NV5V1-0005I3-V5 for gsmc-categories@m.gmane.org; Wed, 13 Jan 2010 16:51:12 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NV5D1-0002pn-CA for categories-list@mta.ca; Wed, 13 Jan 2010 11:32:35 -0400 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5515 Archived-At: Dear Andr=E9,=0A=0AI do not understand the point of your "test". =0A=0AWha= t Bob Par=E9 said, and which I agree with, is that equality =0Ais "okay" fo= r small categories. And as Paul Taylor wrote, =0Aby "small" what one reall= y means is "internal". =0A=0ASo, of course, it makes sense that V-internal= categories =0A(where V is a not-necessarily-braided monoidal category with= =0Adistributive coreflexive equalisers) should have a comonoid =0Aof objec= ts. =0A=0ABut this in no way contradicts the assertion that _large_ =0Acat= egories should not have an equality relation between =0Aobjects---internal = categories are tautologously small! =0A=0ANot being as familiar with index= ed categories/fibrations as=0AI ought to be, I tend to think of large categ= ories in terms =0Aof enriched category theory. Here, we see very clearly t= hat =0Athe collection of objects has nothing whatsoever to do with =0Athe e= nriching category V, and this is as it should be.=0A=0AIn fact, I suppose t= hat it probably would make sense to =0Ageneralise enriched categories by ta= king a (large) groupoid =0Aof objects (and _canonical_ isos, in the spirit = of Par=E9 and =0ASchumacher) instead of a mere class. I don't know if this= =0Ahas ever been done. =0A=0AMy main point is that you are right in asser= ting that a set =0Awithout an equality relation is not a set. But the exac= t =0Ameaning of large category is one whose objects do not =0Anecessarily f= orm a set! =0A=0AMorally speaking, "set" does mean "collection that has an= =0Aequality predicate", but this leaves open the possibility =0Athat there= are collections which do not have such a =0Apredicate, and which are there= fore not sets. These suffice =0Afor the purpose of large category theory--= -for example, they =0Asuffice for the purpose of enriched categories; moreo= ver, =0AFOLDS is explicitly based on these principles. =0A=0ACheers,=0AJeff= .=0A=0A----- Original Message ----=0A> From: "Joyal, Andr=E9" =0A> To: categories@mta.ca=0A> Sent: Tue, January 12, 2010 4:24:39 = PM=0A> Subject: categories: A challenge to all=0A> =0A> Dear All,=0A> =0A> = I cannot imagine a category without an equality relation between the object= s of =0A> this category.=0A> Ok, I may have been brainwashed by my training= in mathematics at an early age.=0A> But more seriously, I think that the e= quality relation is inseparable =0A> from the idea of a set. I do not under= stand what a preset is:=0A> =0A> http://ncatlab.org/nlab/show/preset=0A> = =0A> Two things are equal if they are the same, if they coincide (whatever = that =0A> mean!).=0A> Without this notion, an element of a set has no ident= ity, no individuality.=0A> Of course, a set is often constructed from other= sets, =0A> as in arithmetic with congruence classes. =0A> I am fully aware= that the equality relation between the objects of a =0A> category is not p= reserved by equivalences in general.=0A> But the art of category theory con= sists partly in knowing=0A> which construction on the objects and arrows of= =0A> a category is invariant under equivalences. =0A> =0A> I would like to = propose a test for verifying if the =0A> notion of category can be freed fr= om the equality relation=0A> on its set of objects. The equality relation o= n an ordinary =0A> set S is defined by the diagonal S-->S times S.=0A> The = objects of a symmetric monoidal category have no diagonal in general,=0A> i= e no coalgebra structure.=0A> =0A> The test: Can we define a notion of cate= gory internal to=0A> a symmetric monoidal category without using a coalgebr= a structure=0A> on the object of objects?=0A> =0A> =0A> Best, Andr=E9 =0A> = =0A> =0A> =0A> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]