From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5518 Path: news.gmane.org!not-for-mail From: burroni@math.jussieu.fr Newsgroups: gmane.science.mathematics.categories Subject: Re: A challenge to all Date: Wed, 13 Jan 2010 01:47:43 +0100 Message-ID: References: Reply-To: burroni@math.jussieu.fr NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset=ISO-8859-1;DelSp="Yes";format="flowed" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1263398098 19166 80.91.229.12 (13 Jan 2010 15:54:58 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 13 Jan 2010 15:54:58 +0000 (UTC) Cc: categories@mta.ca To: "Joyal, =?iso-8859-1?b?QW5kcuk=?=" Original-X-From: categories@mta.ca Wed Jan 13 16:54:51 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 1NV5YY-00074k-J4 for gsmc-categories@m.gmane.org; Wed, 13 Jan 2010 16:54:50 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NV5Bv-0002iC-Io for categories-list@mta.ca; Wed, 13 Jan 2010 11:31:27 -0400 In-Reply-To: Content-Disposition: inline Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5518 Archived-At: Dear Andr=E9, > 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? This is possible with this conception : A "monoid" in a monoidal category is what Benabou call *monad* in his =20 paper on bicat=E9gories. The more general notion of "category", in the =20 same vein, is the notion of *polyad* (in the same paper of Benabou --- =20 apologize me : I have not the reference whith me): the objets are =20 given externally. I don't know if my remark you satisfie, I think not (you have had =20 certainly yourself this idea), but, for me, the moral of this fact =20 seems to say that the equality of objets have something of =20 intrinsically very different of morphisms. (This meets perhaps the ideas of Bob Par=E9 on index categories) I am convided of this (If I have the occasion, I will explain this). Best, Albert "Joyal, Andr=E9" a =E9crit=A0: > Dear All, > > I cannot imagine a category without an equality relation between the =20 > objects of this category. > Ok, I may have been brainwashed by my training in mathematics at an =20 > 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 =20 > (whatever that mean!). > Without this notion, an element of a set has no identity, no individuality= . ... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]