From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5494 Path: news.gmane.org!not-for-mail From: burroni@math.jussieu.fr Newsgroups: gmane.science.mathematics.categories Subject: Re: Small is beautiful Date: Thu, 07 Jan 2010 23:24:51 +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=UTF-8;DelSp="Yes";format="flowed" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1262917929 903 80.91.229.12 (8 Jan 2010 02:32:09 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 8 Jan 2010 02:32:09 +0000 (UTC) To: Zinovy Diskin Original-X-From: categories@mta.ca Fri Jan 08 03:32:02 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 1NT4dt-0002H0-MH for gsmc-categories@m.gmane.org; Fri, 08 Jan 2010 03:32:01 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NT4Ft-0003jV-Bl for categories-list@mta.ca; Thu, 07 Jan 2010 22:07:13 -0400 In-Reply-To: Content-Disposition: inline Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5494 Archived-At: Dear categorists, The question for me is not : small or large categories, but small or =20 large structures. The Bourbaki's time was the time of small structures (groups, mono=C3=AFds, = =20 rings, spaces, etc), the "categorical time" is the time of structures =20 of structures,i.e. large structures (groupoids, complete categories, =20 abelian cat=C3=A9gories, topos, etc.). The bridge beetwen the firsts and the seconds is the Yoneda lemma, =20 that is to say the introduction of logic, thus of the only true evil =20 category : the category of sets. All the other large categorical =20 stuctures introduced by categorists are deduced from category Set, by =20 abstraction, generalisations, constructions or restrictions. In fact =20 "category theory" is an (wonderfull but) inappropriate name (here the =20 word category is only an important and historical keyword): the reason =20 is that a lot of data (limits, classifiant object, etc) appear as =20 properties because of their universal properties (unicity up to =20 isomorphism). But "category theory" is an illusion, nobody studies =20 seriouly the categorical structures (I don't know any structure =20 theorem on the categorical structures without additional properties). That is my starting point of reflexion on this subject. I think there is a true theory of the small categories, but it is not =20 yet born, if it must ever exist. This theory should be, not only for =20 the 1-categories, but also for the n and omega-categories (strict or =20 not possibly). And, for me, the Yoneda lemma is an important tool (but =20 only a tool). Such a theory may be eventually important for the =20 computer science (particularly for the formal languages). My best wishes for the new year, Albert burroni [For admin and other information see: http://www.mta.ca/~cat-dist/ ]