From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5798 Path: news.gmane.org!not-for-mail From: Andre Joyal Newsgroups: gmane.science.mathematics.categories Subject: Re: calculus, homotopy theory and more (corrected) Date: Thu, 13 May 2010 22:53:33 -0400 Message-ID: References: <4BE81F26.4020903@dm.uba.ar> <4BEC846B.5050000@ics.mq.edu.au> Reply-To: Andre Joyal 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: dough.gmane.org 1273843667 4379 80.91.229.12 (14 May 2010 13:27:47 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 14 May 2010 13:27:47 +0000 (UTC) To: Original-X-From: categories@mta.ca Fri May 14 15:27:45 2010 connect(): No such file or directory 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 4.69) (envelope-from ) id 1OCuvZ-0006pf-0X for gsmc-categories@m.gmane.org; Fri, 14 May 2010 15:27:45 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OCuQt-0002b8-R9 for categories-list@mta.ca; Fri, 14 May 2010 09:56:03 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5798 Archived-At: Dear Michael, A basic ingredient in my approach to higher categories is the notion of complete Segal space introduced by Rezk. I have learned Rezk theory in proving the Quillen equivalence=20 between quasi-categories and complete Segal spaces. In my "Notes on Quasi-categories" I am introducing an abstract notion of complete Segal space called *Rezk category*, or *reduced category*. A category object (internal to a quasi-category) is said=20 to be *reduced* if its object of objects is=20 *isomorphic* to its object of isomorphisms via the unit map.=20 (an isomorphism in a quasi-category is an arrow which is invertible in the homotopy category). An ordinary category (in set) is reduced iff every isomorphism is a unit, a very stringent condition. Ordinary categories are seldom reduced (posets are). Every reduced category is skeletal. An equivalence between reduced categories is necessarly=20 an isomorphism. In contrast, there are plenty of=20 reduced categories in homotopy theory. In fact every category internal to the quasi-category of spaces is *equivalent* to a reduced category (via a fully faith ess surj = functor). This key result was proved by Rezk for complete Segal spaces: every Segal category is *equivalent* to a complete Segal space. The theory of reduced categories is essentially (homotopy) algebraic=20 (unlike ordinary category theory in which we need to expand the notion=20 of isomorphism (of categories) with that of equivalence). I do not have the time to explain more of the idea of my proof now.=20 A sketch can be found in my "Notes on Quasi-categories". You wrote: >I can not reproduce it in this post because it requires some=20 >pictures. But I remember, Andre, we discussed it with you in Montreal = in >2004. I'll be happy to explain it again in Genoa. =20 I hope I will understand this time! I always find our conversation very stimulating! See you in Genoa, Andr=E9 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]