From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9337 Path: news.gmane.org!.POSTED!not-for-mail From: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= Newsgroups: gmane.science.mathematics.categories Subject: RE: "op"_Fred_and_Thurston Date: Sat, 9 Sep 2017 04:33:28 +0000 Message-ID: References: , Reply-To: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1505137776 9407 195.159.176.226 (11 Sep 2017 13:49:36 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Mon, 11 Sep 2017 13:49:36 +0000 (UTC) To: "Eduardo J. Dubuc" , Emily Riehl , "categories@mta.ca" Original-X-From: majordomo@mlist.mta.ca Mon Sep 11 15:49:27 2017 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.40]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1drP5J-0001IK-Eb for gsmc-categories@m.gmane.org; Mon, 11 Sep 2017 15:49:09 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:38330) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1drP6V-0006Td-Tr; Mon, 11 Sep 2017 10:50:23 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1drP4q-0004RP-FX for categories-list@mlist.mta.ca; Mon, 11 Sep 2017 10:48:40 -0300 Thread-Topic: categories: "op"_Fred_and_Thurston Thread-Index: AQHTKNLnZ/dikNECu0Ouu0W0sUVHX6Kr29ZL In-Reply-To: Accept-Language: en-US, en-CA Content-Language: en-US Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9337 Archived-At: Dear Eduardo,=0A= =0A= Thank you for recalling this remarkable article by Thurston.=0A= It contains profound observations on the role of *communities* in the creat= ion of mathematics.=0A= Mathematical research is about developing *human understanding* of mathemat= ics.=0A= =0A= Thurston does not mention category theory.=0A= I remember trying to learn algebraic topology by reading the =0A= "Foundations of Algebraic Topology" by Eilenberg and Steenrod. =0A= It is a great book, but not the right place to learn the subject.=0A= I also tried to learn algebraic geometry by reading the =0A= "Elements de Geometrie Algebrique" by Grothendieck and Dieudonn=E9.=0A= I never became an algebraic-geometer.=0A= It is very difficult to learn anything without direct access to the people = who knows.=0A= =0A= Best,=0A= Andr=E9=0A= =0A= ________________________________________=0A= From: Eduardo J. Dubuc [edubuc@dm.uba.ar]=0A= Sent: Friday, September 08, 2017 12:03 PM=0A= To: Emily Riehl; categories@mta.ca=0A= Subject: categories: "op"_Fred_and_Thurston=0A= =0A= 1) Two days ago by chance I come across an article of Bill Thurston:=0A= =0A= https://arxiv.org/pdf/math/9404236.pdf=0A= =0A= and seeing his name mentioned in this thread it occurs to me that=0A= everybody in this list should read it. In my opinion it is an=0A= extraordinary document about mathematics, mathematical activity and=0A= mathematicians.=0A= =0A= 2) Respect to to subject of this thread, the formal opposite of a=0A= category, denoted "op", is simply a notation very useful to work with=0A= functors which are contravariant in some variables, either with the "op"=0A= in the domain or the codomain of the functor arrow.=0A= =0A= Notations are important, and the "op" notation is essential in the=0A= language of categories and functors.=0A= =0A= 3) Finally, concerning Fred Linton, his death sadness me, he did=0A= important work in the early days of category theory, but more important,=0A= he was one of us, it was always a pleasure to encounter him, an he was a=0A= good guy.=0A= =0A= all the best e.d.=0A= =0A= =0A= On 07/09/17 14:03, Emily Riehl wrote:=0A= >> There is one other anecdote about UACT, nothing to do with Fred, that I= =0A= >> have always loved. In the course of MSRI director Bill Thurston's=0A= >> opening remarks, he said words to the effect that the notion of the=0A= >> opposite of a category made him nauseous. This was the only meeting I=0A= >> have ever attended where fully half the attendees drew in enough breath= =0A= >> to drop the air pressure by an audible amount.=0A= >=0A= > I?ll confess that the idea of an opposite category appearing as the codom= ain of a functor also makes me somewhat nauseated (the domain of course is = no problem).=0A= >=0A= > But this said, in the interest of full disclosure, I should admit that in= a joint paper with Cheng and Gurski someone ? Eugenia, I believe? ? convin= ced us that the easiest way to think of a functor=0A= >=0A= > C x D ?> E=0A= >=0A= > admitting right adjoints in both variables is as a functor=0A= >=0A= > C x D ?> (E^op)^op=0A= >=0A= > because in this way (writing E? for E^op) the other two adjoints also hav= e the form=0A= >=0A= > D x E? ?> C^op=0A= >=0A= > and=0A= >=0A= > E? x C ?> D^op.=0A= >=0A= > Such two-variable adjunctions form the vertical binary morphisms in a ?cy= clic double multi category? of multivariable adjunctions and parametrized m= ates:=0A= >=0A= > https://arxiv.org/abs/1208.4520=0A= >=0A= > Regards,=0A= > Emily=0A= >=0A= > ?=0A= > Assistant Professor, Dept. of Mathematics=0A= > Johns Hopkins University=0A= > www.math.jhu.edu/~eriehl=0A= >=0A= [For admin and other information see: http://www.mta.ca/~cat-dist/ ]