From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9365 Path: news.gmane.org!.POSTED!not-for-mail From: =?iso-8859-1?Q?Ren=E9_Guitart?= Newsgroups: gmane.science.mathematics.categories Subject: Re: Fred Date: Wed, 27 Sep 2017 11:10:25 +0200 Message-ID: References: Reply-To: =?iso-8859-1?Q?Ren=E9_Guitart?= NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 (Apple Message framework v1085) Content-Type: text/plain; charset=windows-1252 Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1506558417 4746 195.159.176.226 (28 Sep 2017 00:26:57 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Thu, 28 Sep 2017 00:26:57 +0000 (UTC) Cc: "categories@mta.ca" To: Emily Riehl Original-X-From: majordomo@mlist.mta.ca Thu Sep 28 02:26:52 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 1dxMfA-0000YV-RQ for gsmc-categories@m.gmane.org; Thu, 28 Sep 2017 02:26:48 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:45950) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1dxMgI-0002BP-FR; Wed, 27 Sep 2017 21:27:58 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1dxMeS-0003lW-N0 for categories-list@mlist.mta.ca; Wed, 27 Sep 2017 21:26:04 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9365 Archived-At: Dear Emily, many thanks for your message. It push me to precise some aspects of my = link with the "op" things, with some words about your joint paper = https://arxiv.org/abs/1208.4520.=20 In fact in a talk at a the Louvain-la-Neuve's meeting in 2011, = Category theory, algebra and geometry, 26-27 may 2011, I spoke on = "Borromean Objects and Trijunctions". I do remember well that Eugenia = was listening to this talk, and so probably her attention was attracted on the notion of a = trijunction (the notion you are explaining in your message). Some times = later this was published in a paper "Trijunctions and Triadic Galois = Connections" (Cahier Top. G=E9o. Diff. Cat, LIV-1 (2013), pp. 13-28) = (accessible on my site : = http://rene.guitart.pagesperso-orange.fr/publications.html). =46rom the = summary, we can learn why I did so :=20 "In this paper we introduce the notion of a trijunction, which = is related to a triadic Galois connection just as an adjunction is to a = Galois connection. We construct the trifibered tripod associated to a = trijunction, the trijunction between toposes of presheaves associated to a = discrete trifibration, and the generation of any trijunction by a = bi-adjoint functor. While some examples are related to triadic Galois = connections, to ternary relations, others are associated to some = symmetric tensors, to toposes and algebraic universes".=20 Now it is interesting to understand how this was achieved, in two steps:=20= 1 - Firstly I read a paper by Biedermann, on triadic Galois connections, = related to ternary relations as Galois connections of Ore between = ordered sets are related to binary relations. Immediately I try to = extend that from order sets to categories. 2 - Fortunately in the same time I was conducted to read again carefully = the famous paper by Kan on Adjoint functor. There I observed that in = fact he he is mainly working with tensors and Gom, i.e. with bifunctors = ; and furthermore in the Mac Lane's book, the convenient lemma for = parameterized adjunctions are reproduced. Then I notice that the perfect = explicit ternary symmetry in Biedermann was in fact also implicit in = Kan, but that only he "missed" to put the accent on it, by introducing = the opposite of the opposite (E^op)^op in your message). So I did, and = then I got the application to the descriptions of trifibrations and = toposes, to the analysis clearly the system of functions or operations = generated by a tripod. So you can see that my motivations (to unified Kan and Biedermann at the = level n =3D 3), in order to produce a functional analysis of tripod), = seems rather different from yours (to enter in a game of general = n-multiadjunctions). Hence finally my question : to analyze the system = of functions or operations generated by an n-pod, and to understand = there the part play by the mysterious "op". with my friendly greetings, Ren=E9. Le 7 sept. 2017 =E0 19:03, Emily Riehl a =E9crit : >> There is one other anecdote about UACT, nothing to do with Fred, that = I >> have always loved. In the course of MSRI director Bill Thurston's = Galois Connections >> opening remarks, he said words to the effect that the notion of the >> opposite of a category made him nauseous. This was the only meeting I >> have ever attended where fully half the attendees drew in enough = breath >> to drop the air pressure by an audible amount. >=20 > I=92ll confess that the idea of an opposite category appearing as the = codomain of a functor also makes me somewhat nauseated (the domain of = course is no problem).=20 >=20 > But this said, in the interest of full disclosure, I should admit that = in a joint paper with Cheng and Gurski someone =97 Eugenia, I believe? =97= convinced us that the easiest way to think of a functor=20 >=20 > C x D =97> E=20 >=20 > admitting right adjoints in both variables is as a functor=20 >=20 > C x D =97> (E^op)^op >=20 > because in this way (writing E=92 for E^op) the other two adjoints = also have the form >=20 > D x E=92 =97> C^op >=20 > and=20 >=20 > E=92 x C =97> D^op. >=20 > Such two-variable adjunctions form the vertical binary morphisms in a = =93cyclic double multi category=94 of multivariable adjunctions and = parametrized mates: >=20 > https://arxiv.org/abs/1208.4520 >=20 > Regards, > Emily >=20 > =97 > Assistant Professor, Dept. of Mathematics > Johns Hopkins University > www.math.jhu.edu/~eriehl >=20 >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]