From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3953 Path: news.gmane.org!not-for-mail From: JeanBenabou Newsgroups: gmane.science.mathematics.categories Subject: "Historical terminology" Date: Fri, 5 Oct 2007 16:52:40 +0200 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 (Apple Message framework v752.2) Content-Type: text/plain;charset=US-ASCII;delsp=yes;format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019624 11071 80.91.229.2 (29 Apr 2009 15:40:24 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:40:24 +0000 (UTC) To: Categories Original-X-From: rrosebru@mta.ca Sat Oct 6 23:19:16 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 06 Oct 2007 23:19:16 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IeLXh-0003Bi-OJ for categories-list@mta.ca; Sat, 06 Oct 2007 23:06:53 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 10 Original-Lines: 42 Xref: news.gmane.org gmane.science.mathematics.categories:3953 Archived-At: Dear colleagues I need your help for the following questions: (i) Who gave the name of "cartesian" to categories with finite limits? When was this name given? What is the first published paper where this name occurs? (ii) Same questions for "cartesian closed" (iii) Same questions again for "locally cartesian closed". Moreover, in this case, does the precise definition imply that such a category has a terminal object? Thanks for your help, Jean --Apple-Mail-1-687879601 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=ISO-8859-1 Dear colleagues=A0

I need your help for the = following questions:

(i) Who gave the name of = "cartesian"=A0=A0to categories with finite limits? When was this = name given? What is the first published paper where this name = occurs?

(ii) Same = questions for "cartesian closed"

(iii) Same questions again for "locally = cartesian closed". Moreover, in this case, does the precise = definition imply that such a category has a terminal = object?

Thanks for your = help,

Jean

= --Apple-Mail-1-687879601-- From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3959 Path: news.gmane.org!not-for-mail From: JeanBenabou Newsgroups: gmane.science.mathematics.categories Subject: Re: "Historical terminology" Date: Sun, 7 Oct 2007 09:48:02 +0200 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 (Apple Message framework v752.2) Content-Type: text/plain;charset=US-ASCII;delsp=yes;format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019627 11089 80.91.229.2 (29 Apr 2009 15:40:27 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:40:27 +0000 (UTC) To: Categories Original-X-From: rrosebru@mta.ca Sun Oct 7 10:53:14 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 07 Oct 2007 10:53:14 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IeWTq-0000fF-Fk for categories-list@mta.ca; Sun, 07 Oct 2007 10:47:38 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 16 Original-Lines: 76 Xref: news.gmane.org gmane.science.mathematics.categories:3959 Archived-At: Cher Fred, Merci pour ta reponse rapide. Although your french is perfect, I shall continue in english, for the persons who are less familiar with french. (i) Your "guess" about cartesian closed categories is most certainly correct. I knew that Eilenberg/Kelly had explicitly used this name in their La Jolla paper, and it is probably the first instance, because "closed", in this sense, was first introduced in that paper, as far as I know.. (ii) Your "guess" about cartesian is not correct. Neither in Tohoku, nor in much later papers of his or any of his students, and also by me, was cartesian used in the sense of category with finite limits. If Grothendieck had used this name, which he has not, my "guess" is that he would have called cartesian categories with pull backs , because he and his students used the name "cartesian square" for square which is a pull back. Moreover this is special case of his notion of cartesian map in a fibration. (iii) I agree with you on the idea that the "natural" definition of locally cartesian closed category should not imply the existence of a terminal object. If I asked the question, it is because in Johnstone's "Elephant" he does assume a terminal object. Has such an assumption become, now, commonly accepted in the definition ? Thanks again, to you of course, and to whoever will help me to clarify (ii) and (iii) Jean > Salut, Jean, > > Without references at hand to consult, other than my failing > memory, I venture to hazard the following GUESSES at answers: > >> (i) Who gave the name of "cartesian" to categories with finite >> limits? When was this name given? What is the first published paper >> where this name occurs? > > This name I thought either you, or perhaps earlier Grothendieck, > had coined. When? Where? no idea (but if Grothendieck, then Tohoku?). > >> (ii) Same questions for "cartesian closed" > > My unverified guess: Eilenberg/Kelly, La Jolla, 1965. > >> (iii) Same questions again for "locally cartesian closed". > > No idea, but rather much later. > >> ... Moreover, >> in this case, does the precise definition imply that such a category >> has a terminal object? > > Here I have no answer at all, sorry, beyond this: IF the > definition of LCC is just that each "slice" category (but > not necessarily the category itself) be cartesian closed, > then most probably NOT. > >> Thanks for your help, > > I can only hope you find my guesses WERE actually of any help. > I fear, though, that they probably weren't at all. I'd be very > interested in learning the outcome of your survey, however. > >> Jean > > Cheers, > > -- Fred > From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3969 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: "Historical terminology" Date: Sun, 7 Oct 2007 22:49:22 +0100 (BST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1241019633 11135 80.91.229.2 (29 Apr 2009 15:40:33 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:40:33 +0000 (UTC) To: Categories Original-X-From: rrosebru@mta.ca Mon Oct 8 10:13:58 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Oct 2007 10:13:58 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IesNG-0003sr-Vc for categories-list@mta.ca; Mon, 08 Oct 2007 10:10:19 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 26 Original-Lines: 41 Xref: news.gmane.org gmane.science.mathematics.categories:3969 Archived-At: On Sun, 7 Oct 2007, Jean Benabou wrote: > (ii) Your "guess" about cartesian is not correct. Neither in Tohoku, > nor in much later papers of his or any of his students, and also by > me, was cartesian used in the sense of category with finite limits. > If Grothendieck had used this > name, which he has not, my "guess" is that he would have called > cartesian categories with pull backs , because he and his students > used the name "cartesian square" for square which is a pull back. > Moreover this is special case of his notion of cartesian map in > a fibration. > I first encountered `cartesian' as a synonym for `having finite limits' in Peter Freyd's unpublished `pamphlet' "On canonizing category theory; or, on functorializing model theory" written in about 1975 (I may have got the title wrong, since I no longer possess a copy). However, that paper made it clear that the word was already in use as a synonym for "having finite products"; in it, Peter argued that Descartes should be given credit for having invented equalizers as well as cartesian products. I suspect that its use to mean `having finite products' was a conscious back-formation from `cartesian closed', which undoubtedly dates from Eilenberg--Kelly 1965; but I don't know who first used it in this sense. > (iii) I agree with you on the idea that the "natural" definition of > locally cartesian closed category should not imply the existence > of a terminal object. If I asked the question, it is because in > Johnstone's "Elephant" he does assume a terminal object. Has such an > assumption become, now, commonly accepted in the definition ? > I did that because it seemed the appropriate convention to adopt in the context of topos theory. I wasn't trying to dictate to the rest of the world what the convention should be. On the other hand, there seem to be remarkably few `naturally occurring' examples of locally cartesian closed categories which lack terminal objects: the category of spaces (or locales) and local homeomorphisms is almost the only one I can think of. Peter Johnstone From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3972 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: "Historical terminology" Date: Sun, 07 Oct 2007 17:12:36 -0700 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019635 11144 80.91.229.2 (29 Apr 2009 15:40:35 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:40:35 +0000 (UTC) To: Categories Original-X-From: rrosebru@mta.ca Mon Oct 8 10:14:11 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Oct 2007 10:14:11 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IesQR-00047n-EG for categories-list@mta.ca; Mon, 08 Oct 2007 10:13:35 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 29 Original-Lines: 28 Xref: news.gmane.org gmane.science.mathematics.categories:3972 Archived-At: JeanBenabou wrote: > (i) Your "guess" about cartesian closed categories is most certainly > correct. I knew that Eilenberg/Kelly had explicitly used this name > in their La Jolla paper, and it is probably the first instance, > because "closed", in this sense, was first introduced in that paper, > as far as I know.. What most impressed my students and me two decades ago, when we were applying the concepts of EK65 to modeling concurrency, was their attempt to define "closed" as a self-contained notion independently of any tensor product as its left adjoint (or so it seemed to us). This defeated us. Has a clearer story of that attempt, or any related story, emerged in the meantime? > (iii) I agree with you on the idea that the "natural" definition of > locally cartesian closed category should not imply the existence > of a terminal object. If I asked the question, it is because in > Johnstone's "Elephant" he does assume a terminal object. Has such an > assumption become, now, commonly accepted in the definition ? Hopefully not. If affine geometry has no origin, why should locally cartesian closed categories have a global reference point? (What would Andy Pitts have decided there, and for that matter the orientation of profunctors in B2.7, which seems backwards from say Borceux?) Vaughan From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3976 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: "Historical terminology" Date: Mon, 08 Oct 2007 11:18:42 -0700 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019638 11160 80.91.229.2 (29 Apr 2009 15:40:38 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:40:38 +0000 (UTC) To: Categories Original-X-From: rrosebru@mta.ca Tue Oct 9 00:04:12 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 09 Oct 2007 00:04:12 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1If5I4-0007ja-GV for categories-list@mta.ca; Mon, 08 Oct 2007 23:57:48 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 33 Original-Lines: 38 Xref: news.gmane.org gmane.science.mathematics.categories:3976 Archived-At: Vaughan Pratt wrote: > JeanBenabou wrote: >> (i) Your "guess" about cartesian closed categories is most certainly >> correct. I knew that Eilenberg/Kelly had explicitly used this name >> in their La Jolla paper, and it is probably the first instance, >> because "closed", in this sense, was first introduced in that paper, >> as far as I know.. > > What most impressed my students and me two decades ago, when we were > applying the concepts of EK65 to modeling concurrency, was their attempt > to define "closed" as a self-contained notion independently of any > tensor product as its left adjoint (or so it seemed to us). This > defeated us. Has a clearer story of that attempt, or any related story, > emerged in the meantime? Meanwhile the following examples occurred to me: 1. Implicational logic without conjunction. 2. The type structure of the pure lambda calculus without products. 3. The subcategory of FinSet consisting of the prime powers. (With regard to 3, Mike Barr mentioned to me that (Eilenberg and?) Kelly had come up with the category "-6" meaning the category of all sets save those with six elements, but this seems less natural than the prime powers, important in ideal theory as we saw in the recent discussion about the division lattice.) The free closed category would be a good example if it had ever been sighted in nature? Has it? (Just because we see initial ring every day in the wild doesn't mean that all free objects arise in nature.) Vaughan