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 I first encountered `cartesian' as a synonym for `having finite limits' in Peter Freyd's unpublished `pamphlet' "On canonizing category theory; or, on functorializing model theory" written in about 1975 (I may have got the title wrong, since I no longer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