From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9113 Path: news.gmane.org!.POSTED!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: Terminology Date: Sat, 11 Feb 2017 15:07:53 +0000 Message-ID: References: Reply-To: Steve Vickers NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 (1.0) Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1486828657 24910 195.159.176.226 (11 Feb 2017 15:57:37 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Sat, 11 Feb 2017 15:57:37 +0000 (UTC) Cc: Categories , To: Jean Benabou Original-X-From: majordomo@mlist.mta.ca Sat Feb 11 16:57:32 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 1cca3H-000666-UP for gsmc-categories@m.gmane.org; Sat, 11 Feb 2017 16:57:32 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:40821) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1cca31-0008Ve-ID; Sat, 11 Feb 2017 11:57:15 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1cca2V-0006Hf-Pg for categories-list@mlist.mta.ca; Sat, 11 Feb 2017 11:56:43 -0400 In-Reply-To: X-Mailer: iPad Mail (13G36) Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9113 Archived-At: Dear Jean, My own understanding (superficial and possibly wrong) of the history is that= since Bourbaki there have been definitions of "structure" with the aim of r= econciling the algebraic examples (where the homomorphisms preserve structur= e) with the topological spaces (where the continuous maps have inverse image= s that preserve structure). Certainly if you look at Joy of Cats, the prime c= lasses of examples are those of topological and algebraic categories. But, as we know from topos theory, it is not foundationally robust to treat t= opological spaces as "sets with structure", i.e. point-set topology. In gene= ral we have to work point-free, at least if we want to save important parts o= f topology from going down the drain. If such an important source of examples, the point-set topological spaces, t= urned out to be misleading, then, in retrospect, any "precise meaning [of st= ructure] on which the community of mathematicians agree", was probably misgu= ided. It's like looking for a definition of "fish", but on the understanding that i= t has to include whales. All the best, Steve. > On 9 Feb 2017, at 16:38, Jean Benabou wrote: >=20 > Dear Christopher, > What I, personally, mean by structure is not the point. This word is used,= very often, in mathematical texts. Sometimes giving the impression that it h= as a precise meaning on which the community of mathematicians agree. And I w= as sure there was at least one definition on which the majority of users did= agree >=20 > Then I received 3 answers all referring to: The joy of Cats, but different= : > For Carsten F=C3=BChrman, only faithfulness is required, which obviously i= s not enough > Jiri Adamek adds: an isomorphism in S is an identity if its image is. I ag= ree with this; but again not enough. > Thomas Streicher adds a third condition, with which I would probably agree= if was sure of the precise meaning of isofibration. Could you please, even a= t the risk of being pedantic say what you mean by that >=20 > Many thanks to all >=20 >=20 >> =C3=A2=E2=82=AC=DA=98Hi Jean - I don't quite understand this question but= would like to. What do you mean by 'structure'? Thanks >>=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]