From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9005 Path: news.gmane.org!.POSTED!not-for-mail From: Marta Bunge Newsgroups: gmane.science.mathematics.categories Subject: Re: Grothendieck toposes Date: Wed, 2 Nov 2016 18:34:19 +0000 Message-ID: References: ,<30618_1477941855_58179A5F_30618_291_1_E1c1IA3-0007Te-Te@mlist.mta.ca>, Reply-To: Marta Bunge NNTP-Posting-Host: blaine.gmane.org Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1478194916 895 195.159.176.226 (3 Nov 2016 17:41:56 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Thu, 3 Nov 2016 17:41:56 +0000 (UTC) To: Olivia Caramello , "categories@mta.ca" Original-X-From: majordomo@mlist.mta.ca Thu Nov 03 18:41:51 2016 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.7.28]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1c2M19-0006jW-2D for gsmc-categories@m.gmane.org; Thu, 03 Nov 2016 18:41:35 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:33956) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1c2M0l-0005Gy-7Y; Thu, 03 Nov 2016 14:41:11 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1c2M0m-0002q9-Ar for categories-list@mlist.mta.ca; Thu, 03 Nov 2016 14:41:12 -0300 Thread-Topic: categories: Re: Grothendieck toposes Thread-Index: AQHSM6xqZsgApB2Qe0GnP/T1jMfzsQGfmH22AbPKCVkBsQ9Iz6EzT9HA/2qItpA= In-Reply-To: <004501d23520$bce007f0$36a017d0$@oliviacaramello.com> Accept-Language: en-CA, en-US Content-Language: en-CA Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9005 Archived-At: Dear Olivia, Could you perhaps explain why you say that the notion of S-bounded elementary topos is "certainly less controversial" than the Grothendieckian notion of category of sheaves on a site? As you know, by a theorem of Diaconescu, the two points of view are equivalent: an elementary topos is S-bounded if and only if it is equivalent to the category of S-valued sheaves on an internal site in S. In light of this result, I find it very natural to refer to bounded S-toposes as "Grothendieck toposes over S", and I have noticed that this use is quite widespread in the literature. Of course agree with you that S-bounded topos and S-valued sheaves on a sit= e in S, for S an elementary topos, are equivalent notions by a theorem of D= iaconescu. By "certainly less controversial" I was referring to a previous = posting of mine in response to Steve Vickers, in which by a Grothendieck to= pos I had meant therein a category of Set-valued sheaves on a site in Set r= ather than on an arbitrary (elementary) topos S. I agree with Andr=E9 that it is important to clearly distinguish the concep= t of Grothendieck topos from that of elementary topos also terminologically, since the presence of sites of definition is a distinctive feature which wa= s central in Grothendieck's view and usage of toposes. Sites (or other kinds of presentations for bounded toposes) are essential for studying 'concrete' mathematical problems (not just in algebraic geometry or topology but in virtually any branch of mathematics) from a topos-theoretic point of view. Whilst general results about bounded toposes should be preferably proved without referring to their presentations and even at the elementary topos level whenever possible, the essential ambiguity given by the fact that a Grothendieck topos admits in general an infinite number of different sites of definition can be exploited to generate a great number of interesting notions and results arising from the 'calculation' of topos-theoretic invariants in terms of these different presentations. Once again, there seems to be a misunderstanding, as I too have pointed out= the distinction between the notion of a Grothendieck topos (as, say, S-val= ued sheaves on a site in S, for S an elementary topos) and that of an eleme= ntary topos, such as S. What I was arguing against was the need to change t= he terminology in such a way that by "topos" one meant "sheaves on a site" = and that "elementary topos" ought to be relabelled "logical topos" conside= ring, according to Joyal, that "the natural notion of morphism between elem= entary toposes is that of a logical morphism", suggesting by it that the no= tion of an elementary topos came from (or is suitable to) logic and not fr= om (or suitable to) geometry. Now, this is simply not the case. The very fa= ct that such categories were called "(elementary) toposes" already suggests= otherwise. Moreover, the discovery (by Lawvere) that all of higher-order l= ogic could be interpreted in an elementary topos came afterwards, and so it= turned out that both geometry and logic were present in it. The only way t= o distinguish them is therefore by means of the morphisms adopted in each c= ase - that is, either geometric or logical morphisms. Best wishes, Marta ________________________________ From: Olivia Caramello Sent: November 2, 2016 11:49:44 AM To: 'Marta Bunge'; categories@mta.ca Subject: R: categories: Re: Grothendieck toposes Dear Marta, Could you perhaps explain why you say that the notion of S-bounded elementary topos is "certainly less controversial" than the Grothendieckian notion of category of sheaves on a site? As you know, by a theorem of Diaconescu, the two points of view are equivalent: an elementary topos is S-bounded if and only if it is equivalent to the category of S-valued sheaves on an internal site in S. In light of this result, I find it very natural to refer to bounded S-toposes as "Grothendieck toposes over S", and I have noticed that this use is quite widespread in the literature. I agree with Andr=E9 that it is important to clearly distinguish the concep= t of Grothendieck topos from that of elementary topos also terminologically, since the presence of sites of definition is a distinctive feature which wa= s central in Grothendieck's view and usage of toposes. Sites (or other kinds of presentations for bounded toposes) are essential for studying 'concrete' mathematical problems (not just in algebraic geometry or topology but in virtually any branch of mathematics) from a topos-theoretic point of view. Whilst general results about bounded toposes should be preferably proved without referring to their presentations and even at the elementary topos level whenever possible, the essential ambiguity given by the fact that a Grothendieck topos admits in general an infinite number of different sites of definition can be exploited to generate a great number of interesting notions and results arising from the 'calculation' of topos-theoretic invariants in terms of these different presentations. Best wishes, Olivia [For admin and other information see: http://www.mta.ca/~cat-dist/ ]