Dear Steve, Clemens and Andre, Grothendieck did not come up with the notion of an elementary topos - Lawvere and Tierney did, so much so that he informally referred to the subobjects classifier as "the Lawvere object", as Clemens observes. Nevertheless, and referring to a remark by Steve, the basic idea of Grothendieck of a category of sheaves on a site is indeed captured by the more general (and certainly less controversial) notion of an S-bounded elementary topos, where S is an arbitrary elementary topos with an NNO. As for the word "topos", I believe that, in view of its many uses and regardless of the meaning "space", it ought to be specified in any context where one uses it. I find this way of proceeding preferable to identifying it with "Grothendieck topos" as Andre suggests. In addition, I see no reason to use "logical" instead of "elementary" since the latter is already in use and means "first-order". Best regards, Marta ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill University Montreal, QC, Canada H3A 2K6 Home: (514) 935-3618 marta.bunge@mcgill.ca http://www.math.mcgill.ca/people/bunge ************************************************ ________________________________ From: Marta Bunge <martabunge@hotmail.com> Sent: October 31, 2016 6:40:16 PM To: Steve Vickers Cc: categories@mta.ca Subject: Re: categories: Re: Grothendieck toposes Dear Steve, Thank you for your interesting comments. Grothendieck did not come up with the notion of an elementary topos - Lawvere and Tierney did, so much so that he informally referred to the subobjects classifier as "the Lawvere object". Nevertheless, the basic idea of Grothendieck of a category of sheaves on a site is indeed captured by the more general (and certainly less controversial) notion of an S-bounded elementary topos, where S is an arbitrary elementary topos with an NNO. I think that this is what you had in mind. As for the word "topos", I believe that it ought to be specified in any context where one uses it. I find this way of proceeding preferable to identifying it with "Grothendieck topos" as Joyal suggests. Best regards, Marta ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill University Montreal, QC, Canada H3A 2K6 Home: (514) 935-3618 marta.bunge@mcgill.ca http://www.math.mcgill.ca/people/bunge ************************************************ [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Let me affirm that, as late as summer of 1971, Grothendieck had never heard of elementary toposes. In fact, he gave a talk in Uldum, Denmark, saying that on the basis of the Verdier axioms the topos axioms looked like a kind of set theory and logicians, to whom he was talking, ought to study that. So I got up and presented the Lawvere-Tierney axioms, which looked a lot more like set theory than the Verdier axioms and Grothendieck seemed impressed. I did add completeness since I wanted to give an equivalent set of axioms. Michael ----- Original Message ----- From: "<Unknown>" <martabunge@hotmail.com> To: categories@mta.ca Sent: Tuesday, November 1, 2016 11:33:22 AM Subject: categories: Re: Grothendieck toposes Dear Steve, Clemens and Andre, Grothendieck did not come up with the notion of an elementary topos - Lawvere and Tierney did, so much so that he informally referred to the subobjects classifier as "the Lawvere object", as Clemens observes. Nevertheless, and referring to a remark by Steve, the basic idea of Grothendieck of a category of sheaves on a site is indeed captured by the more general (and certainly less controversial) notion of an S-bounded elementary topos, where S is an arbitrary elementary topos with an NNO. As for the word "topos", I believe that, in view of its many uses and regardless of the meaning "space", it ought to be specified in any context where one uses it. I find this way of proceeding preferable to identifying it with "Grothendieck topos" as Andre suggests. In addition, I see no reason to use "logical" instead of "elementary" since the latter is already in use and means "first-order". Best regards, Marta ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill University Montreal, QC, Canada H3A 2K6 Home: (514) 935-3618 marta.bunge@mcgill.ca http://www.math.mcgill.ca/people/bunge ************************************************ ________________________________ From: Marta Bunge <martabunge@hotmail.com> Sent: October 31, 2016 6:40:16 PM To: Steve Vickers Cc: categories@mta.ca Subject: Re: categories: Re: Grothendieck toposes Dear Steve, Thank you for your interesting comments. Grothendieck did not come up with the notion of an elementary topos - Lawvere and Tierney did, so much so that he informally referred to the subobjects classifier as "the Lawvere object". Nevertheless, the basic idea of Grothendieck of a category of sheaves on a site is indeed captured by the more general (and certainly less controversial) notion of an S-bounded elementary topos, where S is an arbitrary elementary topos with an NNO. I think that this is what you had in mind. As for the word "topos", I believe that it ought to be specified in any context where one uses it. I find this way of proceeding preferable to identifying it with "Grothendieck topos" as Joyal suggests. Best regards, Marta ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill University Montreal, QC, Canada H3A 2K6 Home: (514) 935-3618 marta.bunge@mcgill.ca http://www.math.mcgill.ca/people/bunge ************************************************ [For admin and other information see: http://www.mta.ca/~cat-dist/ ] [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 André that it is important to clearly distinguish the concept of Grothendieck topos from that of elementary topos also terminologically, since the presence of sites of definition is a distinctive feature which was 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 -----Messaggio originale----- Da: Marta Bunge [mailto:martabunge@hotmail.com] Inviato: martedì 1 novembre 2016 16:33 A: categories@mta.ca Oggetto: categories: Re: Grothendieck toposes Dear Steve, Clemens and Andre, Grothendieck did not come up with the notion of an elementary topos - Lawvere and Tierney did, so much so that he informally referred to the subobjects classifier as "the Lawvere object", as Clemens observes. Nevertheless, and referring to a remark by Steve, the basic idea of Grothendieck of a category of sheaves on a site is indeed captured by the more general (and certainly less controversial) notion of an S-bounded elementary topos, where S is an arbitrary elementary topos with an NNO. As for the word "topos", I believe that, in view of its many uses and regardless of the meaning "space", it ought to be specified in any context where one uses it. I find this way of proceeding preferable to identifying it with "Grothendieck topos" as Andre suggests. In addition, I see no reason to use "logical" instead of "elementary" since the latter is already in use and means "first-order". Best regards, Marta ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill University Montreal, QC, Canada H3A 2K6 Home: (514) 935-3618 marta.bunge@mcgill.ca http://www.math.mcgill.ca/people/bunge ************************************************ ________________________________ [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (3)
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Marta Bunge -
Michael Barr -
Olivia Caramello