Re: Grothendieck toposes

[Michael Barr <barr@math.mcgill.ca>]

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
************************************************

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