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/ ]