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 ************************************************ ________________________________ From: Steve Vickers <s.j.vickers@cs.bham.ac.uk> Sent: October 31, 2016 7:27:57 AM To: Marta Bunge Cc: categories@mta.ca Subject: categories: Re: Grothendieck toposes Dear Marta, Thanks for your reply. My question was a survey of usage, so what follows is not meant disputatiously, but I thought your reasoning raised some interesting issues. As I see it you give two reasons here for taking Grothendieck toposes as being over Set = ZFC, both pertinent: (1) what Grothendieck meant, and (2) the (essential) uniqueness of geometric morphisms to Set. (1) is delicate, given Grothendieck's underlying implicit definition of topos as "that of which topology is the study". My understanding of this (you may know more - my knowledge of Grothendieck's work is almost all second-hand) is that there are two ways of viewing it, somewhat akin (respectively) to algebraic and general topology. The first is that he meant a topos to be a category with which to do sheaf cohomology, by forming an (exact) injective resolution of Abelian groups, taking global sections (becoming non-exact), and then extracting the ker/im cohomology groups. This is perhaps an algebraic topologist's idea of what topology means. Classicality of Set seems then to be needed in order to get injective hulls in the category of sheaves over a site. On the other hand, there is also the idea that global points can be recovered as sections of the geometric morphism to Set, and the topos simultaneously embodies both the points and the topology on them. Is this also part of what Grothendieck meant? It is closer to general topology, points and their continuous transformations. This idea then generalizes well to elementary toposes, replacing Set by S. Elementary toposes are not generalized spaces in themselves, but bounded geometric morphisms between them are, and many topological properties are redefined for geometric morphisms. (2) raises the question of why Set should have its privileged property. Technically, it is that every object of Set is a colimit of copies of 1, and that is preserved up to unique isomorphism by the inverse image part of any geometric morphism. But is that not because ZFC provides a 2-level structure of sets and classes, and we are implicitly using ZFC classes for our toposes? As we explore the foundational options then we should expect this uniqueness property for Set = ZFC to evaporate. All the best, Steve.

On 30 Oct 2016, at 20:17, Marta Bunge <martabunge@hotmail.com> wrote:

Dear Steve,

When an elementary (base) topos S is specified, I use "S-bounded topos" to mean the pair (E, e), with E an elementary topos and e: E--> S a (bounded) geometric morphism. When S = Set (a model of ZFC) and E an arbitrary elementary topos, then there is a most one geometric morphism e:E---> Set, so in that case the latter need not be specified. I therefore use "E is a Grothendieck topos" to mean "E is an elementary topos bounded over Sets". The latter has been shown to be equivalent to what Grothendieck meant by it.

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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