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 ************************************************ From: Steve Vickers <s.j.vickers@cs.bham.ac.uk> Sent: October 27, 2016 7:07:52 AM To: Categories Subject: categories: Grothendieck toposes
For some years now, I have been using the phrase "Grothendieck topos" - category of sheaves over a site - to allow the site to be in an arbitrary base elementary topos S (often assumed to have nno). Hence "Grothendieck topos" means "bounded S-topos". The whole study of Grothendieck toposes, as of geometric logic, is parametrized by choice of S.
That's presumably not how Grothendieck understood it, and I know some of his results assumed S = Set, some classical category of sets. Moreover, the Elephant defines "Grothendieck topos" that way.
On the other hand, if a topos is a generalized space, with a classifying topos being the space of models of a geometric theory, then that surely meant Grothendieck topos; and there are various reasons for wanting to vary S. For example, using Sh(X) as S gives us a generalized topology of bundles, fibrewise over X.
I'm coming to suspect my usage may confuse.
How do people actually understand the phase "Grothendieck topos"? Do they hear potential for varying an implicit base S, or do they hear a firm implication that S is classical?
Steve Vickers.
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