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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Dear Marta and Steve, I'm not at all a specialist in topos theory, and learned about them quite late when I was already engaged in algebraic topology ... but let me give an "external" point of view on the subject. I believe that Grothendieck did never have the "modern point of view" of elementary toposes, but he always thought of a topos as being a category of sheaves on a site. For instance, in "Pursuing Stacks" he calls the subobject classifier the "Lawvere object" and he uses the latter with homotopical purposes (as a good interval object) but he never uses it as a foudational structure for defining a topos. Therefore, I find it a little artificial to employ the term "Grothendieck topos" in contexts where the topos cannot be represented as a category of sheaves on a site. It might of course be possible to give a sense ``internal to a given elementary topos S'' of what it means to be a category of sheaves on a site, but this is beyond my competence. Working over S=Sets I always found it quite nice that Grothendieck toposes can be characterized among elementary toposes as those which are accessible, because this gives in a very precise sense what is needed in order to represent the elementary topos as a category of sheaves on a site. Moreover, there is an analogous characterization of accessible quasi-toposes due to Borceux et al. Of course, when working over an arbitrary elementary topos S, one should define what it means to be S-accessible, but I guess that this has already been done. All the best, Clemens. Le 2016-10-31 12:27, Steve Vickers a ??crit??:
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.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
As to the question brought up by Steve I want to remark that Grothendieck always spoke about U-topos for some (Grothendieck) universe U. To write simply Set instead of U is covenient but slightly misleading if one takes logical foundations seriously since what after all is this Set? One shouldn't forget that ZFC has many models even when one adds the axiom that every set is element of a Grothendieck universe (it has a countable model by downward Loewenheim-Skolem). If one uses an extended set-theoretic foundation as Grothendieck did Set is just a name for a generic Grothendieck universe. With the advent of elementary topos theory one wanted to forget about set theory since one thought that category theory provide its own foundation via elementary toposes. This certainly makes sense but what then is Set? Well, one may choose some (unspecified) base topos SS and consider categories relative to SS as Grothendieck fibrations over SS. The role of "Set" is then taken by the fundamental ("codomain") fibration P_SS = cod : SS^2 -> SS (where 2 is the ordinal 2). From this relative point of view Grothendieck toposes over SS correspond to bounded geometric morphisms to SS as worked out in detail in Johnstone's 1977 book. But, of course, there may be many non isomorphic g.m.s from EE to SS. However, in Top/SS there is a (kind of) terminal object, the identity g.m. on SS. As explained in Moens's 1982 Thesis g.m.s to SS correspond to cocomplete locally small fibrations of toposes over SS (he assumed that the internal sums were stable and disjoint which 6 years later was shown by Jibladze to be the case for all cocomplete fibered toposes. But if one has a Grothendieck topos EE over SS the internal language of SS doesn't allow one to speak about EE in all relevant respects. In particular, one cannot quantify over the objects of EE within the internal language of SS. However, one may "blow up" SS so that one can. It is an old observation by Benabou that *split* fibrations over SS correspond to categories internal to presheaves over SS (for a large enough "Set"). This, however, is not possible for non-split fibrations like P_SS. Different ways of overcoming this problem have been found by Awodey, Butz, Simpson and myself "Relating first-order set theories, toposes and categories of classes" (APAL 2014) and in an unpublished paper by Mike Shulman arXiv:1004.3802. The restrictions of the internal language of the base topos w.r.t. speaking about a fibration over it can be overcome when one admits universes in the base topos. These universes are less set-theoretic than Grothendieck's ones but play a similar role. Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (3)
-
Clemens.BERGER＠unice.fr -
Steve Vickers -
Thomas Streicher