Re: Grothendieck toposes

[Steve Vickers <s.j.vickers@cs.bham.ac.uk>]

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 interest=
ing issues.

As I see it you give two reasons here for taking Grothendieck toposes as bei=
ng over Set =3D ZFC, both pertinent: (1) what Grothendieck meant, and (2) th=
e (essential) uniqueness of geometric morphisms to Set.

(1) is delicate, given Grothendieck's underlying implicit definition of topo=
s 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, s=
omewhat akin (respectively) to algebraic and general topology.

The first is that he meant a topos to be a category with which to do sheaf c=
ohomology, by forming an (exact) injective resolution of Abelian groups, tak=
ing global sections (becoming non-exact), and then extracting the ker/im coh=
omology groups. This is perhaps an algebraic topologist's idea of what topol=
ogy means. Classicality of Set seems then to be needed in order to get injec=
tive hulls in the category of sheaves over a site.=20

On the other hand, there is also the idea that global points can be recovere=
d 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 wha=
t Grothendieck meant? It is closer to general topology, points and their con=
tinuous transformations.

This idea then generalizes well to elementary toposes, replacing Set by S. E=
lementary toposes are not generalized spaces in themselves, but bounded geom=
etric morphisms between them are, and many topological properties are redefi=
ned for geometric morphisms.

(2) raises the question of why Set should have its privileged property. Tech=
nically, it is that every object of Set is a colimit of copies of 1, and tha=
t is preserved up to unique isomorphism by the inverse image part of any geo=
metric 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? A=
s we explore the foundational options then we should expect this uniqueness p=
roperty for Set =3D ZFC to evaporate.

All the best,

Steve.

> On 30 Oct 2016, at 20:17, Marta Bunge <martabunge@hotmail.com> wrote:
>=20
> Dear Steve,
>=20
> 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) g=
eometric morphism. When S =3D Set (a model of ZFC) and E an arbitrary elemen=
tary topos, then there is a most one geometric morphism e:E---> Set, so in t=
hat case the latter need not be specified. I therefore use "E is a Grothendi=
eck topos" to mean  "E is an elementary topos bounded over Sets".  The latte=
r has been shown to be equivalent to what Grothendieck meant by it. =20
>=20
> Best regards,
> Marta=20
> ************************************************
> Marta Bunge
> Professor Emerita
> Dept of Mathematics and Statistics=20
> McGill University=20
> Montreal, QC, Canada H3A 2K6
> Home: (514) 935-3618
> marta.bunge@mcgill.ca=20
> 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
> =20
> 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=
.
>=20
> That's presumably not how Grothendieck understood it, and I know some of
> his results assumed S =3D Set, some classical category of sets. Moreover,
> the Elephant defines "Grothendieck topos" that way.
>=20
> 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.
>=20
> I'm coming to suspect my usage may confuse.
>=20
> 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?
>=20
> Steve Vickers.

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