Re: Grothendieck toposes

Re: Grothendieck toposes

[Steve Vickers]

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.

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Grothendieck toposes

[Marta Bunge]

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
> ************************************************

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Grothendieck toposes

[Clemens.BERGER]

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/ ]

Re: Grothendieck toposes

[Thomas Streicher]

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


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Re: Grothendieck toposes

[Marta Bunge]

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/ ]

Re: Grothendieck toposes

[Michael Barr]

Let me affirm that, as late as summer of 1971, Grothendieck had never 
heard of elementary toposes.  In fact, he gave a talk in Uldum, Denmark, 
saying that on the basis of the Verdier axioms the topos axioms looked 
like a kind of set theory and logicians, to whom he was talking, ought to 
study that.  So I got up and presented the Lawvere-Tierney axioms, which 
looked a lot more like set theory than the Verdier axioms and Grothendieck 
seemed impressed.  I did add completeness since I wanted to give an 
equivalent set of axioms.

Michael

----- Original Message -----
From: "<Unknown>" <martabunge@hotmail.com>
To: categories@mta.ca
Sent: Tuesday, November 1, 2016 11:33:22 AM
Subject: categories: Re:  Grothendieck toposes

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/ ]


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Re: Grothendieck toposes

[Marta Bunge]

Dear Olivia,



Could you perhaps explain why you say that the notion of S-bounded
elementary topos is "certainly less controversial" than the Grothendieckian
notion of category of sheaves on a site? As you know, by a theorem of
Diaconescu, the two points of view are equivalent: an elementary topos is
S-bounded if and only if it is equivalent to the category of S-valued
sheaves on an internal site in S. In light of this result, I find it very
natural to refer to bounded S-toposes as "Grothendieck toposes over S", and
I have noticed that this use is quite widespread in the literature.



Of course agree with you that S-bounded topos and S-valued sheaves on a site in S, for S an elementary topos, are equivalent notions by a theorem of Diaconescu. By "certainly less controversial" I was referring to a previous posting of mine in response to Steve Vickers, in which by a Grothendieck topos I had meant therein a category of Set-valued sheaves on a site in Set rather than on an arbitrary (elementary) topos S.




I agree with André that it is important to clearly distinguish the concept
of Grothendieck topos from that of elementary topos also terminologically,
since the presence of sites of definition is a distinctive feature which was
central in Grothendieck's view and usage of toposes. Sites (or other kinds
of presentations for bounded toposes) are essential for studying 'concrete'
mathematical problems (not just in algebraic geometry or topology but in
virtually any branch of mathematics) from a topos-theoretic point of view.
Whilst general results about bounded toposes should be preferably proved
without referring to their presentations and even at the elementary topos
level whenever possible, the essential ambiguity given by the fact that a
Grothendieck topos admits in general an infinite number of different sites
of definition can be exploited to generate a great number of interesting
notions and results arising from the 'calculation' of topos-theoretic
invariants in terms of these different presentations.


Once again, there seems to be a misunderstanding, as I too have pointed out  the distinction between the notion of a Grothendieck topos (as, say, S-valued sheaves on a site in S, for S an elementary topos) and that of an elementary topos, such as S. What I was arguing against was the need to change the terminology in such a way that by "topos" one meant "sheaves on a site" and that  "elementary topos" ought to be relabelled "logical topos" considering, according to Joyal, that "the natural notion of morphism between elementary toposes is that of a logical morphism", suggesting by it that the notion of an elementary topos came from (or is suitable to) logic and  not from (or suitable to) geometry. Now, this is simply not the case. The very fact that such categories were called "(elementary) toposes" already suggests  otherwise. Moreover, the discovery (by Lawvere) that all of higher-order logic could be interpreted in an elementary topos came afterwards, and so it  turned out that both geometry and logic were present in it. The only way to distinguish them is therefore by means of the morphisms adopted in each case - that is, either geometric or logical morphisms.



Best wishes,

Marta



________________________________
From: Olivia Caramello <olivia@oliviacaramello.com>
Sent: November 2, 2016 11:49:44 AM
To: 'Marta Bunge'; categories@mta.ca
Subject: R: categories: Re: Grothendieck toposes

Dear Marta,

Could you perhaps explain why you say that the notion of S-bounded
elementary topos is "certainly less controversial" than the Grothendieckian
notion of category of sheaves on a site? As you know, by a theorem of
Diaconescu, the two points of view are equivalent: an elementary topos is
S-bounded if and only if it is equivalent to the category of S-valued
sheaves on an internal site in S. In light of this result, I find it very
natural to refer to bounded S-toposes as "Grothendieck toposes over S", and
I have noticed that this use is quite widespread in the literature.

I agree with André that it is important to clearly distinguish the concept
of Grothendieck topos from that of elementary topos also terminologically,
since the presence of sites of definition is a distinctive feature which was
central in Grothendieck's view and usage of toposes. Sites (or other kinds
of presentations for bounded toposes) are essential for studying 'concrete'
mathematical problems (not just in algebraic geometry or topology but in
virtually any branch of mathematics) from a topos-theoretic point of view.
Whilst general results about bounded toposes should be preferably proved
without referring to their presentations and even at the elementary topos
level whenever possible, the essential ambiguity given by the fact that a
Grothendieck topos admits in general an infinite number of different sites
of definition can be exploited to generate a great number of interesting
notions and results arising from the 'calculation' of topos-theoretic
invariants in terms of these different presentations.

Best wishes,
Olivia




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Re: Grothendieck toposes

[Patrik Eklund]

Heraclitus of Ephesus used 'logos' for what is pretty well-known. Even
more known is that 'logos' is in the first sentences of the Gospel of
John. I'm not claiming anybody should care, but using 'logos' for a math
concept must be justified, not for what I mentioned, but for what it is.
What is this discussion really about?

Just wondering.

Patrik



On 2016-11-09 04:35, Marta Bunge wrote:
> Dear all,
>
>
> Indeed, as pointed out by Bill Lawvere, the term "logos" was
> introduced and  is central to the book by Freyd and Scedrov. In
> addition to that of Walter  Tholen there is a review of it by myself
>
>
> Categories, Allegories, by Peter J. Freyd; Andrej Scedrov
>
> Review by Marta C. Bunge,
>
> The Journal of Symbolic Logic  56-1 (March 1993) 352-354
>
>
> Best wishes,
>
> Marta
>
>
>
>
>
>
>
>
>
>
>
>
>
>
> ________________________________
> From: wlawvere <wlawvere@buffalo.edu>
> Sent: November 8, 2016 8:32:16 AM
> To: categories@mta.ca
> Subject: categories: Re: Grothendieck toposes
>
>
> The term 'logos' already has a well-established meaning.
> See Tholen's review of the 1990 book by Freyd and Scedrov:
> Categories, allegories
>
> ...(a logos is a regular category in which the subobjects of
> an object form a lattice, and in which each inverse-image map
> has a right adjoint)
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: Grothendieck toposes

[Thomas Streicher]

On Wed, Nov 09, 2016 at 02:44:45PM +0000, Paul B Levy wrote:
>
>
> On 09/11/16 10:48, Thomas Streicher wrote:
>> What Eduardo means by "Giraud topos" is a category validating all
>> conditions of the Giraud theorem with the exception of having a small
>> generating family. These guys can be elementary toposes or not.
>> There is Freyd's example where objects are set X with an ordinal
>> indexed family of bijections from X to X
>
> So they don't form a class.
>
> Is there an example where the objects form a class?
>
> (I like categories to have this property!)

That's not the problem. Objects do form a class! Eduardo rather
thought that Freyd's example were not locally small. But it is!

Thomas


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Re: Grothendieck toposes

[Thomas Streicher]

What Eduardo means by "Giraud topos" is a category validating all
conditions of the Giraud theorem with the exception of having a small
generating family. These guys can be elementary toposes or not.
There is Freyd's example where objects are set X with an ordinal
indexed family of bijections from X to X and morphisms are maps commuting with
all these ordinal many maps. This is a locally small cocomplete
elementary topos lacking a s mall generating family.
In 2011 on this list Johnstone gave the example of a category which
superficially is like Freyd's example but the class many endomaps are
not required to be bijections. This category is also bicomplete and
locally small but it hasn't got a subobject classifier.

Thomas


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Re: Grothendieck toposes

[Marta Bunge]

Dear all,


Indeed, as pointed out by Bill Lawvere, the term "logos" was introduced and  is central to the book by Freyd and Scedrov. In addition to that of Walter  Tholen there is a review of it by myself


Categories, Allegories, by Peter J. Freyd; Andrej Scedrov

Review by Marta C. Bunge,

The Journal of Symbolic Logic  56-1 (March 1993) 352-354


Best wishes,

Marta














________________________________
From: wlawvere <wlawvere@buffalo.edu>
Sent: November 8, 2016 8:32:16 AM
To: categories@mta.ca
Subject: categories: Re: Grothendieck toposes


The term 'logos' already has a well-established meaning.
See Tholen's review of the 1990 book by Freyd and Scedrov:
Categories, allegories

...(a logos is a regular category in which the subobjects of
an object form a lattice, and in which each inverse-image map
has a right adjoint)


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Re: Grothendieck toposes

[wlawvere]

The term 'logos' already has a well-established meaning.
See Tholen's review of the 1990 book by Freyd and Scedrov:
Categories, allegories

...(a logos is a regular category in which the subobjects of
an object form a lattice, and in which each inverse-image map
has a right adjoint)


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Re: Grothendieck toposes

[wlawvere]

Dear friends and colleagues,

In Spring 1981, near a lavender field in Southern France,
Alexander Grothendieck greeted me at the door of his home.
He wasted no time and immediately put the question:

'What is the relationship between the two uses of the term 'topos'?'

This led to a very interesting discussion.The first thing that
was established as a basis was that SGA4 never defined 'topos',
but rather spoke always of 'U-topos',  where U was a certain
kind of model of set theory. All the categories so arising have
common features, such as cartesian closure, and the U itself can
be construed as such a category. (TAC Reprints no. 11).

Thus we arrived at the notion of 'U-topos' as a special geometric
morphism E →U  of 'elementary' toposes. Grothendieck's
general method of relativization suggests the usefulness
of a general topos as a codomain or base U. (see Giraud, SLN 274).
But to focus more specifically on the original case, various special
properties of the base U could also be considered:
Booleanness (note for example, that Booleanness distinguishes
algebraic points among algebraic figures)
Axiom of choice;
Lack of measurable cardinals; et cetera.

One of the many topics we discussed was the
'Medaille de Chocolat' exercise in SGA4, and its basic importance
for understanding applications of topos theory: the gros and
petit sheaves of an object point out that there should be a
qualitative distinction between a topos of SPACES and a topos
of set-valued sheaves on a generalized space. I believe that
considerable progress is now being made on the characterization
of 'gros' toposes under the name of Cohesion. Grothendieck made
a big step towards  the characterization of 'petit' under the name
of  'etendu'  (sometimes known as 'locally localic'). Concerning
Grothendieck's most famous contribution, the 'petit etale' topos,
what are it's distinguishing properties as a topos?

We also discussed the Grauert direct image theorem as a
relativization of the Cartan-Serre theorem. It is important to
note that Grothendieck's work was not limited to the Weil
conjectures but, for example, involved around 1960 several
categories related to complex analysis which were perhaps
part of his inspiration for the notion of topos.


Separation?
Actually, separation has been one of the main sources of confusion.
I wish that someone with internet confidence would correct the
Wikipedia article that claims that pre-1970 toposes were about
geometry, but that post-1970 toposes were about logic. Certainly,
that discourages students from studying either.
Omitted was the fact that logic has always been used to sharpen the
study of geometry; in the last 50 years we have been able to make
this relation more explicit, with the help of categories.

Of course, separating a certain kind of object from a certain kind
of map would be basic 'grammar'.
But we cannot separate the legacy of Grothendieck from the
inspiration it gives to the continuing development of topos theory.

Best wishes
Bill Lawvere





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Re: Grothendieck toposes

[Joyal, André]

Dear

Dear Marta, Martin, Steve and Thomas,

I am considering using the word "logos" instead of "elementary topos".

https://en.wikipedia.org/wiki/Logos

The word "logos" has seldom been used in mathematics.
It is a noble word in philosophy where it means: reason, discourse, logic, knowledge, principle of order.

The category of sets is a boolean logos with natural number objects.
Every (Grothendieck) topos has the structure of a logos, but not every logos is a topos.
The initial logos with natural number object and the effective logos are not toposes.

A topos E over a base topos S can be regarded as a topos internal to the logos defined by S.
More generally, there is a notion of topos relative to a logos.

Best regards,
André

________________________________
From: Marta Bunge [martabunge@hotmail.com]
Sent: Friday, November 04, 2016 8:34 PM
To: Martin Escardo; Joyal, André; categories@mta.ca; Steve Vickers
Cc: Thomas Streicher
Subject: Re: categories: Re: Grothendieck toposes


Dear Martin, Andre, and Steve,


I will abstain from commenting on the "mystery" of univalence at least until I study the paper by Martin and Thomas (Streicher) which was made  kindly  made available to me but have not yet found the time to do so.


Since Martin in his latest mail goes back to the original question by Steve  which prompted some of this correspondence on and off categories, I point out that I had already accepted regarding Grothendieck toposes as S-bounded  elementary toposes for a given elementary topos S, as the idea is the same  as that of Grothendieck - namely sheaves on a site, without, however, specifying a set theory to be given by Set but by some unspecified elementary (base) topos S. It is not a matter of honesty but of revising the notion of a Grothendieck topos which arose before elementary toposes were introduced.


There was, however, another, in my view still not totally settled question in this forum, of whether the notion of an elementary topos ought to be equated with that of a "logical" topos as proposed by Andre, so that the "natural" morphisms bewteen them, according to him,  would be the logical morphisms. As I already argued in this forum, the notion of an elementary topos is no more logical than it is geometric, and the way to specify which will it be in any given context is by the choice of morphisms - logical or geometrical, with possibly further conditions in each case.


In connection with the latter, I came upon an old paper by Colin McLarty, "The uses and abuses of the history of topos theory", British J. Phil Sc. 41  (1990) 351--375, in which this issue is discussed at length. I reproduce the abstract here:


"The view that toposes originated as a generalized set theory is a figment of set theoretically educated common sense. This false history obstructs understanding of category theory and specially of categorical foundations for  mathematics. Problems in geometry, topology, and related algebra led to categories and toposes. Elementary toposes arose when Lawvere's interest in the foundations of physics and Tierney's in the foundations of topology led both to study Grothendieck's foundations for algebraic geometry. I end with  remarks on a categorical view of the history of set theory, including a false history plausible from that point of view that would make it helpful to  introduce toposes as a generalization of set theory."


I recommend readin this very interesting article.



Very 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: Martin Escardo <m.escardo@cs.bham.ac.uk>
Sent: November 4, 2016 7:17:43 PM
To: Joyal, André; Marta Bunge; Steven Vickers
Subject: Re: categories: Re: Grothendieck toposes

And, to continue, I think it is slightly dishonest to say that
Grothendiek toposes are defined over Set, as if Set were one uniquely
determined thing. It is not, and we know that there are many things (as
soon as there is as least one) satisfying the axioms of set theory. Why
do we still speak of "the" category of sets, as if we would be able to
magically pin one "intended model" (with no precise specification)?

Martin

On 04/11/16 22:56, Martin Escardo wrote:
> But here is a more mundane question.
>
> Mathematical language is precise. Natural language is not. How can we
> define a precise mathematical language using an imprecise natural language?
>
> When two mathematicians disagree in their chosen foundations, they will
> nevertheless be able to understand each other's rules and be able to
> follow them (if they are willing to, or if they are pressed to do so).
> At some level, the "very basic" foundation seems to be universal (but is
> it?). So, for example, if I disagreed with type theory as a foundation,
> I would nevertheless be able to understand its rules of operation
> (either at a rigorous informal level, or at a formal level if the theory
> is formalized) and follow its formal proofs.
>
> In particular, how can the precise notion of formal system be defined
> using imprecise natural language (as we do).
>
> Martin
>
>

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Re: Grothendieck toposes

[Joyal, André]

Dear All,

The distance between the notion of elementary "topos"
and the notion of space is particularly striking with Hyland's effective "topos". 
The effective "topos" is inspired by constructive set theory (Kleene recursive realizability)
and it is among the important constructions in the theory of elementary "toposes".
But it has no effect on topology! (ask your favorite topologist).
We should probably stop saying that the theory of elementary "topos" (generalised) is topology!
It appears to be more a branch of higher order logic formulated in the language of category theory. 
We can hope that the theory of  elementary "toposes" will eventually find 
applications to topology (something recognised by topologist). The fact that a  (Grothendieck) topos E
over a base (Grothendieck) topos S can be regarded as a  (Grothendieck) topos 
internal to the set theory defined by S could play a role. 

Homotopy type theory is a new avenue for the applications  of higher order logic to topology. 
But the notion of elementary higher "topos" has not yet been formalised precisely.

Best regards,
André


________________________________________
From: Eduardo Julio Dubuc [edubuc@dm.uba.ar]
Sent: Thursday, November 03, 2016 12:45 AM
To: categories@mta.ca
Subject: categories: Re: Grothendieck toposes

The very notion of Grothendieck topos involves in fact two topos
(sheaves over a site on the base). That is, it is a notion of geometric
morphism, it does not make sense as a single category.

The notion of elementary topos is a completely different notion which
involves a single topos, it does perfect sense as a single category.

This observation was made to me by Jacques Penon about 36 years ago.

I agree with Joyal that the two notions should be kept apart. Not doing
so has created a lot of confusion.

best  e.d.


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Re: Grothendieck toposes

[Townsend, Christopher]

Dear all,

The fact that the 'right' morphisms of topos theory (geometric morphisms) are not structure preserving maps gives rise to the debate below. With an entirely localic approach the situation can be recovered; but at the cost of moving from 'elementary topos' as primitive to 'category of locales'. Here geometric morphisms can be represented as adjunctions that commute with the  double power locale monad and it is the double power locale monad that gives the structure in categories of locales. So there are avenues worth exploring which may provide a way of marrying up the structural disconnect between objects and morphisms that is implicit in topos theory. 

Christopher




Subject: Re: categories: Re: Grothendieck toposes

On 2016-11-01 17:16, Joyal wrote:

> It is marvelous that the two notions should be so related.
> But it is be better to keep them appart before uniting them.
> Otherwise the miracle disappear in confusion.

The "miracle disappears in confusion" is an important observation, as is the need to "keep apart before uniting".

Syntax and semantics is like that, or meta and object language.
Foundations of mathematics without categorical consideration is basically then over  Set, naively speaking. Logic is similar. Fons et origo logic from  late 19th century and decades after is confused about being before set theory or after. Topos internalizes logic but is different from the Goguen-Meseguer approach to institutions and entailment systems. The apples and pears  of logic should not be seen as a fruit salad.

I've sometimes thought (and written some pieces about, e.g. to be found under www.glioc.com<http://www.glioc.com>) what if Gödel's Incompleteness  Theorem wasn't a Theorem but a Paradox. After all, Gödel basically transforms the Liar Paradox to a Liar Theorem, and logicians at that time (except maybe Hilbert, but he was too old to quarrel) found it to be very smart.
However, if we use underlying categories and functors to start from signatures, then create terms, then sentences, then entailments, then models, then  proof strategies, and so on, it means we close doors behind us, so that we  disable ourselves to mix truth and provability as being "of the same kind or type", which Gödel did. Categorically, terms come from monads, because  they enable substitution, but sentences just from functors, because otherwise everything is 'term'. The functorial description and generality of entailment and model is of course more tricky, in particular if the underlying category is something more elaborate (like monoidal closed categories) than  just Set.

In this (heretic?) view, Gödel's Theorem/Paradox is actually an example where that miracle appears because of the unintended (?) confusion, so this is why I sometimes think what if it was ween as a Paradox.

Best,

Patrik

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Re: Grothendieck toposes

[Patrik Eklund]

On 2016-11-03 12:17, Steve Vickers wrote:

> Actually, maths is _strong_ on types.

Yes, and I would say mathematics in particular as used within
theoretical computer science (my chair is in that area), and how it
focuses on types and type constructors, but the

> ... way "HoTTematics" doesn't comply with mathematics.

to me is a concern, e.g., since 'type constructors' are brought in from
the outside, so as to say, and this is a bit of that "doesn't comply",
isn't it, Steve? We've tried out something to keep those constructors
"inside", not (yet) for that empty type, but some small pieces can be
found following the links under GLIOC. It shows e.g. how we try to
relate things back to Sch??nfinkel, Church and Curry and their G??ttingen
times. Church's 'o' type in his 1940 paper is still not well understood,
I think. His 'iota' is Martin-L??f's 'type', and we know how things went
wrong when they started to say "type is a type". It was fixed but that
fixing indeed "doesn't comply", does it?

Best,

Patrik


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Re: Grothendieck toposes

[Steve Vickers]

On 03/11/2016 08:45, Patrik Eklund wrote:
> ...
>
> PS Why not also try something out about the empty type. The only thing
> in mathematics that axiomatically is, not just exists, but is, is the
> empty set, and then we put brackets around, and create natural
> numbers. From natural numbers we then create everything else in
> mathematics. Couldn't the empty type be something similar? But we
> shouldn't lean on the HoTT community because they don't comply with
> mathematics in their HoTTematics. They are two dofferent worlds. Set
> theory is untyped, so math is weak on types.
Dear Patrik,

You're too embedded in the world-view that maths is set theory.

It is a miracle* that maths could be built up from the empty set using
just axioms for collections, so that every mathematical construct is a
set of sets of sets of ... .

A century later, some of us look back on the miracle of set theory and
try to appraise it with the benefit of more hindsight.

Actually, maths is _strong_ on types. We implicitly use types all the
time. If we talk about natural numbers, we don't assume any particular
way of _constructing_ them within set theory. We work with rules _using_
them, and that is a type-theoretic standpoint.

If set theory is weak on types (and it is), then that is showing the
limitations of set theory, not that in some way "HoTTematics" doesn't
comply with mathematics.

All the best,

Steve.

* Apologies to Marta here: but a miracle is just something to be
marvelled at because you'd previously thought it impossible, right?
Minor miracles are the stuff of life, since our rationality is never
complete enough to know for sure what's possible and what's not. In
mathematics we can see just from the historical evidence that Bishop's
constructive analysis was a miracle. Toposes are a miracle - in fact
topologists still think it's impossible, or they would have changed
their definition of topology. And I think elementary topos theory is a
bit of a miracle too - that the infinitary constructions inherent in
topology can be captured in finitely axiomatizable systems.


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Re: Grothendieck toposes

[Eduardo Julio Dubuc]

The very notion of Grothendieck topos involves in fact two topos
(sheaves over a site on the base). That is, it is a notion of geometric
morphism, it does not make sense as a single category.

The notion of elementary topos is a completely different notion which
involves a single topos, it does perfect sense as a single category.

This observation was made to me by Jacques Penon about 36 years ago.

I agree with Joyal that the two notions should be kept apart. Not doing
so has created a lot of confusion.

best  e.d.


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Grothendieck toposes

[Marta Bunge]

Dear Andre,


Of course I agree with you in that the natural world is only partially explained by science. I also believe that much of mathematics has been inspired  by a desire to apply it to the natural sciences. Notice, however, that I said "much" and not "all",  and here is where mathematics and the natural sciences differ. Mathematicians have a freedom not afforded to scientists.  It is this freedom which allows the invention of objects such as the complex  numbers or of the infinitesimals. Now, is it a mystery that such products of the human mind find applications in scientific theories, or is it rather  that the latter themselves are also the product of the human mind? After all, it is only through rational thinking (including intuition) that we are able to (believe we) understand the natural world. Now, what about actual applications of science? Those are not just the product of the theories themselves, but also of experimentation and of successive approximations. It is  for this reason that I see no mystery in that certain scientific theories can sometimes be succesfully applied. Whether this is or is not a metaphysical point of view it is not for me to say.


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: Joyal, André <joyal.andre@uqam.ca>
Sent: November 2, 2016 1:50:08 PM
To: Marta Bunge; categories@mta.ca
Cc: Steve Vickers; Patrik Eklund
Subject: RE: categories: Re: Grothendieck toposes

Dear Marta,

Mathematics and science are very often regarded
as the pure product of human rationality.
I can agree with the importance of rationality, except that humanity
is as much the product of nature as it is of rational choices.
You will agree that the natural world is only partially explained by science.
The rest is a big mystery. Not that the mystery is absolutly impenetrable.
I feel compelled to recognize the presence of mysteries even
in mathematics. The history of complex numbers, from the
discovery by Cardano to their applications in quantum physics is bewildering.
They belong to this universe as much as the electron and the human mind.
The fact that we human can understand  complex numbers may
have a metaphysical meaning.
What is it?

Best,
André



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Re: Grothendieck toposes

[majordomo]

Dear Marta,

Mathematics and science are very often regarded
as the pure product of human rationality.
I can agree with the importance of rationality, except that humanity
is as much the product of nature as it is of rational choices.
You will agree that the natural world is only partially explained by science.
The rest is a big mystery. Not that the mystery is absolutly impenetrable.
I feel compelled to recognize the presence of mysteries even
in mathematics. The history of complex numbers, from the
discovery by Cardano to their applications in quantum physics is bewildering.
They belong to this universe as much as the electron and the human mind.
The fact that we human can understand  complex numbers may
have a metaphysical meaning.
What is it?

Best,
André


________________________________
From: Marta Bunge [martabunge@hotmail.com]
Sent: Wednesday, November 02, 2016 7:18 AM
To: categories@mta.ca
Cc: Steve Vickers; Patrik Eklund; Joyal, André
Subject: Re: categories: Re: Grothendieck toposes


Dear all,



> It is marvelous that the two notions should be so related.
> But it is be better to keep them appart before uniting them.
> Otherwise the miracle disappear in confusion.



The above is a quotation from a recent posting by Andre Joyal.  To the risk  of boring everyone I offer the following comment on it here. There is no need to talk about miracles in mathematics, not even as some sort of analogy. Why not instead give credit to the very important insight of an elementary topos as embodying both the logic and the geometry? There are two notions  of morphism between elementary toposes, not a preferred one - the geometric and the logical. One structure - to wit that of an elementary topos, can be seen in two different ways depending on what the mathematical uses one wants to give it. There is no confusion here  - just richness. Let me be more specific.


Thinking of an elementary topos S as the chosen "set theory", a Grothendieck topos (including any category of the form Sh(X) for X a locale in S, but more generally as a category of sheaves on a site in S) can be recovered as  a pair (E, e) where E is another elementary topos and e: E -> S a bounded geometric morphism. Thinking of elementary toposes from the logical point of view, and so of logical morphisms between them, there are other ideas and  constructions that profit from this point of view - for instance a formulation and proof of realizability by means of Artin-Wraith glueing.


Both the geometric and the logical are sides of the same coin. The notion of an elementary topos (or "topos" for short) is simple yet powerful and until now it has served most of the mathematical purposes for which it was intended and more.


Best wishes,

Marta




[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Grothendieck toposes

[Townsend, Christopher]

Dear all,

The fact that the 'right' morphisms of topos theory (geometric morphisms) a=
re not structure preserving maps gives rise to the debate. With an entirely=
  localic approach the situation can be recovered; but at the cost of moving=
  from 'elementary topos' as primitive to 'category of locales'. Here geomet=
ric morphisms can be represented as adjunctions that commute with the doubl=
e power locale monad and it is the double power locale monad that gives the=
  structure in categories of locales. So there are avenues worth exploring w=
hich may provide a way of marrying up the structural disconnect between obj=
ects and morphisms that is implicit in topos theory. =


Christopher =



  =


-----Original Message-----
From: Marta Bunge [mailto:martabunge@hotmail.com] =

Sent: 02 November 2016 11:18
To: categories@mta.ca
Cc: Steve Vickers; Patrik Eklund; Joyal, Andr=E9
Subject: categories: Re: Grothendieck toposes

Dear all,



> It is marvelous that the two notions should be so related.
> But it is be better to keep them appart before uniting them.
> Otherwise the miracle disappear in confusion.



The above is a quotation from a recent posting by Andre Joyal.  To the risk=
   of boring everyone I offer the following comment on it here. There is no =
need to talk about miracles in mathematics, not even as some sort of analog=
y. Why not instead give credit to the very important insight of an elementa=
ry topos as embodying both the logic and the geometry? There are two notion=
s  of morphism between elementary toposes, not a preferred one - the geomet=
ric and the logical. One structure - to wit that of an elementary topos, ca=
n be seen in two different ways depending on what the mathematical uses one=
  wants to give it. There is no confusion here  - just richness. Let me be m=
ore specific.


Thinking of an elementary topos S as the chosen "set theory", a Grothendiec=
k topos (including any category of the form Sh(X) for X a locale in S, but =
more generally as a category of sheaves on a site in S) can be recovered as=
   a pair (E, e) where E is another elementary topos and e: E -> S a bounded=
  geometric morphism. Thinking of elementary toposes from the logical point =
of view, and so of logical morphisms between them, there are other ideas an=
d  constructions that profit from this point of view - for instance a formu=
lation and proof of realizability by means of Artin-Wraith glueing.


Both the geometric and the logical are sides of the same coin. The notion o=
f an elementary topos (or "topos" for short) is simple yet powerful and unt=
il now it has served most of the mathematical purposes for which it was int=
ended and more.


Best wishes,

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/ ]

Re: Grothendieck toposes

[Marta Bunge]

Dear all,



> It is marvelous that the two notions should be so related.
> But it is be better to keep them appart before uniting them.
> Otherwise the miracle disappear in confusion.



The above is a quotation from a recent posting by Andre Joyal.  To the risk  of boring everyone I offer the following comment on it here. There is no need to talk about miracles in mathematics, not even as some sort of analogy. Why not instead give credit to the very important insight of an elementary topos as embodying both the logic and the geometry? There are two notions  of morphism between elementary toposes, not a preferred one - the geometric and the logical. One structure - to wit that of an elementary topos, can be seen in two different ways depending on what the mathematical uses one wants to give it. There is no confusion here  - just richness. Let me be more specific.


Thinking of an elementary topos S as the chosen "set theory", a Grothendieck topos (including any category of the form Sh(X) for X a locale in S, but more generally as a category of sheaves on a site in S) can be recovered as  a pair (E, e) where E is another elementary topos and e: E -> S a bounded geometric morphism. Thinking of elementary toposes from the logical point of view, and so of logical morphisms between them, there are other ideas and  constructions that profit from this point of view - for instance a formulation and proof of realizability by means of Artin-Wraith glueing.


Both the geometric and the logical are sides of the same coin. The notion of an elementary topos (or "topos" for short) is simple yet powerful and until now it has served most of the mathematical purposes for which it was intended and more.


Best wishes,

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: Patrik Eklund <peklund@cs.umu.se>
Sent: November 2, 2016 6:13:30 AM
To: Joyal, André
Cc: Marta Bunge; categories@mta.ca; Steve Vickers
Subject: Re: categories: Re: Grothendieck toposes

On 2016-11-01 17:16, Joyal wrote:

> It is marvelous that the two notions should be so related.
> But it is be better to keep them appart before uniting them.
> Otherwise the miracle disappear in confusion.

The "miracle disappears in confusion" is an important observation, as is
the need to "keep apart before uniting".

Syntax and semantics is like that, or meta and object language.
Foundations of mathematics without categorical consideration is
basically then over  Set, naively speaking. Logic is similar. Fons et
origo logic from late 19th century and decades after is confused about
being before set theory or after. Topos internalizes logic but is
different from the Goguen-Meseguer approach to institutions and
entailment systems. The apples and pears of logic should not be seen as
a fruit salad.

I've sometimes thought (and written some pieces about, e.g. to be found
under www.glioc.com<http://www.glioc.com>) what if Gödel's Incompleteness  Theorem wasn't a
Theorem but a Paradox. After all, Gödel basically transforms the Liar
Paradox to a Liar Theorem, and logicians at that time (except maybe
Hilbert, but he was too old to quarrel) found it to be very smart.
However, if we use underlying categories and functors to start from
signatures, then create terms, then sentences, then entailments, then
models, then proof strategies, and so on, it means we close doors behind
us, so that we disable ourselves to mix truth and provability as being
"of the same kind or type", which Gödel did. Categorically, terms come
from monads, because they enable substitution, but sentences just from
functors, because otherwise everything is 'term'. The functorial
description and generality of entailment and model is of course more
tricky, in particular if the underlying category is something more
elaborate (like monoidal closed categories) than just Set.

In this (heretic?) view, Gödel's Theorem/Paradox is actually an example
where that miracle appears because of the unintended (?) confusion, so
this is why I sometimes think what if it was ween as a Paradox.

Best,

Patrik

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Re: Grothendieck toposes

[Joyal, André]

Dear All,

Just a few remarks:

Grothendieck was always very careful with terminology.
The name "topos" is an explicit reference to the idea of space.
The notion of geometric morphism between toposes is
taken from the notion of continuous maps between topological spaces.

The idea of an "elementary topos" is a child of categorical logic,
especially of the axiomatisation of the category of sets by Lawvere.
The natural notion of morphism between elementary toposes is that of logical functor.

It is marvelous that the two notions should be so related.
But it is be better to keep them appart before uniting them.
Otherwise the miracle disappear in confusion.

  
Best,
André


________________________________________
From: Marta Bunge [martabunge@hotmail.com]
Sent: Sunday, October 30, 2016 4:17 PM
To: categories@mta.ca
Cc: Steve Vickers
Subject: categories: Re: Grothendieck toposes

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
************************************************

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Grothendieck toposes

[Marta Bunge]

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.


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Grothendieck toposes

[Joyal, André]

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Dear All,

The difference between a Grothendieck topos and an elementary topos
is like the difference between a "frame"(= a locale) and a Heyting algebra.

A Grothendieck topos should be called a "topos",
whereas an elementary topos could be called a "logical topos".

Best regards,
André

________________________________________
From: Michael Shulman [shulman@sandiego.edu]
Sent: Saturday, October 29, 2016 11:06 PM
To: David Yetter
Cc: categories@mta.ca
Subject: categories: Re: Grothendieck toposes

I would tend to assume that a "Grothendieck topos" is one bounded over
"Set", whatever the current meaning of "Set" is, and in particular
whether or not "Set" is classical.  Thus, when working in the internal
language of an arbitrary topos S, I would say "Grothendieck topos" to
mean what *externally* to S would be called a bounded S-topos.

On Fri, Oct 28, 2016 at 12:08 PM, David Yetter <dyetter@ksu.edu> wrote:
> I, for one, would assume the same meaning for the phrase "Grothendieck topos" as is used in the Elephant.
>
> David Yetter
>
>

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Re: Grothendieck toposes

[Michael Shulman]

I would tend to assume that a "Grothendieck topos" is one bounded over
"Set", whatever the current meaning of "Set" is, and in particular
whether or not "Set" is classical.  Thus, when working in the internal
language of an arbitrary topos S, I would say "Grothendieck topos" to
mean what *externally* to S would be called a bounded S-topos.

On Fri, Oct 28, 2016 at 12:08 PM, David Yetter <dyetter@ksu.edu> wrote:
> I, for one, would assume the same meaning for the phrase "Grothendieck topos" as is used in the Elephant.
>
> David Yetter
>
>


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Grothendieck toposes

[David Yetter]

I, for one, would assume the same meaning for the phrase "Grothendieck topos" as is used in the Elephant.

David Yetter



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Grothendieck toposes

[Steve Vickers]

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.


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]