Re: Reply to Eduardo Dubuc

Re: Reply to Eduardo Dubuc

[Thomas Streicher]

Dear Eduardo,

I understand your point better now. For you FX is already given as the
respective colimit. This was shown by Celeyrette in his 1974 Th'ese
directed by B'enabou (Ch.VI Theorem de Kan where you also find a proof
of Diaconescu's theorem over arbitrary bases).
But you have to rely on a theorem characterizing colimits of small
diagrams in SS from which it follows that the source tupling (in the
sense of internal sums) of the cone is epic.
Even if you think it is splitting hairs you cannot express the
property of being epic in SS within the internal language of SS. You
can express in the internal language of SS that a map is surjective
and show externally that SS validates f being surjective iff f is epic
in SS. (For expressing "epic" in SS one would have to quantify over
all objects in SS which one cannot do in the internal language of SS.)

So my conclusion is that the argument you give is ok for most purposes.
What I wrote above (and previously) gives some additional justification
of its correctness.

I will be offline till next weekend and therefore have no time to
continue a discussion with many mails back and forth.
Moreover, I think things have been clarified to quite some extent.

Best, Thomas

PS "purely external categorical construction in SS"
is a contradiction in terms
formulations like these have given me the impression that you have
wrong reason for a correct result



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Re: Reply to Eduardo Dubuc

[David Roberts]

Hi,

Since my answer on MathOverflow has been invoked a couple of times,
let me expand on it a little.

There is a notion of a topos being *constructively* well-pointed,
which I learned from Mike Shulman (or rather, his paper Stack
semantics and the comparison of material and structural set theories).
This is the sort of thing that the category of CZF-sets has. And Mike
proved that in the internal logic (in fact in an extension of the
usual one, to cope with unbounded quantifiers), every topos is
constructively well-pointed.

Since Eduardo, at the beginning of his question stated that he was
going to work in the internal logic of the base topos, I took this as
given. Then one can reason as if one has an object * playing the rôle
of the point which picks out elements of 'sets'. The definition of
epimorphism in the internal logic is just that any generalised element
in the codomain lifts to the domain, and this reduces to checking for
elements using *. The coproduct causes no trouble, being indexed by a
'set', and disjoint union doesn't go awry in the intuitionistic
setting, by virtue of the fact coproducts are disjoint in a topos.

To address the point about unbounded quantifiers, Mike gives the
notion of a topos being *autological*, which means that in the
internal logic one has unbounded separation, and so as much of set
theory as one gets from e.g. ZF in the classical case. Given an
autological base topos SS, all locally small cocomplete toposes over
SS are autological, including unbounded ones.

Mike might be able to answer more specific questions people have, but
that is my line of reasoning.

Best,

David

On 29 July 2013 00:50, Thomas Streicher
<streicher@mathematik.tu-darmstadt.de> wrote:
> Dear David,
>
> indeed Michael's work is an alternative to the AST approach taylored
> towards a setting where it is forbidden to speak of equality of sets.
> He doesn't construct the category of classes around a topos but rather
> works with Kripke Joyal semantics for it.
>
> Thomas


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Re: Reply to Eduardo Dubuc

[Marta Bunge]

Eduardo, 

We (Thomas andI) are not answering a different question - at least I was not. Maybe the notation used by Thomas made you think so. Still it is the case that you start with a bounded S-topos EE-->SS (unnamed)) and then consider a point SS --->EE over SS, whose inverse image you denote by F:EE -->SS. I understood that. Now there are the following considerations. 


(a) From EE bounded over SS you get a site CC (with a topology j, say) which is internal to SS. This gives you, for an object X of EE, the epimorphism  from the appropriate coproduct in EE to X. The remarks Thomas made (at least some of them) correspond to this basic construction you definitely use. I totally agree with that. You need an object G of generators and the epi in question expressed by the diagram Thomas included in his answer, and which you  can also see in page 51-52 of my book with Jonathon  Singular Coverings of Toposes, SLNM 1890, 2006. 



(b) Now you resort to F:EE ---> SS which is left exact and has an SS-indexed  right adjoint. This is why you may infer that F preserves the given diagram  of an epi into X from a coproduct involving CC, which is again an epi from a  coproduct (with the same indexing "set"!) to FX. Cannot draw these diagrams  in an iPad, at least not easily. Here you argue "as in Set", but the part that needs the interplay between EE and SS us the first one and you cannot avoid that. 

Without Joyal-Tierney's Memoir at hand I cannot comment on your assertion about it except by recollection. I will do so in Montreal next week. 

As for Diaconescu's theorem, it too requires the notion of an internal object of generators, which is one of of his main contributions in lifting a notion well understood for Grothendieck toposes over Set to "Grothendieck toposes" over an arbitrary base topos SS. Another is of course to express the correct lifting of "flatness". 

I am aware that there are attempts to describe a logic that extends the internal logic of a topos SS but in which all of this can be expressed including  fibrations over SS and, in the process, getting rid of all the neat constructions involving category theory. In my view, the goal of reducing topos theory to a certain "naive constructive set theory", even if succesful, will not aid in the formulation of new concepts or their use. As a matter of pride,  logicians may be able to say "we did it",  but category theorists in general and topos theorists in particular will not be attracted by it. Hopefully, that is. Finally, everyone knows that the internal logic of a topos is intuititionistic  But who exhibited the object \Omega which permits such a conclusion? Grothendieck calls \Omega "the Lawvere object". 

Please revise carefully your construction and argument, and then tell me if I  am not right. I close shop now until further notice.


Marta

Sent from my iPad

On 2013-07-27, at 8:18 PM, "Eduardo J. Dubuc" <edubuc@dm.uba.ar> wrote:

> Well, I intent to work as Joyal-Tierney do in their memoir, justified by
> the internal logic of the topos (e.g. Joyal-Boileau). In this way,
> internal sites have objects, internal lattices have elements, presheaves
> H can be evaluated in an object C of the site CC yielding an object HC
> in SS, etc etc. Sheaves are presheaves satisfying the usual
> requirements, they form a Grothendieck (or bounded) topos
> sh(CC) = EE ---> SS over SS, etc etc. In this context I asked:
> 
>> Consider F: CC ---> SS  to be
>> the inverse image of a point. Then the family Ff: FC ---> FX 
>> epimorphic in SS.
>> 
>> My question is:
>> 
>> Can I do the following ? (meaning, is it correct the following arguing,
>> certainly valid if SS is the topos of sets):
> I clarify now, given a fix (but arbitrary) object X in EE, and
> 
>> Given a in FX, take f:C ---> X and c in FC such that a = Ff(c).
> 
> 
> The answers of David Roberts and Andreas Blass on MathOverflow (that I
> quoted in a previous posting) say the arguing (in the internal language)
> was correct, and this seems quite satisfactory to me. But there was no
> explicit or detailed justification.
> 
> Thomas Streicher (and Marta Bunge making Thomas answer her own) answer
> bringing into play the theory of fibrations/indexed. I appreciate this
> answer because made me think different aspects of the question, and for
> the references given.
> 
> But fibrations/indexed are not necessary when working as in
> Joyal-Tierney Memoir (as they are not used in that memoir, which by the
> way, is extremely carefully written)
> 
> In this last posting, I realized they are answering a different
> question, it seems quite extraordinary! that we were taking of different
> things without realizing it !! (I reading their answers, and they
> reading my question and my expanded clarification)
> 
> I say F: CC ---> SS the inverse image of a point. Abusing notation as
> usual, we also write F: EE ---> SS.
> 
> Now, as it follows from this last posting, they (for some reason that I
> ignore) interpret that F is the inverse image SS ---> EE of the g.m.
> EE ---> SS defining EE over SS (g.m. that I did not even denoted by any
> letter since it does not play a role in the formulation of the
> question). The g.m whose inverse image I denoted F is that of a point SS
> ---> EE, as I explicitely say it, and as I thought it was clear in my
> question.
> 
> I should have realized in the original Thomas answer that he was
> thinking in a different question, but did not !.
> 
> All this is mathematics, so I hope it should be completely clarified.
> 
> The question and its derivations may be of interest to members of the
> list that do not post about it, but if the moderator thinks this is not
> so, I will pursue my search for understanding these matters privately.
> 
> best   e.d.
> 
> On 27/07/13 05:56, Marta Bunge wrote:
>> Dear Thomas,
>> 
>> What you wrote in the first part of your letter ( before the last
>> paragraph) is precisely what I wanted to write but decided to wait
>> until Montreal also to answer Eduardo's other questions, for which I
>> needed to have the references at hand. Also a better computer than my
>> laptop here in Greece. But I shall be home next week.
>> 
>> In short, I fully agree with you. Thanks for answering Eduardo's
>> question to me with such precision. I could not have done it better
>> myself.
>> 
>> Best regards, Marta
>> 
>> Sent from my iPad
>> 
>> On 2013-07-27, at 4:33 AM, "Thomas
>> Streicher"<streicher@mathematik.tu-darmstadt.de>  wrote:
>> 
>>> Dear Eduardo,
>>> 
>>> let's carefully look how internal to SS your statement is. You
>>> start from a bounded geometric morphism F -| U : EE ->  SS and an
>>> object X of EE. What ensures your claim is that there is a diagram
>>> (in EE)
>>> 
>>> e X<<--- u^*G ----->  G |         | |   p.b.  | V         V FJ
>>> ---->  FI Fu
>>> 
>>> where G -->  FI is a generic family for P_F (the fibered topos
>>> associated with F : SS ->  EE). Here you have an EXTERNAL
>>> existential quantification over J, u  and e. But using local
>>> smallness of P_F you can concretely witness J as \coprod_{i \in I}
>>> hom_EE(G_i,X) and u as first projection on I. From this you also
>>> get e. But that means that one proves the statement on the
>>> metalevel which allows one to speak about SS, EE and functors
>>> between them.
>>> 
>>> In particular, you have to argue how you can express within SS that
>>> e (a morphism of EE) is epic. How can you do this in the internal
>>> language of SS? You have to quantify over arbitrary maps in EE with
>>> source X.
>>> 
>>> Moreover, you do not want to have your claim valid for just a
>>> single particular external X in EE but you want to have it "for all
>>> X in EE". Kripke-Joyaling this amounts to considering all external
>>> families a : A -->  FK (as on p.51 of Marta and Jonathon's book
>>> though they use different letters).
>>> 
>>> There is some possibility of giving a coherent account of this. If
>>> you have your base topos SS then split fibrations over SS are the
>>> same as categories internal to Psh(SS) (where Psh(SS) = Set^{SS^op}
>>> for Set big enough to contain SS as an internal category). Now you
>>> can reason in the internal language of Psh(SS) but not in the
>>> internal language of SS. That I have learnt from B'enabou some
>>> years ago but it's unpublished (as usual). Now the problem is that
>>> Psh(SS) is too weak a logic and one might want to work rather in
>>> sheaves over SS w.r.t. regular cover topology.
>>> 
>>> In the work of Awodey, Forssell and Warren on their variant of
>>> algebraic set theory
>>> (http://www.phil.cmu.edu/projects/ast/Papers/afw_06.pdf) this is
>>> preformed to some extent. They consider Idl(SS), the "ideal
>>> completion of SS" within which SS appears as a small category.
>>> 
>>> Best regards, Thomas
> 

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Re: Reply to Eduardo Dubuc

[Eduardo J. Dubuc]

Well, I intent to work as Joyal-Tierney do in their memoir, justified by
the internal logic of the topos (e.g. Joyal-Boileau). In this way,
internal sites have objects, internal lattices have elements, presheaves
H can be evaluated in an object C of the site CC yielding an object HC
in SS, etc etc. Sheaves are presheaves satisfying the usual
requirements, they form a Grothendieck (or bounded) topos
sh(CC) = EE ---> SS over SS, etc etc. In this context I asked:

> Consider F: CC ---> SS  to be
> the inverse image of a point. Then the family Ff: FC ---> FX is
> epimorphic in SS.
>
> My question is:
>
> Can I do the following ? (meaning, is it correct the following arguing,
> certainly valid if SS is the topos of sets):
>
I clarify now, given a fix (but arbitrary) object X in EE, and

> Given a in FX, take f:C ---> X and c in FC such that a = Ff(c).


The answers of David Roberts and Andreas Blass on MathOverflow (that I
quoted in a previous posting) say the arguing (in the internal language)
was correct, and this seems quite satisfactory to me. But there was no
explicit or detailed justification.

Thomas Streicher (and Marta Bunge making Thomas answer her own) answer
bringing into play the theory of fibrations/indexed. I appreciate this
answer because made me think different aspects of the question, and for
the references given.

But fibrations/indexed are not necessary when working as in
Joyal-Tierney Memoir (as they are not used in that memoir, which by the
way, is extremely carefully written)

In this last posting, I realized they are answering a different
question, it seems quite extraordinary! that we were taking of different
things without realizing it !! (I reading their answers, and they
reading my question and my expanded clarification)

I say F: CC ---> SS the inverse image of a point. Abusing notation as
usual, we also write F: EE ---> SS.

Now, as it follows from this last posting, they (for some reason that I
ignore) interpret that F is the inverse image SS ---> EE of the g.m.
EE ---> SS defining EE over SS (g.m. that I did not even denoted by any
letter since it does not play a role in the formulation of the
question). The g.m whose inverse image I denoted F is that of a point SS
---> EE, as I explicitely say it, and as I thought it was clear in my
question.

I should have realized in the original Thomas answer that he was
thinking in a different question, but did not !.

All this is mathematics, so I hope it should be completely clarified.

The question and its derivations may be of interest to members of the
list that do not post about it, but if the moderator thinks this is not
so, I will pursue my search for understanding these matters privately.

best   e.d.

On 27/07/13 05:56, Marta Bunge wrote:
> Dear Thomas,
>
> What you wrote in the first part of your letter ( before the last
> paragraph) is precisely what I wanted to write but decided to wait
> until Montreal also to answer Eduardo's other questions, for which I
> needed to have the references at hand. Also a better computer than my
> laptop here in Greece. But I shall be home next week.
>
> In short, I fully agree with you. Thanks for answering Eduardo's
> question to me with such precision. I could not have done it better
> myself.
>
> Best regards, Marta
>
> Sent from my iPad
>
> On 2013-07-27, at 4:33 AM, "Thomas
> Streicher"<streicher@mathematik.tu-darmstadt.de>  wrote:
>
>> Dear Eduardo,
>>
>> let's carefully look how internal to SS your statement is. You
>> start from a bounded geometric morphism F -| U : EE ->  SS and an
>> object X of EE. What ensures your claim is that there is a diagram
>> (in EE)
>>
>> e X<<--- u^*G ----->  G |         | |   p.b.  | V         V FJ
>> ---->  FI Fu
>>
>> where G -->  FI is a generic family for P_F (the fibered topos
>> associated with F : SS ->  EE). Here you have an EXTERNAL
>> existential quantification over J, u  and e. But using local
>> smallness of P_F you can concretely witness J as \coprod_{i \in I}
>> hom_EE(G_i,X) and u as first projection on I. From this you also
>> get e. But that means that one proves the statement on the
>> metalevel which allows one to speak about SS, EE and functors
>> between them.
>>
>> In particular, you have to argue how you can express within SS that
>> e (a morphism of EE) is epic. How can you do this in the internal
>> language of SS? You have to quantify over arbitrary maps in EE with
>> source X.
>>
>> Moreover, you do not want to have your claim valid for just a
>> single particular external X in EE but you want to have it "for all
>> X in EE". Kripke-Joyaling this amounts to considering all external
>> families a : A -->  FK (as on p.51 of Marta and Jonathon's book
>> though they use different letters).
>>
>> There is some possibility of giving a coherent account of this. If
>> you have your base topos SS then split fibrations over SS are the
>> same as categories internal to Psh(SS) (where Psh(SS) = Set^{SS^op}
>> for Set big enough to contain SS as an internal category). Now you
>> can reason in the internal language of Psh(SS) but not in the
>> internal language of SS. That I have learnt from B'enabou some
>> years ago but it's unpublished (as usual). Now the problem is that
>> Psh(SS) is too weak a logic and one might want to work rather in
>> sheaves over SS w.r.t. regular cover topology.
>>
>> In the work of Awodey, Forssell and Warren on their variant of
>> algebraic set theory
>> (http://www.phil.cmu.edu/projects/ast/Papers/afw_06.pdf) this is
>> preformed to some extent. They consider Idl(SS), the "ideal
>> completion of SS" within which SS appears as a small category.
>>
>> Best regards, Thomas



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Re: Reply to Eduardo Dubuc

[Thomas Streicher]

Dear Eduardo,

let's carefully look how internal to SS your statement is. You start
from a bounded geometric morphism F -| U : EE -> SS and an object X of EE.
What ensures your claim is that there is a diagram (in EE)

             e
       X <<--- u^*G -----> G
                 |         |
                 |   p.b.  |
                 V         V
                 FJ ----> FI
                     Fu

where G --> FI is a generic family for P_F (the fibered topos
associated with F : SS -> EE). Here you have an EXTERNAL existential
quantification over J, u  and e. But using local smallness of P_F you
can concretely witness J as \coprod_{i \in I} hom_EE(G_i,X) and u as
first projection on I. From this you also get e. But that means that
one proves the statement on the metalevel which allows one to speak
about SS, EE and functors between them.

In particular, you have to argue how you can express within SS that e
(a morphism of EE) is epic. How can you do this in the internal language
of SS? You have to quantify over arbitrary maps in EE with source X.

Moreover, you do not want to have your claim valid for just a single
particular external X in EE but you want to have it "for all X in EE".
Kripke-Joyaling this amounts to considering all external families
a : A --> FK (as on p.51 of Marta and Jonathon's book though they use
different letters).

There is some possibility of giving a coherent account of this. If you
have your base topos SS then split fibrations over SS are the same as
categories internal to Psh(SS) (where Psh(SS) = Set^{SS^op} for Set
big enough to contain SS as an internal category). Now you can reason
in the internal language of Psh(SS) but not in the internal language of SS.
That I have learnt from B'enabou some years ago but it's unpublished
(as usual). Now the problem is that Psh(SS) is too weak a logic and
one might want to work rather in sheaves over SS w.r.t. regular cover
topology.

In the work of Awodey, Forssell and Warren on their variant of algebraic
set theory (http://www.phil.cmu.edu/projects/ast/Papers/afw_06.pdf) this is
preformed to some extent. They consider Idl(SS), the "ideal completion of SS"
within which SS appears as a small category.

Best regards, Thomas


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Re: Reply to Eduardo Dubuc

[Eduardo J. Dubuc]

On 26/07/13 17:08, Thomas Streicher wrote:
>>> 2) If on the other hand you work in a topos EE that is bounded
>>> over SS, then you cannot work "as in Sets" concerning
>>> constructions in EE that involve SS. Here it is necessary to
>>> resort to either the "teoria delle categorie sopra un topos di
>>> base" (indexed categories) or to the theory of fibrations.
>>>
>>> [[1===1]]
>>> Concerning your question, let me point out that, although you
>>> pretend to work entirely inside SS, you do not, as your
>>> construction involves EE. It certainly makes sense, since EE is
>>> locally small, and CC is small ( meaning internal to SS). What I
>>> want to warn you about is how you proceed from there without
>>> resorting to EE as a bounded SS- topos.
>
> I fully subscribe to this, That's what I also wanted to say but
> formulated less clearly.
>
> The saying that "toposes correspond to naive set theory without
> principle and excluded middle" is right only if "naive set theory"
> is identified with higher order logic. But in "naive category
> theory" you all the time have to quantify over families of objects
> and morphisms of a category where these collections don't form sets.
> In particular, you have to quantify over all objects of the base
> topos which you can't do inside the base topos.
>
> As Marta has said one has to resort to fibered/indexed reasoning
> which also cannot be (fully) expressed in the internal language of
> the base topos.
  >
  > [[2===2]]
  > As I wrote to Eduardo in a non-public mail I consider
> the "naive set theory" style arguments one often find in the
> literature as kind of "blueprints" for the precise fibered/indexed
> arguments.
>
> An alternative is using Algebraic Set Theory which allows one to
> build around every topos EE a category of classes in such a way that
> EE appears as internal to it. This, among other things, is the aim of
> a recent paper by Awodey, Butz, Simpson and myself which has is to
> appear at APAL and a preliminary version of which can be obtained
> from Alex's homepage
> (http://homepages.inf.ed.ac.uk/als/Research/Sources/set-models.pdf).
>
> Sorry for speaking pro domo but it is relevant here when one wants
> to fully stay within an internal language.
>
> Thomas

Thanks Marta an Thomas, I am learning a lot !.

I would like to comment in the items marked [[1===1]] and [[2===2]] of
your answers. It seems to me now that the argument in my question does
not need any theory of indexing categories.
See, concerning [1===1] above I can say:

My construction does not involve EE. It involves an internal site CC in
SS, a functor CC ---> SS, a coproduct in SS indexed by an object in SS,
and a SINGLE object X in EE, which is sheaf CC^op ---> SS. All this
involves ONLY SS, and no forbiden quantifications are used. So the
internal logic of SS should suffice.

Concerning [2===2], you are probably right in many cases, precisely
cases that I distrust because I feel the authors are not carefully
enough (this due to my ignorance together with a tendency to be as Saint
Thomas, "have to see to belive"). But in other cases not. For example,
the entire paper Joyal-Tierney, An extension of the galois theory of
grothendieck, Memoirs 309, is no "blue print" for nothing, it is in
final form, very careful, and entirely justified (even their change of
base) without any need of fibered/indexed arguments, as they say, (e.g.
Boileau-Joyal) suffices.

For example, they consider p: EE ---> SS, and say p_* sends sup-lattices
to sup-lattices, they consider a left adjoint  p^# (between the
categories of sup-lattices) and stablish an isomorphism

                Hom(L, p_*(M)) = p_*Hom(p^#(L), M)  in  SS

and say "expresses the (strong) adjointness p^# --| p_*" (page 44).

They use "strong", no mention at all to "indexed adjoint", and this
isomorphism is just a definition of strong.

But, is not true that a topos p: EE ---> SS is canonically indexed and
that for many things you can proceed tacitly without need to mention any
theory of indexed categories ?. Even for constructions in EE ?. For
example, a family in EE indexed by I in SS is just an arrow X ---> p*I,
and the COPRODUCT_{i \in I} X_i in EE is just X.

Please, do not think I am against fibered/indexed arguments, I can not
be either against or in favor, since I do not know the theory.

After your answers I understand now the theory is useful when you need
to use quantifications not avaiable in the internal logic of a topos and
other things of this sort.

Best  Eduardo.



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Re: Reply to Eduardo Dubuc

[Thomas Streicher]

>> 2) If on the other hand you work in a topos EE that is bounded over SS, then you cannot work "as in Sets" concerning constructions in EE that involve SS. Here it is necessary to resort to either the "teoria delle categorie sopra un topos di base" (indexed categories) or to the theory of fibrations.
>>
>> Concerning your question, let me point out that, although you pretend to work entirely inside SS, you do not, as your construction involves EE. It certainly makes sense, since EE is locally small, and CC is small ( meaning internal to SS). What I want to warn you about is how you proceed from there without resorting to EE as a bounded SS- topos.

I fully subscribe to this, That's what I also wanted to say but
formulated less clearly.

The saying that "toposes correspond to naive set theory without
principle and excluded middle" is right only if "naive set theory" is
identified with higher order logic. But in "naive category theory"
you all the time have to quantify over families of objects and
morphisms of a category where these collections don't form sets. In
particular, you have to quantify over all objects of the base topos
which you can't do inside the base topos.

As Marta has said one has to resort to fibered/indexed reasoning which
also cannot be (fully) expressed in the internal language of the base
topos. As I wrote to Eduardo in a non-public mail I consider the
"naive set theory" style arguments one often find in the literature as
kind of "blueprints" for the precise fibered/indexed arguments.

An alternative is using Algebraic Set Theory which allows one to build
around every topos EE a category of classes in such a way that EE
appears as internal to it. This, among other things, is the aim of a
recent paper by Awodey, Butz, Simpson and myself which has is to
appear at APAL and a preliminary version of which can be obtained from
Alex's homepage
(http://homepages.inf.ed.ac.uk/als/Research/Sources/set-models.pdf).

Sorry for speaking pro domo but it is relevant here when one wants to
fully stay within an internal language.

Thomas


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Reply to Eduardo Dubuc

[Marta Bunge]

> Dear Eduardo, 
> 
> Thank you for keeping me au courant. Difficult as it is to write in my iPad, I want to tell you my position and what I think of your question.
> 
> My position (as that of any topos theorist) is that
> 
> 1) If you work in an elementary topos SS, then it is true that you can do so "as in Sets", provided you do not use excluded middle ( as the internal logic is intuitionistic) or use Choice (as the set theory intrinsic to SS is constructive).
> 
> 2) If on the other hand you work in a topos EE that is bounded over SS, then you cannot work "as in Sets" concerning constructions in EE that involve SS. Here it is necessary to resort to either the "teoria delle categorie sopra un topos di base" (indexed categories) or to the theory of fibrations. 
> 
> Concerning your question, let me point out that, although you pretend to work entirely inside SS, you do not, as your construction involves EE. It certainly makes sense, since EE is locally small, and CC is small ( meaning internal to SS). What I want to warn you about is how you proceed from there without resorting to EE as a bounded SS- topos. 

>  I apologize for previous (private) irrelevant remarks as I misunderstood your question - until now, that is.
> 
> Concerning Diaconescu's theorem, AC implies Booleaness can be done "as in Sets" but the proof is not trival. If you mean the theorem which gives a characterization of bounded SS-toposes as a classifying topos, it requires fibrations over SS. 
> 
> Regards,
> Marta
> 
> 


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