Re: Reply to Eduardo Dubuc

[Marta Bunge <martabunge@hotmail.com>]

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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