Fibred 2-category of Grothendieck toposes?

Fibred 2-category of Grothendieck toposes?

[Steve Vickers]

If "Grothendieck topos" means bounded geometric morphism into a given
base S, then by allowing the base to vary we can get a 2-category GTop
of Grothendieck toposes, fibred over some form of ETop (elementary
toposes). This is because pseudopullbacks of bounded geometric morphisms
along arbitrary geometric morphisms exist and are still bounded. (I say
"some form of" ETop because it may be better to restrict the 2-cells
downstairs to be isos, even though we certainly don't want to do the
same upstairs. Also nnos are needed if classifying toposes are to exist.)

Has anyone worked on that particular combination of bounded and unbounded?

Steve Vickers.


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Re: Fibred 2-category of Grothendieck toposes?

[Thomas Streicher]

Steve,

we are speaking about different things. Pullbacks of bgm's along
arbitrary gm's are described sufficiently well [Joh77] and in the
Elephant.

But what I have commented on was how to appropriately captured change
of base when studying topose sover a base topos.

I am working on the "algebraic" side whereas you think on the
"geometric" side. Reindexing in terms of the algebraic view is given
by precomposition but reindexing in terms of the geometric view is
postcomposition.

What I want to say is that pullbacks in Top are very different from
the change of base for relative topos theory.

Thomas


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Re: Fibred 2-category of Grothendieck toposes?

[Thomas Streicher]

Dear Steve,

> How shall we wrap this up?

the point is that postcomposition with a gm F-|U is tanatamount to
change of base long F.

Thomas


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Re: Fibred 2-category of Grothendieck toposes?

[Marta Bunge]

Dear Steve,

The setting of a 2-category GTop bounded over an elementary topos S has been  extensibly worked out in practice in (for instance) the book by Marta Bunge  and Jonathon Funk, Singular Coverings of Toposes, LNM 1890, Springer 2006, and in several papers by myself or with collaborators which you can look up in my Research Gate page. The terminology that I have used for it everywhere (including lectures) is Top_S, by which it is not meant Top/S but the sub 2-category of it whose objects are bounded geometric morphisms between elementary toposes, with codomain S. In particular It often becomes necessary to consider change of base. The terminology is well adapted to the consideration of certain distinguished sub 2-categories of Top_S - for instance LTop_S denotes that whose objects are geometric morphisms with codomain S and a locally  connected elementary topos. I hope this is useful to you. 

Cordial regards,
Marta

> On Dec 2, 2016, at 5:20 AM, Steve Vickers <s.j.vickers@cs.bham.ac.uk> wrote:
> 
> If "Grothendieck topos" means bounded geometric morphism into a given
> base S, then by allowing the base to vary we can get a 2-category GTop
> of Grothendieck toposes, fibred over some form of ETop (elementary
> toposes). This is because pseudopullbacks of bounded geometric morphisms
> along arbitrary geometric morphisms exist and are still bounded. (I say
> "some form of" ETop because it may be better to restrict the 2-cells
> downstairs to be isos, even though we certainly don't want to do the
> same upstairs. Also nnos are needed if classifying toposes are to exist.)
> 
> Has anyone worked on that particular combination of bounded and unbounded?
> 
> Steve Vickers.


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Fibred 2-category of Grothendieck toposes?

[Steve Vickers]

This picks up a thread from a month ago, regarding an idea that
Grothendieck toposes, interpreted generally as geometric morphisms
bounded over the codomain base topos, might be treated in a 2- (or
bi-)fibrational way over a variable base.

I have now completed a paper "Arithmetic universes and classifying
toposes" that does this. It is submitted to arXiv, and meanwhile
available at

   http://www.cs.bham.ac.uk/~sjv/papersfull.php#AUClTop

The arithmetic universes enter in via a logic interpretable in any
elementary topos with nno. The "arithmetic" reasoning gives topos
results that are base-independent and also "geometric", in the sense
that when the base varies along a geometric morphism, the classifying
topos transforms by pseudopullback.

Steve.




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