Re: Fibred 2-category of Grothendieck toposes?

[Marta Bunge <martabunge1@me.com>]

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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