Dear Marta, Yes, I'm familiar with large parts of your book in some detail, and it inspired my 2012 paper on cosheaves. The Elephant also has a whole section B4 on GTop/S, appearing there as BTop/S. But what I was interested in was the possibility of understanding the dependence of GTop/S on S as a fibration or an indexed 2-category - in other words as one single gadget that covers all bases. (This is because I'm trying to develop base-independent ways to state and prove results on classifying toposes.) Is there somewhere in your book that might cast light on that? All the best, Steve. On 02/12/2016 13:30, Marta Bunge wrote:
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.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]