Dear all, Steve Vickers wrote:
We need to be clear what you mean by "fibred topos". I would take it as bounded geometric morphism E --> S, thinking of topos as generalized space, but Streicher has used it in a sense of functor P: X -> B that is the fibrational analogue of B^op -> (Topos, logical functors).
Depending on what is actually intended in the application one could also consider variants such as B^op -> (Topos, geometric morphism) or even such presheaves of toposes that in addition satisfy some descent property. Along these lines Joost Nuiten has an interesting observation in Nuiten Bohrification of local nets of observables http://ncatlab.org/schreiber/show/bachelor+thesis+Nuiten where he shows that a certain presheaf of toposes satisfying descent by local geometric morphisms encodes certain locality structure in the "total space" topos that is of interest in the study of local nets of algebras. This is probably not related to what David is looking for, but maybe it serves as an example point in the space of interesting variants. Best, Urs [For admin and other information see: http://www.mta.ca/~cat-dist/ ]