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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]