Dear David, 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). If it's the first kind, the usual approach is to use classifying toposes. So as analogue of Set_* --> Set you let S be the object classifier (classifies the geometric theory with a single sort and nothing else) and E classify the theory with a single sort and a constant with that sort. The geometric morphism E --> S in effect forgets the constant. For a generic fibred topos you might try to have S classify sites, and E classify sites equipped with a distinguished model. E --> S forgets the model. But I would conjecture it's not possible to get either a fibration or an opfibration in general - in general a site morphism does not give a functor in either direction between the fibres (the model categories) - and that's a nuisance. Johnstone ("Fibrations and partial products in a 2-category", section 7) has written about generic ones of more restricted kinds where you do get either fibrations or opfibrations. For example, the one analogous to Set_* --> Set is an opfibration. A lot of my own work has been about situations where the geometric morphism E --> B is localic - see, e.g., my "Topical categories of domains". Best wishes, Steve Vickers. David Roberts wrote:
Dear all,
I'm thinking about fibred toposes, and I was wondering if there any references people can suggest? The following are some pitifully vague thoughts.
One particular problem I'm thinking about is whether there is a generic fibred topos, which is the analogue of the generic discrete fibration Set_* --> Set or the generic fibration 1 / Cat --> Cat.
Something like the 2-category Topos of bounded toposes and geometric morphisms (and whatever 2-arrows are appropriate). The objects of this are bounded geometric morphisms, arrows are 2-commutative squares. Then take the 2-category over this where the objects are bounded toposes E --> S with a point Set --> E, or possibly an S-point S --> E, and arrows those geometric morphisms which preserve the point up to natural transformation.
Ideally I'd then like to consider 2-functors T^op -->Topos to be equivalent to (bounded) fibred toposes over T.
Best,
David Roberts
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