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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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/ ]
Fibered toposes are discussed in detail in SGA 4, expose VI as well as in Illusie's thesis. Bill Messing [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Already in 1962-63 in SGA4 was considered and developed the concept of Fibred Topos (SLN 270 VI 7. p 273). Recall that in SGA4, Topos (or U-topos) = your Bounded Topos. If you want to consider a different concept of fibred topos, it is not correct to use the same name. best eduardo On 18/12/12 03:28, 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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (5)
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David Roberts -
Eduardo J. Dubuc -
Steve Vickers -
Urs Schreiber -
William Messing