Fibred toposes

Fibred toposes

[David Roberts]

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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Re: Fibred toposes

[Steve Vickers]

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/ ]

Re: Fibred toposes

[Urs Schreiber]

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


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Re: Fibred toposes

[Eduardo J. Dubuc]

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
>


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Re: Fibred toposes

[William Messing]

Fibered toposes are discussed in detail in SGA 4, expose VI as well as
in Illusie's thesis.

Bill Messing


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