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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 8th 2011
    • (edited Jun 8th 2011)

    In some application I am running into 2-sheaves (genuine category-valued analog of stacks) that happen to take values in (Grothendieck) toposes.

    I am wondering if there is any existing work on such gadgets that I should be aware of. Is there anything useful that has been said about topos-valued 2-sheaves? Did anyone do anything nontrivial with such?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 8th 2011

    maybe I should clarify: I am after something aking to indexed toposes, but maybe a bit different and/or a bit more general: 2-sheaves that factor through the forgetful 2-functor ToposCatTopos \to Cat.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJun 8th 2011

    Which forgetful functor? The covariant one or the contravariant one?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 8th 2011

    Which forgetful functor? The covariant one or the contravariant one?

    The covariant one.

    Here is one more detail on the actual example that I am looking at:

    we happen to have a presheaf of toposes on a site, for which the descent morphisms (those which should be equivalences for an actual 2-sheaf) are geometric surjections. So it’s like a “topos-valued epi-presheaf”. In case that rings any bell.

    In principle I can just accept that this is the way it happens to be in this example. But I am wondering if this is secretly telling me that there ought to be a connection to some sort of more general theory.

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeJun 8th 2011

    Is it more general ? If a site is SS and we have presheaves into another Grothendieck topos HH then HShv(S H)H \cong Shv(S_H); so that we deal with presheaves of sets on the product site S×S HS\times S_H; now only one has to figure out how to do the “pseudo”part (in 2-sheaf condition) consistently with this viewpoint.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 8th 2011

    Wait, Zoran, maybe there was a misunderstanding:

    I am looking at pseudofunctors

    C opToposCat C^{op} \to Topos \to Cat

    to the 2-category of all toposes, not into a single topos.

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeJun 8th 2011
    • (edited Jun 8th 2011)

    OK, I misunderstood “takes values in toposes”. But now if it is taking values in 2-category of toposes, then you talk about stacks of topoi on usual 1-site.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJun 8th 2011

    Yes. That’s why I wrote:

    In some application I am running into 2-sheaves (genuine category-valued analog of stacks) that happen to take values in (Grothendieck) toposes.

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeJun 8th 2011

    Are you sure you mean “2-sheaves [of categories] that factor through the forgetful 2-functor Topos→Cat” rather than “2-sheaves taking values in the 2-category Topos”? That forgetful 2-functor doesn’t preserve limits, so the two notions are different.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJun 9th 2011
    • (edited Jun 9th 2011)

    Yes, I think I need to compute my limits in CatCat.

    But also I am struggling a bit with that, so maybe I am making a mistake. Is there anywhere a collection of helpful statements about limits of presheaf categories (as categories, not as toposes)? Specifically I have diagrams of posets with left adjoint functors between them and I am looking at the limits of the corresponding diagrams of presheaf toposes.

    I should just post my computation in detail here. But I need to bring it into better shape first…

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeJun 9th 2011

    I wouldn’t expect limits of presheaf categories to be special in any particular way, aside from being accessible (since Acc has 2-limits).

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJun 30th 2011

    Further investigation shows that my above hunch was not quite right:

    what we do see is pseudofunctors C opToposC^{op} \to Topos that satisfy descent not by an equivalence but by a local geometric surjection .

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJun 30th 2011

    Also the toposes in question are locally ringed. Currently I am discarding the local rings before computing the limits.

    I need to think again: how do I compute limits in the 2-category of locally ringed toposes? It seems I knew that. But maybe not at this time of night…

    • CommentRowNumber14.
    • CommentAuthorzskoda
    • CommentTimeJul 1st 2011

    Where in the nLab is this 2-category defined as a 2-category ? What are the compatibilities of a geometric morphism with the line object etc. ?

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJul 1st 2011

    Where in the nLab is this 2-category defined as a 2-category ? What are the compatibilities of a geometric morphism with the line object etc. ?

    At ringed topos. A quick way to state the 2-category structure: it is Topos/ℛ𝒾𝓃ℊTopos/\mathcal{Ring} where ℛ𝒾𝓃ℊ\mathcal{Ring} is the classifying topos for rings.

    • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeJul 1st 2011

    I presume you mean that Ring is the classifying topos for local rings, since your toposes are locally ringed. Also, I think you mean the lax slice 2-category, not the ordinary one. But I don’t know offhand how to compute limits in lax slice 2-categories.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeJul 1st 2011

    just very briefly: actually currently in my toy appliucation my toposes are “globally ringed”. for the locally ringed case one also needs to restrict the morphisms in the slice to local morphisms

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeJul 5th 2011
    • (edited Jul 5th 2011)

    Maybe slowly approaching a more complete picture:

    the latest version of the sheaf condition that I believe we are running into is:

    a presheaf of ringed toposes such that the descent morphism of ringed toposes is

    1. a local geometric morphism on the underlying toposes;

    2. an isomorphism between the corresponding ring objects.

    So, in some sense, on the ring objects it is a genuine sheaf after all, only that the sheaf condition does not hold globally but only in an appropriately adjusted ambient topos.

    Hm…

    • CommentRowNumber19.
    • CommentAuthorMike Shulman
    • CommentTimeJul 5th 2011

    Would it be possible to define some sort of “semidirect product” site which amalgamates all the ringed toposes into one big one, and consider a ring object therein?