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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?
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 $Topos \to Cat$.
Which forgetful functor? The covariant one or the contravariant one?
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.
Is it more general ? If a site is $S$ and we have presheaves into another Grothendieck topos $H$ then $H \cong Shv(S_H)$; so that we deal with presheaves of sets on the product site $S\times S_H$; now only one has to figure out how to do the “pseudo”part (in 2-sheaf condition) consistently with this viewpoint.
Wait, Zoran, maybe there was a misunderstanding:
I am looking at pseudofunctors
$C^{op} \to Topos \to Cat$to the 2-category of all toposes, not into a single topos.
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.
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.
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.
Yes, I think I need to compute my limits in $Cat$.
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…
I wouldn’t expect limits of presheaf categories to be special in any particular way, aside from being accessible (since Acc has 2-limits).
Further investigation shows that my above hunch was not quite right:
what we do see is pseudofunctors $C^{op} \to Topos$ that satisfy descent not by an equivalence but by a local geometric surjection .
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…
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. ?
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 $Topos/\mathcal{Ring}$ where $\mathcal{Ring}$ is the classifying topos for rings.
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.
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
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
a local geometric morphism on the underlying toposes;
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…
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?
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