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?

]]>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…

]]>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

]]>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.

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.

]]>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. ?

]]>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…

]]>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* .

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

]]>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…

]]>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. 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.

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.

]]>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.

]]>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.

]]>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.

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

]]>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 $Topos \to Cat$.

]]>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?

]]>