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started stubs for (infinity,n)-sheaf and (infinity,n)-topos; for the moment mostly as receptors and donors of cross-links, only.
I guess that’s not a definition of $(\infty, n)$-topos, unless the first part of the sentence tells us the only way to form them. Is it more a case of whatever they are, these things must be examples?
I think whatever they are, they must be equivalent to this. We had had a long discussion about that for $n=2$ a while back, and there it comes out precisely this way. So this is the essentially unique way to get them, I suppose. Or rather: by extrapolating from the cases of $n=1$ and $n=2$ where alternative definitions exist and agree, we will want to define that a (Grothendieck) $(\infty,n)$-topos is an $(\infty,n)$-category of $(\infty,n-1)$-category objects in some $(\infty,1)$-topos.
(Also, this is the natural perspective if HoTT is indeed a good foundations.)
By the way, it’s easy to make $\pm 1$-offset typos when writing about $(\infty,n)$-sheaves being $(\infty,n-1)$-category objects and forming an $(\infty,n)$-topos. I think I have fixed now most of these that my fingers had introduced. But if you spot remaining ones, please fix, if you have a second.
Where is the n=2 discussion? I’d like to go look at that…
Right, I should have pointed to it earlier:
it’s a consequence of a theorem of Mike Shulman, see at 2-topos – In terms of internal categories.
Wait, what? Even in the 2-dimensional case, not every 2-topos arises as internal categories in a (2,1)-topos, only the (2,1)-localic ones.
In the entry it does say “$(\infty,1)$-site”. That’s what I am talking about:
For $\mathbf{H}$ an (∞,1)-topos and $n \in \mathbb{N}$, an $(\infty,n+1)$-sheaf on (an (∞,1)-site of definition of) $\mathbf{H}$ is an n-fold category object in $\mathbf{H}$, $X \in n Cat(\mathbf{H})$.
Ah, I see, David’s #2 could be interpreted more generally. I was thinking of the statement as I have it in the entry.
Hrmmm…. So I can imagine the following higher category, which is similar to sheaves of (infty,1)-cats on the circle. In fact locally it is isomorphic to that, but the global sections consist of (infty,1)-categories which have an equivalence with their opposite. Is this a higher topos in the sense linked above?
@Urs: At (infinity,n)-topos you wrote
For $\mathbf{H}$ an (∞,1)-topos and $n \in \mathbb{N}$, the collection $n Cat(\mathbf{H})$ of n-category object in $\mathbf{H}$, hence of (∞,n+1)-sheaves on $\mathbf{H}$ is an $(\infty,n+1)$-topos.
in the “Definition” section. I was saying that isn’t the right Definition, since it doesn’t get you all of them.
@Chris: An interesting question, but my imagination doesn’t seem to be up to the task. Can you describe more explicitly what you mean? Maybe leave out the infinities and just talk about 1-categories?
At (infinity,n)-topos you wrote
Ah, okay, I have added “$(\infty,1)$-localic”
Chris,
are you thinking of bundles of categories over the circle that are equipped with a local trivialization such that on one semi-circle they are constant on a given categoy $C$ (I’ll suppress $\infty$s), on the other they are constant on $C^{op}$ and the transition function on the overlap is constant on an equivalence $C \stackrel{\simeq}{\to} C^{op}$?
If so, then the choice of local trivialization will need to be part of the data in order to have that equivalence be part of the data, it seems to me. Or else, I am misunderstanding what you are hinting at.
Maybe here’s a way to put it: for any 2-topos the global section geometric morphism lands in $Cat$ (unless we are working explicitly over a different base 2-topos, of course). So global sections will necessarily be categories, not categories equipped with some extra data.
But let me know if I am completely missing your point.
Hi Mike and Urs,
Yes Urs is getting close to what I was thinking. The point, I guess, is that, unlike SET, CAT has automorphisms (Z/2 many to be exact). I am wondering if these come into the picture for higher topoi. If I have a Z/2-torsor on a space (or topos) I would like to be able to tensor that with CAT to get a bundle of 2-categories over that space, an “associated bundle” if you will. Locally the sections just look like sheaves of categories, but globally the whole thing is twisted by the torsor.
I guess in the kind of higher topoi you are talking about, this is not an example?
Let me try to describe my circle example more precisely. We can first form the 2-category of sheaves of categories on the unit interval [0,1]. If I have such a sheaf F and a point x in [0,1], then I can pull back F to the point x. This is just an ordinary category. At the endpoints we get categories $F_0$ and $F_1$
So now I can consider a new 2-category where the objects are pairs consisting of such an F and an equivalence of categories $F_0 \simeq F_1^{op}$. Ordinary sheaves of sets on the circle can be regarded as discrete objects of this category.
This construction is local in the circle. For any open set U we can consider the subcategory of those F which are supported in U. If U is not the whole circle, then this restricted 2-category is just the category of sheaves of categories on U. So locally this category is just sheaves of categories, but globally it twisted.
Okay, I think I see. And no, I don’t think that would be a higher topos. I bet it will fail the exactness properties.
I don’t entirely understand how the automorphism of Cat affects 2-topos theory. Not every 2-topos inherits a similar automorphism; in general this seems to be the case only for (pre)sheaves on 2-categories that are isomorphic to their “co” duals (such as (2,1)-categories). But there is instead something like an induced automorphism of the 3-category of 2-topoi, which sends $Cat^{C}$ to $Cat^{C^co}$.
I am in a seminar right now, so just very briefly:
it seems to me Chris’s question lives rather in 3-topos theory: we are looking at the 2-category valued sheaf of sections of the 2-category bundle
Möbius $\times_{\mathbb{Z}_2} Cat \to S^1$ .
The 2-category of global sections of this 3-bundle is that of categories equipped with an equivalence to their opposite.
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