The n-Category Café

What is a point of the site $SmthMfd$?

There is, up to equivalence, one for every $n \in \mathbb{N}$, given by the stalk at the origin of $\mathbb{R}^n \simeq D^n$.

The proof reduces via slicing to the observation that the topos of sheaves over any fixed manifold has enough points, each of which looks like one of these.

Okay; that helps a little. I don’t have a very good intuition for what a “point” of a general site looks like, aside from the two special cases of (1) a locale, and (2) the classifying site of a geometric theory (where points are models of the theory in Set).

I guess you would say that as a “space”, this topos consists of countably many “fat points” each of which is fat with a different number of dimensions? I’m assuming that SmthMfd is at least approximately a cohesive site, and as such is supposed to have “the shape of a single point” — but I guess that that is a homotopy-theoretic shape, and it can still have many different actual points as long as they are connected together in a “contractible way”. Is that the right way to think?

I don’t have a very good intuition for what a “point” of a general site looks like, aside from the two special cases of (1) a locale, and (2) the classifying site of a geometric theory (where points are models of the theory in Set).

So and here we can reduce to the first case. We are asking if two sheaves on $SmthMfd$ have the same value on every single smooth manifold. Since both restrict to sheaves over that given smooth manifold regarded as a topological space, we can now check that they agree there by checking if they agree as objects of a localic topos. So we know we need to check that they agree on every germ of disks around every point of the manifold. But since the two sheaves on this fixed manifold are restrictions of sheaves on all manifolds, their value on any disk is the same, no matter whether we think of this disk now as being a neighbourhood of one point or other in the given manifold. So we finally conclude that the two original sheaves are isomorphic already if they have the same value on all germs of abstract disks.

I guess you would say that as a “space”, this topos consists of countably many “fat points” each of which is fat with a different number of dimensions?

Yes, that’s exactly how I think of it. The “fat cohesive point” that smooth geometry is modeled on is something like the inductive limit over all germs of abstract smooth $n$-disks. Which intuitively makes perfect sense, doesn’t it?

I’m assuming that SmthMfd is at least approximately a cohesive site

Yes, exactly. The precise statement is that $CartSp_{smooth} \hookrightarrow SmthMfd$ exhibits a dense subsite, and that $CartSp_{smooth}$ is an $\infty$-cohesive site. That’s one reason for going on about that site $CartSp_{smooth}$. For the purposes of sheaves over it it is equivalent to $SmthMfd$, but it has some better properties as compared to the latter.

as such is supposed to have “the shape of a single point”

Yes, every cohesive $\infty$-topos is pointlike in the following respects (many of which imply each other):

• it is locally and globally $\infty$-connected and local (by definition);

• it has the shape of the point;

• it has homotopy dimension 0;

• it has cohomology dimension 0;

• it is hypercomplete.

However, it does not necessarily need to have “a unique $\infty$-topos point”, up to equivalence.

— but I guess that that is a homotopy-theoretic shape, and it can still have many different actual points as long as they are connected together in a “contractible way”. Is that the right way to think?

Yes, I guess so. I haven’t thought about how to formally think of the fact that there may be many different “topos points” in a cohesive $\infty$-topos.

Let’s see. So I think here the thing is that all these points “sit inside each other” like a Matryoshka doll. Not by invertible morphisms, though. Still, there is this inductive system of points of $Smooth \infty Grpd$. I should think about formalizing that somehow only the single colimiting point of this system is the “actual” single point that we would like to see. Hm…

Thanks! Of course, in general the real question to ask about the points of a topos is, what is the (higher) category of its points, rather than just the set or (higher) groupoid of them. Can you say precisely what the category of points of Smooth∞Grpd is?

Still, there is this inductive system of points of Smooth∞Grpd.

The category of points of a 1-topos has filtered colimits (B3.4.8); is an analogous statement true for $(\infty,1)$-toposes?

I should think about formalizing that somehow only the single colimiting point of this system is the “actual” single point that we would like to see.

In the 1-topos case, traditionally people have thought about “topos homotopies” as geometric morphisms $\mathcal{E}\times Sh([0,1]) \to \mathcal{F}$. Since there is a map from $Sh([0,1])$ to $Sh(\Sigma)$ where $\Sigma$ is the Sierpinski space, and the latter is the incarnation of the interval category as a topos, any natural transformation gives rise to a “topos homotopy”, so that even noninvertible maps between topos-points induce paths between “points” of the “topos homotopy type”. Does that have an analogy from the $\Pi_\infty$/shape-theory perspective on topos homotopy types?

For example, I believe we can say that $\Pi_\infty$ of an $(\infty,1)$-presheaf topos $\infty Gpd^C$ is the $\infty$-groupoid reflection of $C$. I’m not sure what more general precise statement could be made though.

Of course, in general the real question to ask about the points of a topos is, what is the (higher) category of its points, rather than just the set or (higher) groupoid of them. Can you say precisely what the category of points of Smooth∞Grpd is?

Good question. I haven’t thought about this! I should, this is an interesting question to ask.

It feels like it should be possible to use the quadruple of adjoint functors on a cohesive topos to show that the category of points in it is contractible in some sense by doing some retract yoga or something. But I don’t see it yet.

What I think can be checked easily over a cup of coffee (now that I had one) is that $Smooth \infty Grpd$ has one point $p(\infty)$ which alone is “enough points” and which is the inductive limit over all the $p(n)$ that are stalks at the origin of the $n$-disk.

I have to run now to catch a train. But I would like to come back to thinking about what can be said generally and specifically about topos points of a cohesive topos. That’s an evident fundamental question that I haven’t thought enough about to date.

It feels like it should be possible to use the quadruple of adjoint functors on a cohesive topos to show that the category of points in it is contractible in some sense

According to C3.6.1(vi), if a topos $\mathcal{E}$ is local, then for every topos $\mathcal{G}$ the hom-category $Topos(\mathcal{G},\mathcal{E})$ has an initial object, and thus a (weakly) contractible nerve. Thus the category of “$\mathcal{G}$-points of $\mathcal{E}$” is (weakly) contractible for any $\mathcal{G}$, not just $\mathcal{G}=Set$.

(There was some vaguely amusing banter at Swansea about whether calling a category “contractible” should mean that it has a weakly contractible nerve, or that it is equivalent as a category to the terminal one.)

What I think can be checked easily over a cup of coffee (now that I had one) is that Smooth∞Grpd has one point p(∞) which alone is “enough points” and which is the inductive limit over all the p(n) that are stalks at the origin of the n-disk.

Hmm, I thought that up here you were saying that Smooth∞Grpd has precisely $\mathbb{N}$-many points. Now are you saying there’s another one too?

I think I have a clearer way of putting this question I keep bugging you with.

I might think to investigate a smooth manifold by taking it as a special kind of object of the cohesive $(\infty, 1)$-topos $Top$. Or I might decide it’s better to look it as an object in a more closely tailored setting where smoothness prevails, i.e., in $Smooth \infty Grpd$.

Now if I’m dealing with, say, a complex or symplectic manifold, should I rest happy with taking it to be a special object of $Smooth \infty Grpd$, or should I seek out a better fitting setting, somewhere where holomorphic or symplectic cohesion prevails?

In the symplectic case you adopt the former stategy.

Hi David,

I see. So when I replied to your question on the $n$Forum, I was not speaking in the context of cohesion, but just of sheaf cohomology in general. So I guess we were talking past each other to some extent.

Your general question here, is a very good one, I think, in the sense that it is very interesting, hence also hard. So I may not have a really good answer. But I can try to indicate better what I know and what I don’t know.

First, it is absolutely true that a single traditional object may naturally present very different objects in various toposes, reflecting very different aspects of the former object. For instance taking a smooth manifold to present an object in $\mathrm{ETop}\infty Grpd \hookrightarrow \Smooth \infty Grpd$ instead of directly in $Smooth \infty Grpd$ means forgetting its smooth structure, regarding it just as a topological space, and then “freely” adding a diffeological structure again (which is a weird thing to do from the point of traditional differential geometry, but it makes sense).

Then concerning the question of complex and analytic geometry: as you know, I don’t know if these geometries are cohesive. For the analytic case it might be true if interpreted suitably, but I still don’t know. Accordingly, I have no intuition really what the difference will be – or would be – between regarding an object as a plain object in an analytic cohesive $\infty$-topos or as an object with extra structure in some other $\infty$-topos. Sorry, I realize that this is a huge frustrating gap from the point of view of getting a general overview idea of cohesion, but for the moment I can’t help it.

On the other hand, concerning your question about symplectic structure: such structure is very naturally and usefully regarded as a special form of differential cohomological structure in $Smooth \infty Grpd$. So while I don’t know if there might be a useful “symplectic cohesive $\infty$-topos”, I first of all doubt it, and secondly it doesn’t bother me at all that I don’t know it, because I see no indication that I need to know it, with instead the theory in $Smooth \infty Grpd$ with extra differential cohomological structure working so very well.

That’s what I can say for the moment. I wish I had a more conceptual answer than these pragmatic attitudes. But maybe we’ll get there in the future.

there’s a typo

Thanks! Fixed now. Though that whole section needs more editorial attention. I’ll come to that in a little while…

Hi Jim

were you thinking of this?

Hmm, no that’s not it. Only Chacholski, not Blomgren. But that is the only other mention of Chacholski on the n-category cafe.

Jim Stasheff asks:

has there been a comment or discussion anywhere on the blog about:

Blomgren and Chacholski - On the classification of fibrations

Thanks for the pointer!

While I had not seen that particular article yet, we have had various discussions here about this topic: all the discussions about univalence for instance are about this result and its generalizations. You might like Ieke Moerdijk’s short and sweet version of the proof of the classification of fibrations here.

Also if you look at the talk handout slides that are announced in the above entry you see that on p. 4 a generalization of this result is stated. The details I had posted just the other day, at Principal ∞-Bundles – general theory and presentations.

It is noteworthy that all of this follows as a fairly straightforward consequence from what in modern language is a central result of $\infty$-topos theory: the existence of “object classifiers”. I once added a little discussion that makes explicit how your classical result and the formulations of it by May, and now Blomgren-Chacholski follow from this here on the $n$Lab.

Which brings me to a question I have for you: can you help me organize the history of who is to be attributed for what? It seems to me that the classification of fibrations is being re-proven various times, and it is not always entirely clear to me to which extent the authors are actively aware of their predecessors.

An attempt on my part to list the relevant literature is in the reference section at associated infinity-bundle here.

You would do me a favour if you could have a look at that list and tell me what you think, which items and/or comments should be added. Thanks!