The n-Category Café

I kind of like “punctually locally connected” in preference to “Nullstellensatz” or “pieces have points” — if nothing else, at least it’s an adjective! It also makes it clear that it’s a strong form of (local) connectedness.

To get a clearer idea of this it might be helpful to see to which extent the relations between the axioms that Peter Johnstone finds have analogs on $(n,1)$-toposes, and if “punctually (locally) connected” does generalize to “punctually (locally) $(n-1)$-connected”.

At some point I had come to decide that the good generalization from topos theory to higher topos theory of the condition that $p_* X \to p_! X$ is an epimorphism is to say that it is an effective epimorphism in an $(n,1)$-category. And in $(n-1)Grpd$ this just means: surjective on connected components! (HTT, cor. 7.2.1.15). So this would mean that the condition does not generalize to something one would want to call “punctually $(n-1)$-connected”. Instead it would mean that for all $n$ the condition really just says: pieces have points .

But I may still be missing the right way to think about the higher analog of this axiom. I certainly feel that I am still missing something important about the dual axiom, discrete objects are concrete , in the higher context.

this just means: surjective on connected components! … So this would mean that the condition does not generalize to something one would want to call “punctually $(n-1)$-connected”.

Why does it mean that? What else would you expect “punctually $(n-1)$-connected” to mean?

Why does it mean that?

Let me know how you are thinking about it, maybe I am thinking about it in the wrong way. I thought: the condition that $\Gamma X \to \Pi X$ is epi on $\pi_0$ just means that every connected component has a point. It is not saying anything about points in relation to higher connectedness. Apart from the fact that $\Pi$ exists in the first place, of course, but why would that need to be restated in connection to the fact that pieces have points?

I hope to find time to read the article in detail later tonight. Have been busy days here. Now off to dinner.

I am thinking about it the same way you are. I just don’t understand why that means you wouldn’t want to call it “punctually $n$-connected” in the locally $n$-connected case.

• “locally $n$-connected” = has a basis consisting of $n$-connected things
• “punctually locally $n$-connected” = has a basis consisting of pointed $n$-connected things.

I don’t know of any notion of “$n$-pointed” that I would want the latter to also imply; do you?

Not sure, maybe a problem on my side with the English.

But how about saying the topos is locally pointed ?

Well, “locally pointed” doesn’t suggest to my intuition that it includes “locally connected”; certainly a space can be pointed without being connected.

I think “locally pointed” and “pieces have points” also share another disadvantage: they are misleading for little topoi. Saying of a single space that “pieces have points” sounds to me like saying that all of its connected components have points, so that for instance any topological space would satisfy this property. But of course the topos of sheaves on a topological space is very rarely “punctually locally connected” — the “points” here refer really to global sections. “Punctually locally connected” conveys a sense that not only does the local connectedness involve points, the topos itself is somehow “punctual” (e.g. it has the “shape of a point”).

Well, “locally pointed” doesn’t suggest to my intuition that it includes “locally connected”;

It wasn’t supposed to, just as pieces have points is not. Both are meant to qualify connected and locally connected .

But I see the problem with the confusion with global points.

“Punctually locally connected” conveys a sense that not only does the local connectedness involve points, the topos itself is somehow “punctual” (e.g. it has the “shape of a point”).

Let’s see, that would suggest we could just say punctual topos ?

I think what’s troubling me is that in punctually locally connected it sounds like the “punctuality” characterizes the local connectedness, while rather it is something on top of local connectedness. But as I said, maybe that’s just a conflict between me and the English language ;-)

Is there a way that a topos can be “locally pointed” without being locally connected? The definition “$p_\ast \to p_!$ is epic” requires the existence of $p_!$. So it makes sense to me to consider “punctuality” as an additional property of the local connectedness.

(The way I parse “punctually locally connected” is that “punctual” is an adverb specifying a way in which a topos can be locally connected. Like “strongly” or “weakly” or “totally”.)

Wait a second, didn’t you write first

Well, “locally pointed” doesn’t suggest to my intuition that it includes “locally connected”;

and now

Is there a way that a topos can be “locally pointed” without being locally connected?

?

But, you know, let’s go with “punctually locally $n$-connected”, if you say that sounds right. I’ll get used to it.

Yes, I did write both of those. The first was about what the words sound to me like they mean; the second was about the actual mathematical definitions. (-:

Good question! Does Lurie address hyperconnectivity anywhere? We can at least define a topos to be $n$-hyperconnected if its $n$-localic reflection is trivial. Then we’d have to figure out whether such topoi can be characterized in any way analogous to 0-hyperconnected 1-topoi. (Plain “localic” is 0-localic, right?)

Does Lurie address hyperconnectivity anywhere?

Not that I am aware of.

We can at least define a topos to be $n$-hyperconnected if its $n$-localic reflection is trivial.

Let me see, for a 1-topos. Assume that $\mathcal{E}$ is hyperconnected. Then for $X$ any locale I find by the hyperconnected/localic factorization system that every geometric morphism $\mathcal{E} \to Sh(X)$, which necessarily sits in a unique diagram

$$\array{ \mathcal{E}&\to& Sh(X) \\ \downarrow && \downarrow \\ Set & = & Set }$$

in $Topos$ (notationally suppressing 2-isomorphisms) factors essentially uniquely through a point $p$ of $Sh(X)$

$$\array{ \mathcal{E}&\to& Sh(X) \\ \downarrow &\nearrow_{\mathrlap{p}}& \downarrow \\ Set & = & Set } \,.$$

This means that $Topos(\mathcal{E},Sh(X)) \simeq LocTopos(Set, Sh(X))$ and hence the localic reflection of $\mathcal{E}$ is the point.

Conversely, assume that the localic reflection is the point. Let then $\mathcal{E} \to L \mathcal{E} = Set$ be the unit of the adjunction. By uniqueness of the terminal geometric morphism this must be the global section geometric morphism. Therefore in a lifting problem

$$\array{ \mathcal{E} &\to& \mathcal{S} \\ \downarrow && \downarrow \\ Set &\to& \mathcal{T} }$$

the left morphism is the unit of the reflective localic topos inclusion.

Hm, now how does the argument continue?

(Plain “localic” is 0-localic, right?)

Right. $\infty$-sheaf toposes over an $(n,1)$-site are precisely the $n$-localic $(\infty,1)$-toposes.

Since localic morphisms are stable under pullback, you can pull $\mathcal{S}\to\mathcal{T}$ back to a localic morphism $Sh(X) \to Set$, and then the fact that $\mathcal{E}\to Set$ is the localic reflection means that the induced map $\mathcal{E}\to Sh(X)$ factors uniquely through $Set$, hence the lifting problem has a unique solution.

This is a standard argument whenever we have a unique factorization system $(E,M)$: the $(E,M)$-factorization of a morphism $x\to y$ is always the same as its reflection into the sub-slice-category $M/y$.

Since localic morphisms are stable under pullback

Ah, right, thanks. I have now recorded the completed derivation here:

hyperconnected geometric morphism.

Do you really expect classifying spaces of cohesive groups to homotopically classify cohesive bundles in generality? I know it’s true that classifying spaces of Lie groups homotopically classify smooth bundles over manifolds, but the only proof of that I know of uses the fact that continuous maps between smooth manifolds can be approximated by smooth maps, which in turn uses a partition of unity argument. Are you thinking of axioms on the cohesive topos which somehow encode some abstract version of “smooth approximation” or “partitions of unity”?

Do you really expect classifying spaces of cohesive groups to homotopically classify cohesive bundles in generality?

No. What I have pointed out (or meant to point out, my apologies if I keep coming across as cryptic) is that this is a property that turns out to be formally similar to the pieces have points axiom. Like that axiom, it may be present in some cases, in others not. Identifying the meaning of such an axiom as asserting something like “cohesive $\infty$-bundles have classifying spaces” would usefully equip such an axiom with useful meaning.

Because I would like to understand the need for the extra punctual connectedness axiom in applications. I know I can go a long, long way with local and global $\infty$-connectedness and locality. So why should I consider the further axiom of punctual connectedness and what should be its $\infty$-version? If I knew that such a further axiom asserted, for instance, that “cohesive $\infty$-bundles have classifying spaces” that would be useful to know. In various applications it would make sense to impose such an axiom.

I know it’s true that classifying spaces of Lie groups

And even before we add smooth structure there is an interesting statement: that theorem by Danny Stevenson and David Roberts asserts that it is true for $\infty$-groups with topological cohesion. This seems to be a nontrivial and technical result and it would seem useful to understand its existence more abstractly, I think.

Are you thinking of axioms on the cohesive topos which somehow encode some abstract version of “smooth approximation” or “partitions of unity”?

What is true is that the constructions via $\infty$-cohesive sites that I am aware of as to date happen to effectively rely on partitions of unity. Another big open question (for me) is to see more generally what necessary and sufficient properties of a site are such that its $\infty$-topos is cohesive.

Okay, interesting, I think I see. So the question is really “what ensures that classifying spaces classify cohesive bundles?” and it just happens to appear that this is similar to punctual local connectedness.

Can you point out exactly where partitions of unity are currently (“effectively”) used? I didn’t find it on a brief glance through ∞-cohesive site.

Can you point out exactly where partitions of unity are currently (“effectively”) used?

All the example of $\infty$-cohesive sites that I have looked at rely on the nerve theorem for paracompact spaces to show that the colimit operation applied to a Čech nerve of the good cover of a topologically contractible object is itself a contractible simplicial set.

The nerve theorem relies crucially on the existence of partitions of unity (see the proof of prop 4G.2 in Hatcher’s book), which is also where the condition of paracompactness comes from.

For the case of $ETop\infty Grpd$ the nerve theorem alone is the essential ingredient to show cohesiveness. For $Smooth\infty Grpd$ the argument needs a slight refinement in that one has to show that there are not only good open covers but “differentiably good open covers” (each non-empty finite intersection being diffeomorphic to an $\mathbb{R}^n$). Proving that these exist again crucially depends on assuming paracompactness, hence more or less explicitly again depends on the existence of partitions of unity (paracompactness is essentially characterized by existence of partitions of unity).

From these examples one can derive a bunch of further ones without much further work by just adding some infinitesimal thickening, for instance synthetic-differential and super-smooth $\infty$-groupoids. So their cohesiveness also comes down to the existence of partitions of unity for the reduced (non-infinitesimal) parts of the geometry.

Here is a related, possibly suggestive but vague observation: as David Speyer once nicely amplified in a blog post titled The Nullstellensatz and Partitions of Unity

the Nullstellensatz is a version of partitions of unity

But Nullstellensatz is what Lawvere proposed to call the axiom that we came to call pieces have points or punctual connectedness . So maybe there is also non-vanishing justification for calling this axioms existence of partitions of unity .

Not sure, but maybe all this is supposed to tell us something.