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February 11, 2011

Punctual Local Connectedness

Posted by Mike Shulman

Peter Johnstone has a new paper out in TAC yesterday called Remarks on punctual local connectedness. Here’s the abstract:

We study the condition, on a connected and locally connected geometric morphism p:𝒮p : \mathcal{E} \to \mathcal {S}, that the canonical natural transformation p *p !p_*\to p_! should be (pointwise) epimorphic — a condition which F.W. Lawvere called the ‘Nullstellensatz’, but which we prefer to call ‘punctual local connectedness’. We show that this condition implies that p !p_! preserves finite products, and that, for bounded morphisms between toposes with natural number objects, it is equivalent to being both local and hyperconnected.

In the terminology currently in use at nLab:cohesive topos, this means that if a topos is connected and locally connected, and pieces have points, then it is necessarily also (1) a local topos, (2) a strongly connected topos (i.e. p !p_! preserves finite products), and hence (3) a cohesive topos. The implication “pieces have points \Rightarrow local” was observed, at least for sites, at nLab:cohesive site — Johnstone gives both proofs using sites and site-free versions.

Moreover, he also proves that such a topos is also (4) hyperconnected and (5) satisfies discrete objects are concrete — and that conversely, a topos which is locally connected, hyperconnected, and local necessarily satisfies pieces have points (hence is also strongly connected, thus cohesive, and satisfies discrete objects are concrete).

Finally, he shows that a topos with pieces have points is totally connected (i.e. p !p_! preserves all finite limits) if and only if p *p !p_\ast\to p_! is an isomorphism (i.e. every piece has exactly one point).

Posted at February 11, 2011 8:27 PM UTC

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Re: Punctual Local Connectedness

There was a discussion of punctual local connectedness — a.k.a. the Nullstellensatz — at MathOverflow recently (including a long message from Lawvere).

Posted by: Tom Leinster on February 12, 2011 12:10 AM | Permalink | Reply to this

Re: Punctual Local Connectedness

Thanks for pointing this out, Mike. It is good to see more relations between the axioms, few as they are.

I’ll try to have a close look tomorrow, in a spare minute. Currently it’s too late at night in between two long days.

It seems that with the constructions that I consider, I need/encounter the axiom pieces have points at one point: it is equivalent to saying that the de Rham object (discussed here)

Π dRX:=(p p !X) X \mathbf{\Pi}_{dR} X := (p^{*}p_! X) \coprod_X {*}

is connected. This is part of what identifies it as the general cohesive analog of the de Rham schematic homotopy type of XX.

Posted by: Urs Schreiber on February 12, 2011 1:55 AM | Permalink | Reply to this

Re: Punctual Local Connectedness

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. One might imagine shortening it to just punctually connected, since it does include connectedness, and we also have names like “strongly connected” and “totally connected” which include local connectedness (Johnstone uses “stably locally connected” instead of “strongly connected”).

Posted by: Mike Shulman on February 12, 2011 3:51 AM | Permalink | Reply to this

Re: Punctual Local Connectedness

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)(n,1)-toposes, and if “punctually (locally) connected” does generalize to “punctually (locally) (n1)(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 *Xp !Xp_* X \to p_! X is an epimorphism is to say that it is an effective epimorphism in an (n,1)(n,1)-category. And in (n1)Grpd(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 (n1)(n-1)-connected”. Instead it would mean that for all nn 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.

Posted by: Urs Schreiber on February 12, 2011 8:39 AM | Permalink | Reply to this

Re: Punctual Local Connectedness

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 (n1)(n-1)-connected”.

Why does it mean that? What else would you expect “punctually (n1)(n-1)-connected” to mean?

Posted by: Mike Shulman on February 12, 2011 7:27 PM | Permalink | Reply to this

Re: Punctual Local Connectedness

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 ΓXΠX\Gamma X \to \Pi X is epi on π 0\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 Π\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.

Posted by: Urs Schreiber on February 13, 2011 7:43 PM | Permalink | Reply to this

Re: Punctual Local Connectedness

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 nn-connected” in the locally nn-connected case.

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

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

Posted by: Mike Shulman on February 13, 2011 9:49 PM | Permalink | PGP Sig | Reply to this

Re: Punctual Local Connectedness

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

But how about saying the topos is locally pointed ?

Posted by: Urs Schreiber on February 13, 2011 10:53 PM | Permalink | Reply to this

Re: Punctual Local Connectedness

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”).

Posted by: Mike Shulman on February 14, 2011 12:39 AM | Permalink | PGP Sig | Reply to this

Re: Punctual Local Connectedness

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 ;-)

Posted by: Urs Schreiber on February 14, 2011 1:14 AM | Permalink | Reply to this

Re: Punctual Local Connectedness

Is there a way that a topos can be “locally pointed” without being locally connected? The definition “p *p !p_\ast \to p_! is epic” requires the existence of p !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”.)

Posted by: Mike Shulman on February 14, 2011 1:23 AM | Permalink | PGP Sig | Reply to this

Re: Punctual Local Connectedness

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 nn-connected”, if you say that sounds right. I’ll get used to it.

Posted by: Urs Schreiber on February 14, 2011 1:34 AM | Permalink | Reply to this

Re: Punctual Local Connectedness

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. (-:

Posted by: Mike Shulman on February 14, 2011 1:44 AM | Permalink | Reply to this

Re: Punctual Local Connectedness

I have recorded some of Peter Johnstone’s results in a new section Relation between the axioms.

I realize that I did not appreciate the characterization involving hyperconnectivity (theorem 3.4) before: I remember now that Jim Dolan had suggested this in previous discussion as a formalization of the maximal non-localicness of cohesive toposes, and only now do I see the relevance of this.

That’s something to think about: whether we have for each nn a factorization system of nn-hyperconnected versus nn-localic \infty-geometric morphisms and what it would mean to define nn-cohesive \infty-toposes as locally \infty-connected and local \infty-toposes that are nn-hyperconnected.

Posted by: Urs Schreiber on February 14, 2011 1:22 AM | Permalink | Reply to this

Re: Punctual Local Connectedness

Good question! Does Lurie address hyperconnectivity anywhere? We can at least define a topos to be nn-hyperconnected if its nn-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?)

Posted by: Mike Shulman on February 14, 2011 10:11 PM | Permalink | Reply to this

Re: Punctual Local Connectedness

Does Lurie address hyperconnectivity anywhere?

Not that I am aware of.

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

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

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

in ToposTopos (notationally suppressing 2-isomorphisms) factors essentially uniquely through a point pp of Sh(X)Sh(X)

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

This means that Topos(,Sh(X))LocTopos(Set,Sh(X))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 L=Set\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

𝒮 Set 𝒯 \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)(n,1)-site are precisely the nn-localic (,1)(\infty,1)-toposes.

Posted by: Urs Schreiber on February 16, 2011 8:36 PM | Permalink | Reply to this

Re: Punctual Local Connectedness

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

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

Posted by: Mike Shulman on February 16, 2011 10:14 PM | Permalink | Reply to this

Re: Punctual Local Connectedness

Since localic morphisms are stable under pullback

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

hyperconnected geometric morphism.

Posted by: Urs Schreiber on February 17, 2011 12:19 AM | Permalink | Reply to this

Re: Punctual Local Connectedness

I feel that in order to make progress with identifying the right way of looking at the axiom pieces have points / puncual connectedness in the ordinary and the higher categorical context we need to have a better idea of what such a kind of axiom can achieve or is supposed to be good for, apart from sounding plausible as an axiom characterizing cohesive nn-groupoids.

I’ll point out what I think is a major property that one would want to ask of a cohesive \infty-topos, which looks like it ought to be controled by such a kind of axiom.

As you know, I am working on a document, titled Differential cohomology in a cohesive \infty-topos, that is meant to show that when applied to \infty-toposes instead of just 1-toposes, the axioms on strongly connected and local \infty-toposes – in the form of a quadruple of adjoint \infty-functors (ΠDiscΓcoDisc):HGrpd(\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : \mathbf{H} \to \infty Grpd with DiscDisc and coDisccoDisc full and faithful and Π\Pi preserving finite products – imply a fairly comprehensive list of abstract structures in differential cohomology on the objects in the \infty-topos.

Among the most basic is the fact that ||:HΠGrpdTop|-| : \mathbf{H} \stackrel{\Pi}{\to} \infty Grpd \stackrel{\simeq}{\to} Top models geometric realizaton of cohesive \infty-groupoids. In particular, for a connected object – denoted BG\mathbf{B}G – in H\mathbf{H} and XX any object, H(X,BG)\mathbf{H}(X,\mathbf{B}G) has the interpretation of the \infty-groupoid of cohesive GG-principal \infty-bundles on XX and we may think of the action of |||-| on this

|| X,BG:H(X,BG)Top(|X|,BG) |-|_{X,\mathbf{B}G} : \mathbf{H}(X, \mathbf{B}G) \to Top(|X|, B G)

with BG:=|BG|B G := |\mathbf{B}G|, as sending such cohesive bundles to their underlying topological bundles (or rather to the classifiying maps of these).

Indeed, using a theorem by Danny Stevenson and David Roberts on simplicial topological bundles, applied in a model category presentation of cohesive \infty-groupoids with underlying topological cohesion, we find that in a large class of examples the functor || X,BG|-|_{X,\mathbf{B}G} is indeed an isomorphism on π 0\pi_0: this means that |BG||\mathbf{B}G| is indeed the classifying space for cohesive GG-principal \infty-bundles.

This is clearly an important statement, and one might hope that it is not only true in the specific class of examples that I have checked, but that general abstract axioms on cohesive \infty-toposes guarantee this property. It looks like a decisive property of the interaction of the notion of cohesion with the homotopy-theory of \infty-toposes.

Now, this cannot be true without further qualification. There are plenty of objects in a cohesive \infty-topos that have nontrivial categorical homotopy types (towers of homotopy sheaves) but trivial geometric homotopy type: they map under |||-| to the point. But these contain intuitively and in classes of examples non-concrete objects . And intuitively one expects the functor || X,BG|-|_{X,\mathbf{B}G} to be an iso on π 0\pi_0 for concrete objects: notably for GG a cohesive \infty-group, an \infty-group equipped with cohesive structure. Indeed, in the classes of examples that are understood, GG is presented by a simplicial topological group or a simplicial Lie group, etc.

So therefore I am wondering if

  1. there is a good notion of concrete objects in a cohesive \infty-topos;

  2. such that restricted to morphisms between these with the codomain connected, |||-| becomes an isomorphism on π 0\pi_0.

The reason why I think this might be relevant to the discussion here is that when unwinding this a little, it begins too look like a condition very similar to that of punctual connectedness.

As arguments by you and Peter Johnstone emphasized, in a cohesive 1-topos we find that the property that Γ\Gamma is a monomorphism on morphisms XYX \to Y between concrete objects may be deduced from looking at the naturality square of the (ΓcoDisc)(\Gamma \dashv coDisc)-unit

X Y coDiscΓX coDiscΓY \array{ X &\to& Y \\ \downarrow && \downarrow \\ coDisc \Gamma X &\to& coDisc \Gamma Y }

and using that by definition of concrete objects, the vertical morphism are monomorphisms.

Now for the problem that I am talking about, the question is whether

Π X,BG:H(X,BG)Grpd(Π(X),Π(BG))H(X,DiscΠBG) \Pi_{X,\mathbf{B}G} : \mathbf{H}(X, \mathbf{B}G) \to \infty Grpd(\Pi(X), \Pi(\mathbf{B}G)) \stackrel{\simeq}{\to} \mathbf{H}(X, Disc \Pi \mathbf{B}G)

is an isomorphism on π 0\pi_0. And by the analogous reasoning, one finds again that this is the operation that forms the composites

X BG DiscΠX DiscΠBG. \array{ X &\to& \mathbf{B}G \\ \downarrow && \downarrow \\ Disc \Pi X &\to& Disc \Pi \mathbf{B}G } \,.

So the situation seems to be similar, even though it’s different (we are not asking faithfulness and the units here are not monos).

So I am trying to see what one can do in the spirit of “punctual connectedness”-axioms to possibly find a characterization of this situation.

For instance since we want to assume that XX and BG\mathbf{B}G here are concrete, we probably need to combine these squares with

X BG coDiscΓX coDiscΓBG \array{ X &\to& \mathbf{B}G \\ \downarrow && \downarrow \\ coDisc \Gamma X &\to& coDisc \Gamma \mathbf{B}G }

and possibly demand that the vertical morphisms are something like monos on all homotopy sheaves or the like (this is where this connects to the discussion at Concrete \infty-categories).

So this begins to look like we’d need to consider natural morphisms between the composite adjoint triple

(DiscΠDiscΓcoDiscΓ) (Disc \Pi \dashv Disc \Gamma \dashv coDisc \Gamma)

(which I usually write (ΠΓ)(\mathbf{\Pi} \dashv \mathbf{\flat} \dashv \mathbf{\Gamma})).

So maybe one needs to consider the squares

X coDiscΓX DiscΠX coDiscΠX, \array{ X &\to& coDisc \Gamma X \\ \downarrow && \downarrow \\ Disc \Pi X &\to& coDisc \Pi X } \,,

where on the right and on the bottom we have the morphisms considered in punctual connectedness, and then consider the cube obtained from all these (which I won’t try to typeset here) and then argue with some of these morphsims being epi in some sense, others mono in some sense, etc.

I do not currently know how to proceed at this point. I only observe that it seems like higher analogs of the conditions pieces have points and punctual connectedness should play a role, and that if they do, the property on |||-| that I am hoping can be implied would be a genuinely interesting and desrireble one, one that would clearly justify the extra axiom.

But I don’t know yet.

Posted by: Urs Schreiber on February 18, 2011 11:34 AM | Permalink | Reply to this

Re: Punctual Local Connectedness

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”?

Posted by: Mike Shulman on February 19, 2011 9:16 PM | Permalink | Reply to this

Re: Punctual Local Connectedness

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.

Posted by: Urs Schreiber on February 19, 2011 10:55 PM | Permalink | Reply to this

Re: Punctual Local Connectedness

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.

Posted by: Mike Shulman on February 20, 2011 1:22 AM | Permalink | Reply to this

Re: Punctual Local Connectedness

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 ETopGrpdETop\infty Grpd the nerve theorem alone is the essential ingredient to show cohesiveness. For SmoothGrpdSmooth\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 n\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.

Posted by: Urs Schreiber on February 21, 2011 10:43 PM | Permalink | Reply to this

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