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I added a definition to locally contractible space, but is it correct?
Thanks, Toby. Discussion of this and many related aspects has been on my to-do list for quite a while, but I have not really gotten around to it.
Let’s first clarify the case of locally contractible topological spaces. I have now added a discussion of this at locally infinity-connected (infinity,1)-site.
So the general statement is that (hypercompleted) $\infty$-sheaves on a 1-site $C$ form a locally $\infty$-connected $(\infty,1)$-topos if all constant $\infty$-presheaves are already sheaves, which is the case if there is for each object $U \in C$ a split hypercover $Y \to U$.
This is a sufficient, not a necessary condition. For $X$ a locally contractible topological space, the condition is not satisfied for the category of open subsets $Op(X)$. Becuse even though every open subset is covered by contractibles, it need not even be connected, and hence constant presheaves are not even sheaves.
But there is another site of definition for $Sh_{(1,1)}(X)$ and $\hat Sh_{(\infty,1)}(X)$: the full subcategrory
$cOp(X) \hookrightarrow Op(X)$on the contractible open subsets: since by assumption every $U \in Op(X)$ has a cover by contractbles, two sheaves are equivalent already when they are equivalent on contractibles.
And the site $cOp(X)$ does have the property that all constant $(n,1)$-presheaves on it are already $(n,1)$-sheaves, for all $n$.
So if $X$ is locally contractible then $Sh_{(\infty,1)}(X)$ is locally $\infty$-connected.
For the converse statement, let me think…
I wrote:
For the converse statement, let me think…
I went back to the 1-categorical statement that the sheaf topos over a topological space is a locally connected topos precisely if the topological space is so. The key step is to prove that if we do have an essential geometric mophism $(\Pi_0 \dashv L Const \dashv \Gamma) : \mathcal{E} \to Set$, then every object is the coproduct of connected objects.
In the Elephant, lemma 3.3.6. this is a 6-line argument. But I am not sure how to generalize that argument to $\infty$-sheaves. (Maybe it’s easy, but I am not sure.)
So what I did instead now is that I re-proved this in a fashion that ought to have a straightforward, almost verbatim, $\infty$-categorical analog.
What i came up with is now the proof here at locally connected topos.
I’ll think a bit more now to see if from this I can really deduce that $Sh_{(\infty,1)}(X)$ is locally $\infty$-connected precisely if $X$ is locally contractible…
I wrote:
I’ll think a bit more now to see if from this I can really deduce that $Sh_{(\infty,1)}(X)$ is locally $\infty$-connected precisely if $X$ is locally contractible…
It seems that the argument that is needed is that which we already discussed once at Whitehead tower in an (infinity,1)-topos. I try to go through this now carefully to see if it serves to prove the above, but I will likely be interrupted before I am done.
But let me just briefly notice the relation to Whitehead towers:
Along the lines of the proof for $n = 1$ we want to say that for $(\Pi_n \dashv \Delta \dashv \Gamma) : \mathbf{H} \to n Grpd$ a locally n-connected (n,1)-topos and $X$ any object, that the $n$-connected objects $U_i$ covering $A$ are the pullbacks
$\array{ U_i &\to& * \\ \downarrow && \downarrow \\ X &\to& \Delta \Pi_n X } \,,$where the bottom morphism is the $(\Pi \dashv \Delta)$-unit. It is clear that this collection of objects does cover:
${\lim_\to}_{i \in \Pi X} U_i \stackrel{\simeq}{\to} X$this follows from universal colimits. The question is if the $U_i$ are really all $n$-connected in that $\Pi_n(U_i) \simeq *$.
But this is exactly the implicit claim at Whitehead tower in an (infinity,1)-topos. Notice that
$\mathbf{\Pi}_n X := \Delta \Pi_n X$is what I call the intrinsic path n-groupoid .
In suitable situations where both notions exist, this does coincide with the $n$-truncation of the standard fundamental infinity-groupoid of a topological space. As observed at universal cover, we have that the universal 1-connected cover of a connected topological space is the $\infty$-pullback
$\array{ \hat X &\to& * \\ \downarrow && \downarrow \\ X &\to& \mathbf{\Pi}_1(X) } \,.$Back then this was the starting point for the idea of defining the whole Whitehead tower of any object in any locally $\infty$-connected $(\infty,1)$-topos by the obvious iteration of this construction.
Now, at Whitehead tower in an (ininity,1)-topos I had once typed the argument for how the internal homotopy groups do behave as expected. If this goes through as one might expect, it would serve to yield a proof that $X$ is locally contracible precisely if $Sh_{(infty,1)}(X)$ is locally $\infty$-connected.
[edit: hm, some $n$s are off by $\pm 1$, as usual…]
Thanks, that sounds reasonable. I’ll read through the proofs that you have so far and see if they make sense to me.
There does seem to be some discussion of locally contractible locales in the literature already, although I can’t read any of it online right now. Ieki Moerdijk in particulary has mentioned them, and presumably he has a definition in mind.
I’ll read through the proofs that you have so far and see if they make sense to me.
Please do. I’d be grateful if you’d tried hard to poke holes into everything.
Meanwhile, I still don’t have a general proof that the $U_i$ in comment #4 are $n$-connected in the $\infty$-topos over a locally $n$-connected topological space, except for $n = 0$.
I do have something different, but related: in the cohesive $\infty$-topos ETop ∞Grpd I think I can show (see there) the analogous statement for the case that $X$ is a paracompact space: that all homotopy fibers of $X \to \Delta \Pi(X)$ are geometrically contractible. In fact, in that case I think I can show that the homotopy fibers of $X \to \Delta \tau_{\leq n} \Pi(X)$ for all $n \in \mathbb{N}$ map under $\Pi$ to the whole standard Whitehead tower of $X$.
But the proof of this is not purely general abstract, so I am not sure right now how to generalize it. Maybe I am just being dense. Or maybe there is more to it.
There does seem to be some discussion of locally contractible locales in the literature already, although I can’t read any of it online right now. Ieke Moerdijk in particulary has mentioned them, and presumably he has a definition in mind.
I’ll see what i can find out next week.
Maybe I missed the mention of something like this, but won’t you need some sort of q- or m-cofibrancy? Since isn’t toposic (local) connectiveness more about trivial homotopy (pro)groups?
but won’t you need some sort of q- or m-cofibrancy? Since isn’t toposic (local) connectiveness more about trivial homotopy (pro)groups?
Hm, let’s see, my argument is as follows:
by Artin-Mazur we have that if $X$ is a locally contractible topological space, then the simplicial set
${\lim_\to}_{(Y \to X)} Y_{contr}$(where the colimit ranges over all hypercovers and $Y_{contr}$ is the simplicial set obtained by contracting degreewise each connected summand to a point) has the same homotopy groups as $X$.
Observe that when switching site of definition $Sh(Op(C)) \simeq Sh(cOp(X))$ (right?) we have that this procedure computes the left derived functor of $\lim_\to : [cOp(X)^op, sSet] \to sSet$;
By Dugger-Hollander-Isaksen this means that the colimit in the above localizes on any split hypercover, degreewise a coproduct of contractibles.
I was talking about the converse: if Sh(X) is locally $\infty$-connected then $X$ is locally contractible.
Hi Mike,
sorry, I had missed that you had already replied here. (That $n$Publication-discussion is drawing a lot of energy and attention…)
First, just for the record, concerning item 2 in #8: I have recorded the standard facts and the example of locally contractible spaces at dense sub-site.
Then finally concerning your comment:
so you are suggesting that for an $\infty$-topos over a topological space $X$ being toposophically $\infty$-connected to imply that $X$ is locally contractible it is necessary that $X$ is an m-cofibrant space?
Not sure. Maybe I need to think more about this.
By the way, I asked about the definition of “locally contractibe locale” as given somewhere by Ieke: presumeably it was by saying there is a cover such that the patches are contractible as seens by homming the locales $\Delta^n$ into them.
so you are suggesting that for an
∞-topos over a topological space X being toposophically [locally] ∞-connected to imply that X is locally contractible it is necessary that X is an m-cofibrant space?
Yes, I’m suggesting that something of the sort at least might be necessary (perhaps even “local m-cofibrancy”). I would be a bit surprised if no conditions at all were required. Of course the toposophic homotopy groups are more refined than the classical ones, so my intuition could be off.
I edited the example of good open covers at dense sub-site to read as follows:
every paracompact manifold has a good open cover by open balls diffeomorphic to a Cartesian space.
I changed homeomorphic to diffeomorphic. Just checking this is correct (not quite sure about exotic smoothness, that’s all)
Just checking this is correct
Yes, it is. A while back I spelled out the full detailed proof for this at good open cover. This statement has a curious status in the literature: in textbooks one finds both the statement that this is a difficult open problem and the statement that this is obvious, but never a statement of the proof.
I’d prefer, though, if we kept topological and smooth manifolds as two distinct cases at dense sub-site. I think I’ll do that now.
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