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I have renamed the entry on the $\infty$-topos on $CartSp_{top}$ into Euclidean-topological infinity-groupoid.
Then in the section Geometric homotopy I have written out statement and proof that
the intrinsic fundamental $\infty$-groupoid functor in $ETop \infty Grpd$ sends paracompact topological spaces to their traditional fundamental $\infty$-groupoid
$\Pi_{ETop \infty Grpd}(X) \simeq \Pi_{Top}(X) \simeq Sing X$;
more generally, for $X_\bullet$ a simplicial topological space we have
$|\Pi_{ETop \infty Grpd}(X_\bullet)| \simeq |X_\bullet|$ ,
where on the left we hve geometric realization of simplicial sets, and on the right of (good) simplicial topological spaces.
have added statement and proof of the corollary that over paracompact spaces nonabelian cohomology in $Top$ coincides with cohomology in $ETop \infty Grpd$ with locally constant coefficients. In the section cohomology.
I have added statement and proof of how the intrinsic fundamental infinity-groupoid in a locally infinity-connected (infinity,1)-topos is indeed presented by a path $\infty$-groupoid:
in the subsection Path oo-groupoid.
help: this discussion works generally and ought to go into infinity-connected site. But I cannot edit that entry. It seems that the cache-bug is at work. I tried to clear the cache, but the cache-clear command also says it cannot recognize the entry…
It looks like you moved that page to infinity-connected (infinity,1)-site, so you can edit that. But that’s strange that you can’t remove the cache. (I can’t either, since I don’t have my key with me.)
I have figured it out. Of course it was my fault. i had typed an incorrect path.
have now moved over from Smooth∞Grpd to ETop∞Grpd statement and proof that in the degreewise paracompact case the intrinsic fundamental $\infty$-groupoid functor $\Pi : ETop \infty Grd \to \infty Grpd \simeq Top$ preserves homotopy fibers and hence principal $\infty$-bundles.
This is a direct corollary of
the previous proposition that $\Pi$ is modeled in this case by geometric realization;
the theorem by Danny Stevenson and David Roberts that geometric realization sends universal simplicial topological bundles to universal topological bundles $(|W G \to \bar W G|) = (E |G| \to B |G|)$
and the observation that all universal bundles are just resolutions of the point inclusion $* \to \mathbf{B}G$ by a fibration.
This is now in the section ETop∞Grpd : Cohomology and principal ∞-bundles.
evident open Question: Does $\Pi : ETop \infty Grpd \to \infty Grpd$ maybe preserve homotopy fibers more generally? I don’t know. It is a desireable property for good cohomology theory in a cohesive $\infty$-topos, because it says that if $\mathbf{G}$ is a cohesive refinement of a discrete $\infty$-group $G$, then cohesive $\mathbf{G}$-principal $\infty$-bundles are cohesive refinements of bare $G$-principal $\infty$-bundles.
I have added statement and proof of the corollary that under $|\Pi(-)| : ETop \infty Grpd \to Top$ the internal geometric Whitehead towers map to the traditional Whitehead towers. In the new section: ETop ∞Grpd: Universal coverings and geometric Whitehead towers.
I have added to Euclidean-topological infinity-groupoid statement and proof that the evident functor
$i : TopologicalManifolds \to ETop \infty Grpd$is a full and faithful $\infty$-functor – which boils down to asserting that
$i : TopologicalManifolds \to Sh(CartSp_{top})$is a full and faithful functor.
have added statement and proof of the assertion that
$ETop\infty Grpd \simeq Sh_{(\infty,1)}(TopMfd) \,.$I have removed at Euclidean-topological infinity-groupoid my previous pedestrian proof that $TopMfd$ is a full sub-$\infty$-category and instead state this now as an immediate corollary of the above proposition and the $\infty$-Yoneda lemma.
have added to Euclidean-topological infinity-groupoid a subsection Model category presentation with some remarks.
(This is a more succinct re-write of stuff already in the intro part of smooth infinity-groupoid. Moreover, I have typed it now twice, since I lost the first version when my browser had a mysterious crash. Oh time, where doest thou disappear to? )
I have strengthened the statement about $\Pi : ETop\infty Grpd \to \infty Grpd$ preserving homotopy fibers of morphism of the form $X \to \bar W G$ (in the section Cohomology and prinicipal oo-bundles).
Previously I was asking not only $G$ to be a simplicial group in manifolds, but also $X$ to be a globally Kan fibrant simplicial manifold. That is unnecessary, we can allow $X$ to be any globally fibrant simplicial presheaf.
In particular it follows without further ado that $\Pi$ preserves the homotopy fiber of the smooth fractional Pontryagin class $\frac{1}{2}\mathbf{p}_1$ and hence sends the string 2-group to the string group, as discussed at string 2-group.
Have added the corresponding corollary for the smooth case to SmoothooGrpd – Geometric homotopy.
I have added to Euclidean-topological infinity-groupoid a remark on presentations of $\mathbf{\Pi}(X)$ by topological Kan complexes of paths: in a new section Presentation of the fundamental path oo-groupoid
why is paracompactness relevant?
Paracompactness of $X$ is a sufficient condition for a previous step: I am referring to a general abstract definition of the $\infty$-groupoid $\Pi(X)$ by a certain left adjoint $\Pi$. For that abstract definition to reproduce the expected object $Sing X$ a sufficient condition is that $X$ is paracompact. This is discussed in the part above the theorem that I pointed to in #14.
Now that I finally looked into Delta-generated topological spaces, I understand that the title of this page here needs to be changed.
$\,$
The history of the naming here, as I remember it (probably visible somewhere in the nForum history) is this:
First, I had called the objects of the cohesive $\infty$-topos $Sh_\infty(EuclideanTopologicalSpaces)$ “topological $\infty$-groupoids”.
Then Mike (Shulman) had complained, rightly, that that name would rather apply to $Sh_\infty(TopologicalSpaces)$ (which however is not cohesive).
Then one of us (I forget who) suggested the current title “Euclidean topological $\infty$-groupoid”. This rhymes (or rather alliterates) on the canonical name of its site.
But it’s still not the right name, I think now.
$\,$
An appropriate name should be $\Delta$-generated $\infty$-groupoids, or maybe “Euclidean generated”.
$\,$
Namely, by that characterizing idempotent adjunction (here) $\Delta$-generated spaces are just those which come from concrete sheaves on Euclidean spaces.
I think the “$\Delta$“-terminology is misleading: It is not the shape of the simplices that matters, but only that they are convex subsets of Euclidean spaces of all finite dimensions (authors routinely use this fact, e.g. in the proof of Prop. 3.2 here).
Hence, I suppose that $\Delta$-generated spaces could just as well be called “Euclidean generated spaces”. But once we are at this point, we see that this sense of “Euclidean generation” is just what the construction of $Sh_\infty(EuclideanTopologicalSpaces)$ promotes to the non-concrete and to the higher homotopical.
In fact, “Euclidean generated” is of course a tautological name for $Sh_\infty(EuclideanTopologicalSpaces)$, since the objects of the site are generators of the $\infty$-topos also in the category-theoretic sense. So if we agree that for topological spaces “$\Delta$-generated” and “Euclidean generated” is equivalent, then it is inevitable that “$\Delta$-generated topological $\infty$-groupoids” would be a proper terminology here. And “Euclidean-generated topological $\infty$-groupoids” would be even better, albeit somewhat revisionistic.
that name would rather apply to $Sh_\infty(TopologicalSpaces)$ (which however is not cohesive).
Is the problem lack of local contractibility in general? Not sure if it’s of any interest, but if so, I suspect that the big topos of topological spaces is cohesive over condensed sets with some kind of ’pro-local homeomorphism topology’, i.e. a topological version of the pro-étale topology in algebraic geometry. I spent a few minutes a few weeks ago trying to think about how this should be defined; it would be some kind of different formulation of shape theory. For locally contractible spaces, everything should be the same as usual.
that name would rather apply to $Sh_\infty(TopologicalSpaces)$ (which however is not cohesive).
Is the problem lack of local contractibility in general?
Yes. That’s what prevents the extra left adjoint to $\infty$-groupoids to exist.
I suspect that the big topos of topological spaces is cohesive over condensed sets with some kind of ’pro-local homeomorphism topology’
That sounds roughly analogous to a similar statement for schemes made by Peter Scholze recently. It would be very useful if any of this would be written down in citable form.
We ought to extract some of the conclusions from that long discussion at the Café. But often it sounded like people didn’t mind that there was no $\sharp$, just whether there was ʃ. So I’m not sure what to say precisely.
I’m not sure what to do with something like
the big pro-etale topos of schemes over a fixed algebraically closed field k is, almost, cohesive over pyknotic sets, …
In the case of topological spaces I think # should exist. It would be an excellent exercise/addition to the nLab to work out how to adapt the definition of the pro-étale topology to topology (the reverse of the development of the étale topology!).
To get started one can observe that a local homeomorphism with target a one point space is discrete. Thus if one works with cofiltered limits of local homeomorphisms, and has a finiteness condition on the fibres, one will get profinite sets. This definition is very close to the pro-étale topology in algebraic geometry.
as per #17 I have re-named to Euclidean-generated $\infty$-groupoids and cross-linked with Euclidean-generated topological space (aka $\Delta$-generated space)
Am changing my mind again:
First, a better-term for Delta-generated topological spaces might be
since:
its’s precisely those that carry a D-topology, hence which are, indeed “D-topological”,
the “D” still alliterates to “Delta-generated”
it’s shorter than “Delta-generated”, even when keeping the adjective “topological”
This way, $Sh_\infty(EuclideanSpaces)$ would be called
which makes clear at once both that it’s a kind of topological $\infty$-groupoid, and that it’s a special kind, coming from “D-topology” and alluding to “Delta-generation”.
I like it!
If I get to do my algebraic topology course again in January as an intensive national summer school offering, then I’m planning to use D[elta generated]-topological spaces, since these are the kinds of things that arise as geometric realisations of Delta-sets/simplicial sets.
On a separate note, am I right in saying that D-topological spaces can also be described as those spaces that are quotients of coproducts of standard simplices? That to me sounds slightly closer to the idea of manifolds being quotients of coproducts of charts, so possibly more accessible to students with less topology background.
Thanks for the feedback! I have made that terminology a remark at Delta-generated space: here.
Regarding your question: I haven’t thought about this, but we have (and had since rev. 8) a remark that claims this is the case: here.
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