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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.
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