Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJan 14th 2011

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

1. the intrinsic fundamental $\infty$-groupoid functor in $ETop \infty Grpd$ sends paracompact topological spaces to their traditional fundamental $\infty$-groupoid

$\Pi_{ETop \infty Grpd}(X) \simeq \Pi_{Top}(X) \simeq Sing X$;

2. 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.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJan 14th 2011

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.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJan 14th 2011
• (edited Jan 14th 2011)

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…

• CommentRowNumber4.
• CommentAuthorTobyBartels
• CommentTimeJan 15th 2011

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.)

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJan 15th 2011

I have figured it out. Of course it was my fault. i had typed an incorrect path.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJan 15th 2011
• (edited Jan 15th 2011)

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

1. the previous proposition that $\Pi$ is modeled in this case by geometric realization;

2. 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.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeJan 15th 2011

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.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJan 17th 2011

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.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeJan 20th 2011

have added statement and proof of the assertion that

$ETop\infty Grpd \simeq Sh_{(\infty,1)}(TopMfd) \,.$
• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeJan 20th 2011

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.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeJan 26th 2011
• (edited Jan 26th 2011)

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? )

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeFeb 23rd 2011

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.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeFeb 23rd 2011

Have added the corresponding corollary for the smooth case to SmoothooGrpd – Geometric homotopy.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeSep 13th 2011
• (edited Sep 13th 2011)

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

• CommentRowNumber15.
• CommentAuthorjim_stasheff
• CommentTimeSep 14th 2011
I could check the proof, but why is paracompactness relevant?
• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeSep 14th 2011
• (edited Sep 14th 2011)

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.