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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 14th 2011

    I have renamed the entry on the \infty-topos on CartSp topCartSp_{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 ETopGrpdETop \infty Grpd sends paracompact topological spaces to their traditional fundamental \infty-groupoid

      Π ETopGrpd(X)Π Top(X)SingX \Pi_{ETop \infty Grpd}(X) \simeq \Pi_{Top}(X) \simeq Sing X;

    2. more generally, for X X_\bullet a simplicial topological space we have

      |Π ETopGrpd(X )||X | |\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 TopTop coincides with cohomology in ETopGrpdETop \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 Π:ETopGrdGrpdTop\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 Π\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 (|WGW¯G|)=(E|G|B|G|)(|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 *BG* \to \mathbf{B}G by a fibration.

    This is now in the section ETop∞Grpd : Cohomology and principal ∞-bundles.

    evident open Question: Does Π:ETopGrpdGrpd\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 G\mathbf{G} is a cohesive refinement of a discrete \infty-group GG, then cohesive G\mathbf{G}-principal \infty-bundles are cohesive refinements of bare GG-principal \infty-bundles.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJan 15th 2011

    I have added statement and proof of the corollary that under |Π()|:ETopGrpdTop|\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:TopologicalManifoldsETopGrpd i : TopologicalManifolds \to ETop \infty Grpd

    is a full and faithful \infty-functor – which boils down to asserting that

    i:TopologicalManifoldsSh(CartSp top) 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

    ETopGrpdSh (,1)(TopMfd). 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 TopMfdTopMfd 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 Π:ETopGrpdGrpd\Pi : ETop\infty Grpd \to \infty Grpd preserving homotopy fibers of morphism of the form XW¯GX \to \bar W G (in the section Cohomology and prinicipal oo-bundles).

    Previously I was asking not only GG to be a simplicial group in manifolds, but also XX to be a globally Kan fibrant simplicial manifold. That is unnecessary, we can allow XX to be any globally fibrant simplicial presheaf.

    In particular it follows without further ado that Π\Pi preserves the homotopy fiber of the smooth fractional Pontryagin class 12p 1\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 Π(X)\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 XX is a sufficient condition for a previous step: I am referring to a general abstract definition of the \infty-groupoid Π(X)\Pi(X) by a certain left adjoint Π\Pi. For that abstract definition to reproduce the expected object SingXSing X a sufficient condition is that XX is paracompact. This is discussed in the part above the theorem that I pointed to in #14.

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