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have started simplicial topological group
Does fat geometic realization send global Kan fibrations to topological fibrations?
For $G$ a simplicial topological group, is $\Vert W G\Vert \to \Vert \bar W G\Vert$ a fibration?
I am thinking about the following:
Claim For $G$ a well-sectioned simplicial topological group, $X$ a good simplicial topological space, every simplicial $G$-principal bundle $P$ over $X$ is also proper as a simplicial topological space.
Idea of the proof
Every such $G$-bundle $P \to X$ arises as the pullback
$\array{ P &\to& W G \\ \downarrow && \downarrow \\ X &\stackrel{\tau}{\to}& \bar W G }$for some morphism $\tau$ of simplicial topological spaces.
Therefore for each $n \in \mathbb{N}$ we have that $P_n$ is given by the pullback
$\array{ P_n &\to& (W G)_n \\ \downarrow && \downarrow \\ X_n &\stackrel{\tau_n}{\to}& (\bar W G)_n }$in $Top$.
Now, $W$ and $\bar W G$ are both proper simplicial topological spaces. And $(W G)_n \to (\bar W )_n$ is a Hurewicz fibration for all $n$. Therefore by the theorem now included at closed cofibration, we have that the degeneracy maps of $P$ are induced by morphisms of pullback diagrams one of whose legs is a fibration along a degreewise closed cofibration. hence are themselves closed cofibrations. Hence $P$ is proper.
Right?
Urs, have you seen the paper that Danny and I are working on? We address very similar points.
Hi David,
so apparently you didn’t get the email that I sent to you. What’s your current working email address?
Yes, I am looking at your paper with Danny as we speak! That’s where this question originates from: I think you provide almost all the statements to show that the left derived functor of geometric realization of simplicial topological spaces preserves homotopy fibers of morphisms of the form $X \to W G$, hence sends topological $\infty$-bundles to their underlying topological bundles.
I need that statement for differential cohomology in a cohesive topos (schreiber). But yesterday I realized that in my proof that the intrinsic $\Pi : ETop\infty Grpd \to Top$ preserves homotopy fibers (see Euclidean topological infinity-groupoid) I had been silently asssuming that with $X$ and $W G$ proper simplicial spaces, also the simplicial bundle $P$ that is defined by $\tau : X \to W G$ has the propety that its geometric realization coincides with its homotopy colimit.
This seems to be an important statement also for the main statement in your article with Danny, and so I tried to check with you and Danny if my idea how to show this is correct. Danny has meanwhile confirmed that this argument is indeed true.
Let me just point out again what that has to do with homotopy fibers:
since for a simplicial topological group $G$ we have
$\bar W G$ is globally fibrant (the maps $(\bar W G)_n \to [\Lambda^n_k, \bar W G]$ have global continuous sections) ;
$W G$ is globally fibrant
$W G \to \bar W G$ is a global fibration
we have that $W G \to \bar W G$ is a presentation by a fibration of the point inclusion $* \to \mathbf{B}G$ in the projective model structure for Euclidean-topological $\infty$-groupoids $[CartSp_{top}^{op}, sSet]_{proj}$.
This means that for all simplicial spaces $X$ and morphisms $\tau : X \to \bar W G$ the ordinary pullback
$\array{ P &\to& W G \\ \downarrow && \downarrow \\ X &\stackrel{\tau}{\to}& \bar W G }$is a model for the homotopy pullback of $* \to \mathbf{B}G$ along $\tau$, hence a model for its homotopy fiber, hence that $P$ is indeed a presentation of the correct topological principal $\infty$-bundle given by $\tau$.
Now assume all topological spaces here are degreewise paracompact and admit good open covers. Then one can prove that the intrinsic fundamental $\infty$-groupoid functor $\Pi : ETop\infty Grpd \to \infty Grpd \simeq Top$ is presented on these by fat geometric realization. But now, since $X$ and $\bar W G$ and $W G$ are assumed to be proper and since we find that then $P$ is implied to be proper, it follows that on our situation here $\Pi$ is already presented by ordinary geometric realization.
Now moreover, by your statement with Danny we have that $|W G| \to |\bar W G|$ is again a fibration resolution of the point inclusion $* \to \mathbf{B}|G|$. This shows (and I would think you could mention this as a nice immediate corollary in your article) that
$\array{ |P| &\to& |W G| \\ \downarrow && \downarrow \\ |X| &\stackrel{\tau}{\to}& |\bar W G| }$which is an ordinary pullback diagram by the general fact that geometric realization preserves pullbacks, is again even a homotopy pullback, hence itself exhibits $|P|$ as the homotopy fiber of $|\tau|$.
So in total this corollary of your work with Danny shows:
Theorem On morphisms of Euclidean-topological $\infty$-groupoids $X \to \mathbf{B}G$ that are presented by morphisms of degreewise paracompact simplicial topological spaces, we have that $\Pi : ETop\infty Grpd \to \infty Grpd$ preserves homotopy fibers.
That’s a very powerful statement for doing refinements of Whitehead towers from Top to cohesive $\infty$-groupoids. For instance there is a refinement of the first fractional Pontryagin class $\frac{1}{2}p_1 : B Spin \to B^4 \mathbb{Z}$ to a morphism of simplicial topological (even Lie) groups $\frac{1}{2}\mathbf{p}_1 : W Spin \to W \Xi U(1)[2]$. Its homotopy fiber will be some topological 2-group: the string 2-group. By the above theorem it is immediate that its geometric realization is the ordinary topological string group. And so on for higher steps in the Whitehead tower.
That’s what I am after. Only that yesterday I reallized that I had missed to argue that $P$ in the above is proper. So therefore I was fishing here for sanity checks that indeed it is.
Ah, I did get that email - after I checked here. So you do have the right email for me. Then I’ve been out all day, so couldn’t fix my error. I’ll have to think about this - I need an early night tonight!
I have written out the argument at geometric realization of simp top spaces – Applications – realization of topological principal oo-bundles.
Have to dash off now. Can try to expand more later.
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