First of all: Exactly when there is a bundle morphism from $E_1$ to the pullback of $E_2$ to $B_1$.

Hence you are really asking: Given two fiber bundles over the same base space, when is there is a morphism between them?

The answer crucially depends on which kind of bundles you consider. For example:

For vector bundles always (the zero map). For principal bundles only if they are isomorphic.

]]>related to (G,\alpha, |Gamma) structures? ]]>

Thanks again. This and some other things fixed now in the pdf here.

]]>Typo in the book #79

]]>Borel modal structure

Thanks, I see. So maybe we would just remove the word “following”. (I have done that in our local copy now, not yet uploaded anywhere.)

]]>Re #79: Concerning the smooth Oka principle and the shape theorem it uses, you cite it as “[BBP19] following [Pv14]”.

The original proof in [Pv14] was similar to the current proof in [BBP19]. Later, the current arXiv version of the [BBP19] manuscript was written, incorporating the proof in [Pv14] with some modifications.

In parallel with this, I wrote up a completely new (and much shorter) proof for the shape theorem, which is available in the current (new) version of [Pv14], which I still hope to publish once I have a bit of time. This new version of [Pv14] no longer overlaps with [BBP19].

So technically, [BBP19] is not following [Pv14] anymore, and the oldest proof of the shape theorem is currently available in [BBP19].

]]>For what it’s worth, I have prepared some talk slides with an introduction: here.

(It’s mainly an elementary exposition of just plain 2-bundles, so far. But there may be a followup talk.)

]]>Thanks! These typos have been fixed now (here), together with a couple more in their vicinity.

]]>Def. 4.2.10 (i), that should be

$(B p)^{\ast}_{\mathcal{X}, \mathcal{A}}$

In (3.9.7)

Chtid

$Dsc Pnt$ lacks a $\circ$

and you have them all as counits.

(3.106) is missing a $\simeq$.

Remark 3.3.43

A necessary condition for a finite group have cover-resolvable singularities

missing ’to’.

Lemma 3.3.47, there’s a singularity before a $G$ rather than beneath it.

]]>I have taken the liberty of adding the pointer to

and referencing this to the corresponding bits of the entry that I had written earlier.

]]>Thanks. These typos fixed now and an example on equivariant bundle gerbes added (currently Ex. 4.1.25).

]]>some quick typos

combinatorial model categorie (in blue on p. 85)

(3.1) $Y$ rather than $A$.

(3.97) Dscid and Chtid

(in the 211219f version).

]]>For what it’s worth, and I don’t want to sound like I’m fishing for a citation, Murray–Roberts–Stevenson–Vozzo exhibits some explicitly-described $String_G$-equivariant $U(1)$-bundle gerbes, in case you want to point out examples that aren’t just $G$-equivariant for a(n internal) 1-group $G$.

]]>Congrats!

I do wonder if this allows one to say anything about the conjecture of Tu, Xu and Laurent-Gengoux about torsion twists of K-theory of proper, cocompact Lie groupoids. Such a groupoid is covered (in a weak sense: essentially surjective and full) by the disjoint union of finitely many action groupoids that are orthogonal actions of compact Lie groups on a unit ball. Note that their suggestion “One possibility to prove this conjecture…” really doesn’t hold water, it ignores the distinction between shape and cohesive stack etc. I spent a lot of time during my PhD thinking about that idea…

]]>With Hisham we have been writing – and are now finalizing – a book, titled:

which means to solve the classification problem for equivariant topological bundles with truncated structure group by embedding into principal $\infty$-bundles internal to the cohesive $\infty$-topos of smooth $\infty$-groupoids and then invoking the “smooth Oka principle”.

We are keeping a pdf with the current draft version here. Comments are welcome.

]]>added pointer to:

- Charles Rezk,
*Classifying spaces for 1-truncated compact Lie groups*, Algebr. Geom. Topol.**18**(2018) 525-546 (arXiv:1608.02999)

Added brief statement (here) of the result of Lashof, May and Segal 1983 (classification of equivariant bundles whose structure group is compact Lie and *abelian*), in its more pronounced form given in May 1990, Thm. 3, Thm. 10 .

added pointer to:

- Wolfgang Lück,
*Survey on Classifying Spaces for Families of Subgroups*, In:*Infinite Groups: Geometric, Combinatorial and Dynamical Aspects*Progress in Mathematics,**248**Birkhäuser (2005) (arXiv:math/0312378, doi:10.1007/3-7643-7447-0_7)

added pointer to:

- Charles Rezk, Sec. 2.3 in:
*Global Homotopy Theory and Cohesion*, 2014 (pdf, Rezk14.pdf:file)

Oh, I think I finally see the abstract story here:

For $\Gamma \!\sslash\! G \;\in\; Groups\big( (SingularSmoothGroupoids_\infty)_{/\mathbf{B}G} \big)$, the equivariant classifying space $\mathcal{B}_G \Gamma$ should simply be taken to be (the shape of) the right derived base change of the plain classifying stack along the unit map of the orbi-singular modality $\prec$, hence:

$\mathcal{B}_G \Gamma \;\coloneqq\; \big( \eta^{\prec}_{\mathbf{B}G} \big)_{\ast} \big( \mathbf{B} (\Gamma \!\sslash\! G) \big)$It then follows by the right base change adjunction that the geometric $H$-fixed points of $\mathcal{B}_G \Gamma$ are the homotopy $H$-fixed points of $\mathbf{B} \Gamma$, which identifies the equivariant homotopy type of $\mathcal{B}_G \Gamma$ with the Murayama-Shimakawa-style equivariant classifying space, by the observation in #73 above.

More generally, it follows by the same right base change adjunction that $G$-equivariant $\Gamma$-principal bundles classified by $\mathcal{B}_G \Gamma$ on a $G$-space $X$ are equivalently $\Gamma \sslash G$-principal bundles on the corresponding orbispace, this being the stack $X \!\sslash\! G$ in the slice over $\mathbf{B}G$ – thus identifying the the traditional theory of equivariant principal bundles with the evident stacky formulation.

]]>added a brief remark (here) that the Murayama-Shimakawa equivariant classifying space $\mathcal{B}\Gamma$ has as $H$-fixed points the $H$-homotopy fixed points of $B \Gamma$.

]]>Hm, on the other hand the construction for $\Gamma = PU(\mathcal{H})$ in

- Noe Barcenas, Jesus Espinoza, Michael Joachim, Bernardo Uribe,
*Universal twist in Equivariant K-theory for proper and discrete actions*(arXiv:1202.1880)

looks just like the Murayama-Shiwakawa construction (not cited as such) but with group homomorphisms $G \to \Gamma$ restricted to “stable” maps.

]]>$\Gamma$ here is the structure group, not the equivariance group.

That’s why we’d rather not have much conditions on this at all, because in practice this needs to allow for choices like $PU(\mathcal{H})$.

]]>