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 $PU(\mathcal{H})$.

]]>So not even a proper action of an arbitrary Lie group?

]]>Oh, I see that Guillou, May & Merling 17, pp. 15 has analogous discussion.

Hm, but so they can’t get around assuming $\Gamma$ to be compact Lie, either? That would be too bad.

]]>starting a section on universal equivariant principal bundles (here):

added the definition of the Murayama-Shiwakawa groupoid (for discrete $G$)

$\mathcal{B}\Gamma \;\coloneqq\; Groupoids(TopSpaces) \big( G \times G \underoverset{pr_2}{pr_1}{\rightrightarrows} G, \; \Gamma \rightrightarrows \ast \big)$with its $G$-action

$(g \cdot F) (g_1, g_2) \;\coloneqq\; \alpha(g) \big( F(g_1 g, g_2 g) \big) \,, {\phantom{AAA}} (g \cdot \eta) (g_1) \;\coloneqq\; \alpha(g)(\eta(g_1))$and then statement of its $H$-fixed loci (as topological groupoids)

$\big( \mathcal{B}\Gamma \big)^H \;\; \simeq \;\; \Big( Groups(TopSpaces)_{/G} \big( G, \, \Gamma \rtimes_\alpha G \big) \Big) \sslash \Gamma \,.$I have written some words indicating the proof, which is essentially an elementary inspection (though one best uses some diagrammatic notation which I haven’t tried to Instikify here, my local version uses equations between tikcz diagrams, which cannot be imported here – and tikzcd itself cannot be nested, unfortunately)

The point is that this gives right away the fixed point behaviour of the classifying space for equivariant principal bundles according to Lashof82 Thm 2.17 and Lashof&May86 Theorem 10 (if we grant that they mean “centralizer” instead of “normalizer” in the first slot!?) – IF we can assume that passage to fixed points commutes with realization.

Now Murayama-Shiwakawa use fat realization, but comment that they could use ordinary realization (which would commute so) at least if both $G$ and $\Gamma$ are compact Lie and possibly more generally, which however they leave open.

]]>Oh, I see. I was misreading the definition of the action in Murayama-Shimakawa, p. 1293. So never mind.

]]>I am puzzled by a statement on universal equivariant principal bundles. Maybe somebody can help me:

A neat explicit construction of the universal equivariant principal bundle is given in Murayama-Shimakawa 95: if the equivariance group is discrete, then (using their remark on the bottom of p. 6) the base of the universal $G$-equivariant $\Gamma$-principal bundle is the realization of the $G$-topological groupoid

$\mathcal{B} \Gamma \;\coloneqq\; TopGroupoids \big( G \times G \rightrightarrows G,\; \Gamma \rightrightarrows \ast \big)$whose $G$-action on functors $F$ and natural transformations $\eta$ is

$(g \cdot F)(g_1, g_2) \;\coloneqq\; F(g_1 g, g_2 g) \,, \;\;\;\;\;\; (g \cdot \eta)(g_1) \;\coloneqq\; \eta( g_1 g ) \,.$It’s a fun fact (which these authors don’t mention, but which one can check) that for $H \subset G$ any subgroup, the $H$-fixed groupoid of this is equivalent, as a topological groupoid (no stackification anywhere), to

$(\mathcal{B} \Gamma)^H \;\;\simeq\;\; TopGroupoids \big( G \rightrightarrows \ast,\, \Gamma \rightrightarrows \ast \big) \,.$This implies at once that the $H$-fixed subspaces of the classifying space $\left\Vert \mathcal{B}\Gamma\right\Vert$ are homotopy equivalent to the disjoint union over conjugacy classes of group homomorphisms $\rho : G \to \Gamma$ of the classifying spaces of the centralizer subgroups $\Gamma^\rho$

$(\mathcal{B} \Gamma)^H \;\;\simeq\;\; \underset{ [\rho] }{\sqcup} B \Gamma^\rho$That this should be the case is Theorem 2.17 in Lashof 82, where this is derived not from inspection of a concrete model, but from more abstract criteria for universal equivariant bundles.

There is a subtlety here in that Lashof 82 considers equivariant bundles where the equivariance group $G$ commutes with the structure group $\Gamma$, while Murayama-Shimakawa 95 mean to consider the case where both jointly act as a semidirect product group.

But in the special case where they do commute, the model of Murayama-Shimakawa 95 makes nicely manifest the fixed point structure of the classifying space for equivariant principal bundles according to Theorem 2.17 in Lashof 82.

So far so good.

But implicit in Murayama-Shimakawa 95 is that a more general action of $G$ on $\Gamma$ (to a direct product group structure) *does not affect* the underlying $G$-space of the universal equivariant bundle which they build.

So their result says – unless I am mixed up, but it seems clear – that the above formula for the fixed point structure actually holds generally.

Now, Lashof-May 86 generalize Lashof 82 to these more general group actions. Their Theorem 10 seems to contradict this conclusion from Murayama-Shimakawa 95:

Namely their Theorem 10 says that for (in particular) semidirect product group action $G \rtimes_\alpha \Gamma$, the $H$-fixed point subspace of the classifying space has connected components not indexed by conjugacy classes of group homomorphisms $G \to \Gamma$, as above, but by conjugacy classes of lifts of $H$ to $G \rtimes_\alpha \Gamma$ (slightly paraphrasing here).

That sounds plausible, because such subgroups are exactly what labels the “local models”, namely the equivariant bundles over $G/H$. But how is this compatible with Murayama-Shimakawa 95?

]]>Yes, i was very surprised to see that at torsor. I had remembered that we had material on empty heaps, and was going to make that comparison, but then double checked…

]]>And, luckily, the empty bundle is also a fibration…

]]>Ah; what you call “a torsor on the nLab” was introduced just days ago by Richard, in rev 40! Before that, from Todd’s rev 13 on, “torsor on the nLab” *was* assumed to be inhabited. :-)

Hmm, interesting! I guess this is the difference between a torsor on the nLab (which doesn’t seem to need to be inhabited), and the more usual definition (which requires the underlying set to be inhabited).

]]>I have added a remark to this effect (here).

]]>Just emerged out of a little paradox crisis, with the following insight (unless I am still confused):

If we define “principal bundle” internally by just demanding the principality condition

$\Gamma \times P \underoverset{\simeq}{ (g,p) \mapsto ( p, g \cdot p ) }{\longrightarrow} P \times_X P$(which is a limit-theory condition)

and *not* explicitly demanding that $X \simeq P/G$ – since that is implied by the principality condition *IF* $P \to X$ is an effective epi — then *the empty bundle is principal*.

!!

This is actually relevant – and resolves an apparent paradox – when thinking about fixed loci of equivariant bundles:

The fixed locus functor is right adjoint and hence preserves internal limit theories such as the above flavour of internal principal bundle. But it also frequently keeps the structure group intact while producing an empty underlying bundle.

First I thought I had discovered a flaw in mathematics and was about to call the Fields Institute (or who you’re gonna call in that case?) but now I see that all is good: The empty bundle is principal.

]]>added this pointer as “precursor discussion” (has essentially all the ingedients, but doesn’t quite articulate a definition of equivariant bundles as such):

- T. E. Stewart,
*Lifting Group Actions in Fibre Bundles*, Annals of Mathematics Second Series, Vol. 74, No. 1 (1961), pp. 192-198 (jstor:1970310)

It’s interesting though, because if we grant that this is the origin of equivariant bundles, then the general tomDieck-definition-rediscovered-by-Lashof-May is already right there on the first page, if we agree that by “invariant subgroup” the author must mean “normal subgroup” (clearly).

]]>added this pointer:

- Eyup Yalcinkaya,
*Equivariant Principal Bundles over the 2-Sphere*(arXiv:1902.06293)

Added statement and proof (here) that Lashof’s local models are locally trivial as ordinary fiber bundles.

(This is inside Lemma 1.1 in Lashof 82, I have just tried to isolate it for emphasis and expand out the argument a little for clarity)

]]>finally added the pertinent diagrams to the definition of the internal principal bundles (here)

]]>added the statement (here) that the $H$-fixed point functor takes $G$-equivariant $\Gamma$-principal bundles to $N(H)/H$-equivariant $\Gamma^H$-principal bundles – as is immediate from the internal perspective and the fact that $(-)^H$ on the ambient category is a right adjoint

]]>I have added statement and proof (here) that equivariant local trivializability in the sense of Lashof implies that in the sense of tom Dieck, for $\alpha = 1$ (i.e. when the equivariance group and the structure group commute, as Lashof assumes).

(Not claiming the writeup is optimal yet. Lots of moving parts here, notation-wise. But we’ll get there.)

[Same idea should work for general $\alpha$ and comparing then to Lashof-May 86. But i’ll call it quits for tonight.]

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