nForum - Discussion Feed (equivariant bundle) 2023-10-03T14:16:28+00:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher Urs comments on "equivariant bundle" (108725) https://nforum.ncatlab.org/discussion/11825/?Focus=108725#Comment_108725 2023-04-20T19:08:02+00:00 2023-10-03T14:16:27+00:00 Urs https://nforum.ncatlab.org/account/4/ First of all: Exactly when there is a bundle morphism from E 1E_1 to the pullback of E 2E_2 to B 1B_1. Hence you are really asking: Given two fiber bundles over the same base space, when is there ...

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.

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jim stasheff comments on "equivariant bundle" (108724) https://nforum.ncatlab.org/discussion/11825/?Focus=108724#Comment_108724 2023-04-20T19:01:57+00:00 2023-10-03T14:16:27+00:00 jim stasheff https://nforum.ncatlab.org/account/1409/ Given two fiber bundles (or principal fiber bundles) for i =1,2, E_i to B_i and a map B_1 to B_2 when can the latter be lifted to E_1 to E_2 making a commutative diagram? Given two fiber bundles (or principal fiber bundles) for i =1,2, E_i to B_i and a map B_1 to B_2 when can the latter be lifted to E_1 to E_2 making a commutative diagram? ]]> jim stasheff comments on "equivariant bundle" (100713) https://nforum.ncatlab.org/discussion/11825/?Focus=100713#Comment_100713 2022-07-09T17:13:48+00:00 2023-10-03T14:16:27+00:00 jim stasheff https://nforum.ncatlab.org/account/1409/ Is there an established meaning of a crossed principal bundle?related to (G,\alpha, |Gamma) structures? Is there an established meaning of a crossed principal bundle?
related to (G,\alpha, |Gamma) structures? ]]>
Urs comments on "equivariant bundle" (100549) https://nforum.ncatlab.org/discussion/11825/?Focus=100549#Comment_100549 2022-07-03T21:16:31+00:00 2023-10-03T14:16:27+00:00 Urs https://nforum.ncatlab.org/account/4/ Thanks again. This and some other things fixed now in the pdf here.

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

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David_Corfield comments on "equivariant bundle" (100527) https://nforum.ncatlab.org/discussion/11825/?Focus=100527#Comment_100527 2022-07-03T15:30:29+00:00 2023-10-03T14:16:27+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ Typo in the book #79 Borel modal structure

Typo in the book #79

Borel modal structure

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Urs comments on "equivariant bundle" (97833) https://nforum.ncatlab.org/discussion/11825/?Focus=97833#Comment_97833 2022-04-12T06:32:56+00:00 2023-10-03T14:16:27+00:00 Urs https://nforum.ncatlab.org/account/4/ 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.)

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.)

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Dmitri Pavlov comments on "equivariant bundle" (97831) https://nforum.ncatlab.org/discussion/11825/?Focus=97831#Comment_97831 2022-04-12T01:41:14+00:00 2023-10-03T14:16:27+00:00 Dmitri Pavlov https://nforum.ncatlab.org/account/356/ 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 ...

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].

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Urs comments on "equivariant bundle" (97522) https://nforum.ncatlab.org/discussion/11825/?Focus=97522#Comment_97522 2022-02-08T08:44:21+00:00 2023-10-03T14:16:27+00:00 Urs https://nforum.ncatlab.org/account/4/ 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.)

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.)

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Urs comments on "equivariant bundle" (97265) https://nforum.ncatlab.org/discussion/11825/?Focus=97265#Comment_97265 2022-01-03T12:33:33+00:00 2023-10-03T14:16:27+00:00 Urs https://nforum.ncatlab.org/account/4/ Thanks! These typos have been fixed now (here), together with a couple more in their vicinity.

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

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David_Corfield comments on "equivariant bundle" (97264) https://nforum.ncatlab.org/discussion/11825/?Focus=97264#Comment_97264 2022-01-03T11:31:28+00:00 2023-10-03T14:16:27+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ Def. 4.2.10 (i), that should be (Bp) &Xscr;,&Ascr; &ast;(B p)^{\ast}_{\mathcal{X}, \mathcal{A}} In (3.9.7) Chtid DscPntDsc Pnt lacks a &SmallCircle;\circ and you have ...

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.

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Urs comments on "equivariant bundle" (97168) https://nforum.ncatlab.org/discussion/11825/?Focus=97168#Comment_97168 2021-12-28T08:09:10+00:00 2023-10-03T14:16:27+00:00 Urs https://nforum.ncatlab.org/account/4/ I have taken the liberty of adding the pointer to Hisham Sati, Urs Schreiber, Equivariant principal &infin;\infty-bundles (arXiv:2112.13654) and referencing this to the corresponding bits of ...

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.

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Urs comments on "equivariant bundle" (97054) https://nforum.ncatlab.org/discussion/11825/?Focus=97054#Comment_97054 2021-12-20T12:41:46+00:00 2023-10-03T14:16:27+00:00 Urs https://nforum.ncatlab.org/account/4/ Thanks. These typos fixed now and an example on equivariant bundle gerbes added (currently Ex. 4.1.25).

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

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David_Corfield comments on "equivariant bundle" (97050) https://nforum.ncatlab.org/discussion/11825/?Focus=97050#Comment_97050 2021-12-20T09:12:18+00:00 2023-10-03T14:16:27+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ some quick typos combinatorial model categorie (in blue on p. 85) (3.1) YY rather than AA. (3.97) Dscid and Chtid (in the 211219f version).

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).

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DavidRoberts comments on "equivariant bundle" (97044) https://nforum.ncatlab.org/discussion/11825/?Focus=97044#Comment_97044 2021-12-20T00:12:07+00:00 2023-10-03T14:16:27+00:00 DavidRoberts https://nforum.ncatlab.org/account/42/ 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 GString_G-equivariant ...

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 $G$.

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DavidRoberts comments on "equivariant bundle" (97043) https://nforum.ncatlab.org/discussion/11825/?Focus=97043#Comment_97043 2021-12-20T00:01:32+00:00 2023-10-03T14:16:27+00:00 DavidRoberts https://nforum.ncatlab.org/account/42/ 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 ...

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…

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Urs comments on "equivariant bundle" (97030) https://nforum.ncatlab.org/discussion/11825/?Focus=97030#Comment_97030 2021-12-19T12:04:52+00:00 2023-10-03T14:16:27+00:00 Urs https://nforum.ncatlab.org/account/4/ With Hisham we have been writing – and are now finalizing – a book, titled: Equivariant principal &infin;\infty-bundles which means to solve the classification problem for equivariant ...

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.

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Urs comments on "equivariant bundle" (97016) https://nforum.ncatlab.org/discussion/11825/?Focus=97016#Comment_97016 2021-12-18T12:50:52+00:00 2023-10-03T14:16:27+00:00 Urs https://nforum.ncatlab.org/account/4/ added pointer to: Charles Rezk, Classifying spaces for 1-truncated compact Lie groups, Algebr. Geom. Topol. 18 (2018) 525-546 (arXiv:1608.02999) diff, v71, current

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Urs comments on "equivariant bundle" (96376) https://nforum.ncatlab.org/discussion/11825/?Focus=96376#Comment_96376 2021-10-30T11:26:46+00:00 2023-10-03T14:16:28+00:00 Urs https://nforum.ncatlab.org/account/4/ 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 ...

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 .

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Urs comments on "equivariant bundle" (95179) https://nforum.ncatlab.org/discussion/11825/?Focus=95179#Comment_95179 2021-09-16T04:12:58+00:00 2023-10-03T14:16:28+00:00 Urs https://nforum.ncatlab.org/account/4/ 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 ...

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Urs comments on "equivariant bundle" (94957) https://nforum.ncatlab.org/discussion/11825/?Focus=94957#Comment_94957 2021-09-03T11:25:53+00:00 2023-10-03T14:16:28+00:00 Urs https://nforum.ncatlab.org/account/4/ added pointer to: Charles Rezk, Sec. 2.3 in: Global Homotopy Theory and Cohesion, 2014 (pdf, Rezk14.pdf:file) diff, v65, current

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Urs comments on "equivariant bundle" (92971) https://nforum.ncatlab.org/discussion/11825/?Focus=92971#Comment_92971 2021-06-04T10:45:39+00:00 2023-10-03T14:16:28+00:00 Urs https://nforum.ncatlab.org/account/4/ Oh, I think I finally see the abstract story here: For &Gamma;&parsl;G&Element;Groups((SingularSmoothGroupoids &infin;) /BG)\Gamma \!\sslash\! G \;\in\; Groups\big( ...

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.

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Urs comments on "equivariant bundle" (92936) https://nforum.ncatlab.org/discussion/11825/?Focus=92936#Comment_92936 2021-06-03T14:46:35+00:00 2023-10-03T14:16:28+00:00 Urs https://nforum.ncatlab.org/account/4/ added a brief remark (here) that the Murayama-Shimakawa equivariant classifying space &Bscr;&Gamma;\mathcal{B}\Gamma has as HH-fixed points the HH-homotopy fixed points of B&Gamma;B ...

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 $B \Gamma$.

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Urs comments on "equivariant bundle" (91990) https://nforum.ncatlab.org/discussion/11825/?Focus=91990#Comment_91990 2021-04-30T15:34:44+00:00 2023-10-03T14:16:28+00:00 Urs https://nforum.ncatlab.org/account/4/ Hm, on the other hand the construction for &Gamma;=PU(&Hscr;)\Gamma = PU(\mathcal{H}) in Noe Barcenas, Jesus Espinoza, Michael Joachim, Bernardo Uribe, Universal twist in Equivariant ...

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.

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Urs comments on "equivariant bundle" (91925) https://nforum.ncatlab.org/discussion/11825/?Focus=91925#Comment_91925 2021-04-28T14:06:26+00:00 2023-10-03T14:16:28+00:00 Urs https://nforum.ncatlab.org/account/4/ &Gamma;\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 ...

$\Gamma$ 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 $PU(\mathcal{H})$.

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