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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 12th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexstarting something, to go with and rhyme on _[[equivariant principal bundle]]_. Not done yet, but need to save. <a href="https://ncatlab.org/nlab/revision/equivariant+group/1">v1</a>, <a href="https://ncatlab.org/nlab/show/equivariant+group">current</a>

   starting something, to go with and rhyme on equivariant principal bundle. Not done yet, but need to save.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 13th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexNow some content in place, spelling out how * groups internal to $G$-spaces are equivalently semidirect products with $G$; * group actions internal to $G$-spaces are equivalently actions of these semidirect product groups. All elementary and essentially trivial, but spelled out, hereby, nonetheless. <a href="https://ncatlab.org/nlab/revision/diff/equivariant+group/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/equivariant+group/8">v8</a>, <a href="https://ncatlab.org/nlab/show/equivariant+group">current</a>

   Now some content in place, spelling out how

   • groups internal to $G$-spaces are equivalently semidirect products with $G$;

   • group actions internal to $G$-spaces are equivalently actions of these semidirect product groups.

   All elementary and essentially trivial, but spelled out, hereby, nonetheless.

   diff, v8, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMar 14th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexsome last polishing. Now I have had enough of this. <a href="https://ncatlab.org/nlab/revision/diff/equivariant+group/10">diff</a>, <a href="https://ncatlab.org/nlab/revision/equivariant+group/10">v10</a>, <a href="https://ncatlab.org/nlab/show/equivariant+group">current</a>

   some last polishing. Now I have had enough of this.

   diff, v10, current

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeMar 14th 2021
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexStrange that internalisation hasn’t been rolled out here before. Presumably it’s what follows from a HoTT rendition in equivariant context.

   Strange that internalisation hasn’t been rolled out here before. Presumably it’s what follows from a HoTT rendition in equivariant context.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMar 14th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexThat's certainly what I am really after behind the scenes: Question: What's a twist in twisted equivariant (differential) $A$-cohomology? Answer: it's an $A$-fiber $\infty$-bundle internal to a singular-cohesive $\infty$-topos. Of course. Question: What kind of topological bundles with bells-and-whistles present these under passage to shape? Answer: Principal bundles internal to $G$-spaces -- hence tomDieck69-bundles. But yeah, it is weird that it takes an undergrad-level entry _[[equivariant group]]_ to fill a gap in the literature. Some 20th century maths has fallen into the 21st century here and needs to be cleaned up now ;-)

   That’s certainly what I am really after behind the scenes:

   Question: What’s a twist in twisted equivariant (differential) $A$-cohomology?

   Answer: it’s an $A$-fiber $\infty$-bundle internal to a singular-cohesive $\infty$-topos. Of course.

   Question: What kind of topological bundles with bells-and-whistles present these under passage to shape?

   Answer: Principal bundles internal to $G$-spaces – hence tomDieck69-bundles.

   But yeah, it is weird that it takes an undergrad-level entry equivariant group to fill a gap in the literature. Some 20th century maths has fallen into the 21st century here and needs to be cleaned up now ;-)

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeMar 15th 2021
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexPresumably [[equivariant connection]] could be given a similar internalised treatment.

   Presumably equivariant connection could be given a similar internalised treatment.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeMar 15th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexI won't be editing that entry for the moment. But, yes, as in #5, differential cohomology in a singular-cohesive $\infty$-topos gives equivariant $\infty$-connections on equivariant $\infty$-bundles. (With the usual caveats: Abelian connections come out on the nose from a Hopkins-Singer style homotopy pullback along a character map, while non-abelian connections come out subject to more identifications, unless one intervenes by hand and uses more properties of a concrete ambient model.)

   I won’t be editing that entry for the moment. But, yes, as in #5, differential cohomology in a singular-cohesive $\infty$-topos gives equivariant $\infty$-connections on equivariant $\infty$-bundles.

   (With the usual caveats: Abelian connections come out on the nose from a Hopkins-Singer style homotopy pullback along a character map, while non-abelian connections come out subject to more identifications, unless one intervenes by hand and uses more properties of a concrete ambient model.)

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJun 7th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexNow finally turning to the abstract $\infty$-topos theoretic perspective on these matters: Let * $\mathbf{H}$ be an $\infty$-topos, * $G \in Groups(\mathbf{H})$ then Prop. 2.102 on [p. 35](https://ncatlab.org/schreiber/files/orbi210607.pdf#page=35) of our *[[schreiber:Proper Orbifold Cohomology|Proper Orbifold Cohomology]]* shows that its homotopy quotient by its group-automorphism group is itself, canonically, a group object in the slice $$ G \!\sslash\! Aut_{Grp}(G) \;\in\; Groups \big( \mathbf{H}_{/\mathrm{Aut}_{Grp}(G)} \big) \,. $$ I expect the converse is true: Conjecture. *For $K$ a group in $\mathbf{H}$, group objects in $\mathbf{H}_{/\mathbf{B}K}$ are equivalent to group objects in $\mathbf{H}$ that are equipped with actions by K via group automorphisms*. But I don't have a proof of this converse statement yet.

   Now finally turning to the abstract $\infty$-topos theoretic perspective on these matters:

   Let

   • $\mathbf{H}$ be an $\infty$-topos,

   • $G \in Groups(\mathbf{H})$

   then Prop. 2.102 on p. 35 of our Proper Orbifold Cohomology shows that its homotopy quotient by its group-automorphism group is itself, canonically, a group object in the slice

   $$G \!\sslash\! Aut_{Grp}(G) \;\in\; Groups \big( \mathbf{H}_{/\mathrm{Aut}_{Grp}(G)} \big) \,.$$

   I expect the converse is true:

   Conjecture. For $K$ a group in $\mathbf{H}$, group objects in $\mathbf{H}_{/\mathbf{B}K}$ are equivalent to group objects in $\mathbf{H}$ that are equipped with actions by K via group automorphisms.

   But I don’t have a proof of this converse statement yet.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeJun 7th 2021
   • (edited Jun 7th 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItex[ ... ]

   [ … ]

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJun 8th 2021
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have put my thoughts so far into the [[Sandbox]] (involves `tikz`, so doesn't render here).

    I have put my thoughts so far into the Sandbox (involves tikz, so doesn’t render here).

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 28th 2021
    • PermaLink
    Author: Urs
    Format: MarkdownItexIf a $G$-equivariant group $\Gamma$ acts on a $G$-space $X$ (i.e. a group action internal to $G$-spaces), then the action does not generally pass to naive quotients -- but: The fixed locus $\Gamma^G$ does canonically act on the naive quotient $X/G$ (essentially since reflexive coequalizers commute with products). Elementary as this is, I seem to have run into amusing example for this situation; now I am wondering if this should be thought of as an example of something that is relevant more generally.

    If a $G$-equivariant group $\Gamma$ acts on a $G$-space $X$ (i.e. a group action internal to $G$-spaces), then the action does not generally pass to naive quotients – but:

    The fixed locus $\Gamma^G$ does canonically act on the naive quotient $X/G$ (essentially since reflexive coequalizers commute with products).

    Elementary as this is, I seem to have run into amusing example for this situation; now I am wondering if this should be thought of as an example of something that is relevant more generally.

12. • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 28th 2021
    • (edited Sep 28th 2021)
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexHow does this look in HoTT in terms of contexts $B G$, $B \Gamma$, dependent product for fixed point, truncation of dependent sum for naive quotient, etc.?

    How does this look in HoTT in terms of contexts $B G$, $B \Gamma$, dependent product for fixed point, truncation of dependent sum for naive quotient, etc.?

13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeSep 28th 2021
    • PermaLink
    Author: Urs
    Format: MarkdownItexYeah, that's the thing, it looks a little mixed: In [[schreiber:Proper Orbifold Cohomology|singular-cohesive HTT]] it's the combination of 1. the geometric fixed locus operation $Maps\big( \prec \mathbf{B}G,\, \prec(-) \big)_{\prec \mathbf{B}G}$ on the group, 1. the naive quotient modality $\lt (-)$ on the object it acts on. It's not something I would have thought of on abstract grounds, and I haven't even tried to think about in which abstract generality this works. But now that I am seeing this one example I feel like there may be more to this (namely I am looking at the canonical $\mathbb{Z}_2$-equivariant action of $SU(3)$ on $\mathbb{C}P^2$. Applying this construction and using the [[AKM-theorem]] makes this reduce to an $SO(3)$-action on $S^4$. Still need to check if it's the canonical $SO(3)$-action at that, but I guess it can't be anything else...)

    Yeah, that’s the thing, it looks a little mixed:

    In singular-cohesive HTT it’s the combination of

    1. the geometric fixed locus operation $Maps\big( \prec \mathbf{B}G,\, \prec(-) \big)_{\prec \mathbf{B}G}$ on the group,

    2. the naive quotient modality $\lt (-)$ on the object it acts on.

    It’s not something I would have thought of on abstract grounds, and I haven’t even tried to think about in which abstract generality this works.

    But now that I am seeing this one example I feel like there may be more to this (namely I am looking at the canonical $\mathbb{Z}_2$-equivariant action of $SU(3)$ on $\mathbb{C}P^2$. Applying this construction and using the AKM-theorem makes this reduce to an $SO(3)$-action on $S^4$. Still need to check if it’s the canonical $SO(3)$-action at that, but I guess it can’t be anything else…)