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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMar 12th 2021

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

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMar 13th 2021

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.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMar 14th 2021

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

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeMar 14th 2021

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

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMar 14th 2021

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 ;-)

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeMar 15th 2021

Presumably equivariant connection could be given a similar internalised treatment.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeMar 15th 2021

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.)

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJun 7th 2021

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.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeJun 7th 2021
• (edited Jun 7th 2021)

[ … ]

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeJun 8th 2021

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