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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.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 13th 2021

    Now some content in place, spelling out how

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

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

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

    diff, v8, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 14th 2021

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

    diff, v10, current

    • 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) AA-cohomology?

    Answer: it’s an AA-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 GG-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

    • H\mathbf{H} be an \infty-topos,

    • GGroups(H)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

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

    I expect the converse is true:

    Conjecture. For KK a group in H\mathbf{H}, group objects in H /BK\mathbf{H}_{/\mathbf{B}K} are equivalent to group objects in H\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).

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