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starting something, to go with and rhyme on equivariant principal bundle. Not done yet, but need to save.
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.
Strange that internalisation hasn’t been rolled out here before. Presumably it’s what follows from a HoTT rendition in equivariant context.
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 ;-)
Presumably equivariant connection could be given a similar internalised treatment.
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.)
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.
[ … ]
I have put my thoughts so far into the Sandbox (involves tikz
, so doesn’t render here).
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.
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.?
Yeah, that’s the thing, it looks a little mixed:
In singular-cohesive HTT it’s the combination of
the geometric fixed locus operation $Maps\big( \prec \mathbf{B}G,\, \prec(-) \big)_{\prec \mathbf{B}G}$ on the group,
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…)
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