Processing math: 100%
Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory object of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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 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

    • 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) A-cohomology?

    Answer: it’s an A-fiber -bundle internal to a singular-cohesive -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 -topos gives equivariant -connections on equivariant -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 -topos theoretic perspective on these matters:

    Let

    • H be an -topos,

    • GGroups(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

    GAutGrp(G)Groups(H/AutGrp(G)).

    I expect the converse is true:

    Conjecture. For K a group in H, group objects in H/BK are equivalent to group objects in 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).

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 28th 2021

    If a G-equivariant group Γ 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 ΓG 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.

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 28th 2021
    • (edited Sep 28th 2021)

    How does this look in HoTT in terms of contexts BG, BΓ, dependent product for fixed point, truncation of dependent sum for naive quotient, etc.?

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeSep 28th 2021

    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(BG,())BG on the group,

    2. the naive quotient modality <() 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 2-equivariant action of SU(3) on P2. Applying this construction and using the AKM-theorem makes this reduce to an SO(3)-action on S4. Still need to check if it’s the canonical SO(3)-action at that, but I guess it can’t be anything else…)