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nLab > Latest Changes: semidirect product group

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  1. Grothendieck Construction

    Ammar Husain

    diff, v18, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJul 19th 2019
    • (edited Jul 19th 2019)

    Thanks for the addition. I have edited a little:

    Have added hyperlinks (just enclose technical keywords in double square brackets!).

    Have changed the notation for the delooping groupoid of GG from GG to 𝔹G\mathbb{B}G.

    Adjusted wording a little.

    Now it reads like so, but be invited to edit further:

    Writing 𝔹G\mathbb{B} G for the category with a single object *\ast and the group GG as its hom set (i.e. the delooping groupoid of GG), define a functor F:𝔹G→F \colon \mathbb{B}G \to Cat to send that single object to the delooping groupoid of Γ\Gamma, i.e. *↦𝔹Γ* \mapsto \mathbb{B}\Gamma and to send the morphisms G→Aut(Γ)G \to Aut(\Gamma) according to the given action of GG on Γ\Gamma.

    Then the delooping of the semidirect product group Γ⋊G\Gamma \rtimes G arises as the Grothendieck construction of this functor:

    𝔹(Γ⋊G)≃∫ 𝔹GF \mathbb{B}( \Gamma \rtimes G) \simeq \int_{\mathbb{B}G}F

    diff, v19, current

  3. This is taken from here: https://mathoverflow.net/a/96256 (with a little bit more detail)

    Anonymous

    diff, v21, current

  4. Add missing prime to \rho’ in section on the semidirect product as a left adjoint.

    Mark John Hopkins

    diff, v24, current

  5. Add the homomorphism f explicitly when defining the forgetful functor from Arr(Grp) to GrpActions.

    Mark John Hopkins

    diff, v24, current

  6. Fixup

    Mark John Hopkins

    diff, v24, current

  7. In the section on internal semidirect products the roles of (\Gamma) and (G) were backward. (\Gamma) should be given as the normal subgroup and (G) as the subgroup isomorphic to the quotient.

    Thomas Hunter

    diff, v26, current

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 30th 2023

    Oh, yes, sure enough – thanks!

    • CommentRowNumber9.
    • CommentAuthorT
    • CommentTimeDec 6th 2023

    In the section on internal semidirect products the roles of (\Gamma) and (G) were backward. (\Gamma) should be given as the normal subgroup and (G) as the subgroup isomorphic to the quotient.

    diff, v27, current

    • CommentRowNumber10.
    • CommentAuthorT
    • CommentTimeDec 6th 2023

    more bibliography formatting

    diff, v27, current

  11. Added a detail about how the semidirect product relates to the self-conjugation action - in this sense, it is the “free” or “initial” way to internalise a group action as conjugation.

    Pseudonium

    diff, v28, current

1 to 11 of 11

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