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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:𝔹GF \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 GAut(Γ)G \to Aut(\Gamma) according to the given action of GG on Γ\Gamma.

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

    𝔹(ΓG) 𝔹GF \mathbb{B}( \Gamma \rtimes G) \simeq \int_{\mathbb{B}G}F

    diff, v19, current

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