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 comma complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite 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 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).
  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

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

    Anonymous

    diff, v21, current

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

    Mark John Hopkins

    diff, v24, current

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

    Mark John Hopkins

    diff, v24, current

  5. Fixup

    Mark John Hopkins

    diff, v24, current

  6. 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