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Mathlib.Algebra.Category.ContinuousCohomology.Basic

Continuous cohomology #

We define continuous cohomology as the homology of homogeneous cochains.

Implementation details #

We define homogenous cochains as g-invariant continuous function in C(G, C(G,...,C(G, M))) instead of the usual C(Gⁿ, M) to allow more general topological groups other than locally compact ones. For this to work, we also work in Action (TopModuleCat R) G, where the G action on M is only continuous on M, and not necessarily continuous in both variables, because the G action on C(G, M) might not be continuous on both variables even if it is on M.

For the differential map, instead of a finite sum we use the inductive definition d₋₁ : M → C(G, M) := const : m ↦ g ↦ m and dₙ₊₁ : C(G, _) → C(G, C(G, _)) := const - C(G, dₙ) : f ↦ g ↦ f - dₙ (f (g)) See ContinuousCohomology.MultiInd.d.

Main definition #

TODO #

@[reducible, inline]

The G representation C(G, rep) given a representation rep. The G action is defined by g • f := x ↦ g • f (g⁻¹ * x).

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    theorem ContinuousCohomology.Iobj_ρ_apply (R : Type u_1) (G : Type u_2) [CommRing R] [Group G] [TopologicalSpace R] [TopologicalSpace G] [IsTopologicalGroup G] (rep : Action (TopModuleCat R) G) (g : G) (f : (Iobj rep).V.toModuleCat) (x : G) :
    ((TopModuleCat.Hom.hom ((Iobj rep).ρ g)) f) x = (TopModuleCat.Hom.hom (rep.ρ g)) (f (g⁻¹ * x))

    The functor taking a representation rep to the representation C(G, rep).

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      @[simp]
      @[simp]
      theorem ContinuousCohomology.I_obj_ρ_apply (R : Type u_1) (G : Type u_2) [CommRing R] [Group G] [TopologicalSpace R] [TopologicalSpace G] [IsTopologicalGroup G] (rep : Action (TopModuleCat R) G) (g : G) :
      ((I R G).obj rep).ρ g = TopModuleCat.ofHom { toFun := fun (f : C(G, rep.V.toModuleCat)) => (↑(TopModuleCat.Hom.hom (rep.ρ g))).comp (f.comp (Homeomorph.mulLeft g⁻¹)), map_add' := , map_smul' := , cont := }

      The constant function rep ⟶ C(G, rep) as a natural transformation.

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        The differential map in MultiInd.complex.

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          The complex of functors whose behaviour pointwise takes an R-linear G-representation M to the complex M → C(G, M) → ⋯ → C(G, C(G,...,C(G, M))) → ⋯ The G-invariant submodules of it is the homogeneous cochains (shifted by one).

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            The functor taking an R-linear G-representation to its G-invariant submodule.

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              homogeneousCochains R G is the functor taking an R-linear G-representation to the complex of homogeneous cochains.

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                continuousCohomology R G n is the functor taking an R-linear G-representation to its n-th continuous homology.

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                  The 0-homogeneous cochains are isomorphic to Xᴳ.

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                    H⁰_cont(G, X) ≅ Xᴳ.

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