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Mathlib.CategoryTheory.Sites.GlobalSections

Global sections of sheaves #

In this file we define a global sections functor Sheaf.Γ : Sheaf J A ⥤ A and show that it is isomorphic to several other constructions when they exist, most notably evaluation of sheaves on a terminal object and Functor.sectionsFunctor.

Main definitions / results #

TODO #

@[reducible, inline]

Typeclass stating that the constant sheaf functor has a right adjoint. This right adjoint will then be called the global sections functor and written Sheaf.Γ.

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    We define the global sections functor as the right-adjoint of the constant sheaf functor whenever it exists.

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      The constant sheaf functor is by definition left-adjoint to the global sections functor.

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        Sites with a terminal object admit a global sections functor.

        On sites with a terminal object, the global sections functor is isomorphic to the functor of sections on that object.

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          Every site C admits a global sections functor for A-valued sheaves when A has limits of shape Cᵒᵖ.

          Global sections of sheaves are naturally isomorphic to the limits of the underlying presheaves. Note that while HasLimitsOfShape Cᵒᵖ A is needed here to talk about lim as a functor, global sections are still limits without it - see Sheaf.isLimitConeΓ.

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            noncomputable def CategoryTheory.Sheaf.ΓHomEquiv {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {A : Type u₂} [Category.{v₂, u₂} A] [HasWeakSheafify J A] [HasGlobalSectionsFunctor J A] {X : A} {F : Sheaf J A} :
            ((Functor.const Cᵒᵖ).obj X F.val) (X (Γ J A).obj F)

            Natural transformations from a constant presheaf into a sheaf correspond to morphisms to its global sections.

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              Naturality lemma for ΓHomEquiv analogous to Adjunction.homEquiv_naturality_left.

              Naturality lemma for ΓHomEquiv analogous to Adjunction.homEquiv_naturality_left_symm.

              Naturality lemma for ΓHomEquiv analogous to Adjunction.homEquiv_naturality_right.

              Naturality lemma for ΓHomEquiv analogous to Adjunction.homEquiv_naturality_right_symm.

              The cone over a given sheaf whose cone point are the global sections and whose components are the restriction maps.

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                The global sections cone Sheaf.coneΓ is limiting - that is, global sections are limits even when not all limits of shape Cᵒᵖ exist in A.

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                  noncomputable def CategoryTheory.Sheaf.ΓRes {C : Type u} [Category.{v, u} C] {J : GrothendieckTopology C} {A : Type u₂} [Category.{v₂, u₂} A] [HasWeakSheafify J A] [HasGlobalSectionsFunctor J A] (F : Sheaf J A) (U : Cᵒᵖ) :
                  (Γ J A).obj F F.val.obj U

                  The restriction map from global sections of F to sections on U.

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                    The natural transformation from the global sections functor to the sections functor on any object U.

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                      Global sections of a sheaf of types correspond to sections of the underlying presheaf.

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                        For sheaves of types, the global sections functor is isomorphic to the sections functor on presheaves.

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                          Global sections of a sheaf of types F correspond to morphisms from a terminal sheaf to F. We use the constant sheaf on a singleton type as a specific choice of terminal sheaf here.

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                            noncomputable def CategoryTheory.Sheaf.ΓNatIsoCoyoneda {C : Type u} [Category.{v, u} C] (J : GrothendieckTopology C) (X : Type (max u v)) [Unique X] :
                            Γ J (Type (max u v)) coyoneda.obj (Opposite.op ((constantSheaf J (Type (max u v))).obj X))

                            For sheaves of types, the global sections functor is isomorphic to the covariant hom functor of the terminal sheaf.

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