functors between categories of sheaves

Show that the pushforward of a sheaf is a sheaf, and define the pushforward functor from the category of C-valued sheaves on X to that of sheaves on Y, given a continuous map between topological spaces X and Y.

Main definitions

• TopCat.Sheaf.pushforward: The pushforward functor between sheaf categories over topological spaces.
• TopCat.Sheaf.pullback: The pullback functor between sheaf categories over topological spaces.
• TopCat.Sheaf.pullbackPushforwardAdjunction: The adjunction between pullback and pushforward for sheaves on topological spaces.

theorem TopCat.Sheaf.pushforward_sheaf_of_sheaf {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : TopCat} (f : X ⟶ Y) {F : Presheaf C X} (h : F.IsSheaf) : ((Presheaf.pushforward C f).obj F).IsSheaf

The pushforward of a sheaf (by a continuous map) is a sheaf.

def TopCat.Sheaf.pushforward (C : Type u) [CategoryTheory.Category.{v, u} C] {X Y : TopCat} (f : X ⟶ Y) : CategoryTheory.Functor (Sheaf C X) (Sheaf C Y)

The pushforward functor.

Equations

• TopCat.Sheaf.pushforward C f = (TopologicalSpace.Opens.map f).sheafPushforwardContinuous C (Opens.grothendieckTopology ↑Y) (Opens.grothendieckTopology ↑X)

theorem TopCat.Sheaf.pushforward_forget (C : Type u) [CategoryTheory.Category.{v, u} C] {X Y : TopCat} (f : X ⟶ Y) : (pushforward C f).comp (forget C Y) = (forget C X).comp (Presheaf.pushforward C f)

def TopCat.Sheaf.pushforwardForgetIso (C : Type u) [CategoryTheory.Category.{v, u} C] {X Y : TopCat} (f : X ⟶ Y) : (pushforward C f).comp (forget C Y) ≅ (forget C X).comp (Presheaf.pushforward C f)

Pushforward of sheaves is isomorphic (actually definitionally equal) to pushforward of presheaves.

Equations

• TopCat.Sheaf.pushforwardForgetIso C f = CategoryTheory.Iso.refl ((TopCat.Sheaf.pushforward C f).comp (TopCat.Sheaf.forget C Y))

@[simp]

theorem TopCat.Sheaf.pushforward_obj_val {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : TopCat} (f : X ⟶ Y) (F : Sheaf C X) : ((pushforward C f).obj F).obj = (Presheaf.pushforward C f).obj F.obj

@[simp]

theorem TopCat.Sheaf.pushforward_map {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : TopCat} (f : X ⟶ Y) {F F' : Sheaf C X} (α : F ⟶ F') : ((pushforward C f).map α).hom = (Presheaf.pushforward C f).map α.hom

noncomputable def TopCat.Sheaf.pullback {X Y : TopCat} (A : Type u_1) [CategoryTheory.Category.{w, u_1} A] {FA : A → A → Type u_2} {CA : A → Type w} [(X Y : A) → FunLike (FA X Y) (CA X) (CA Y)] [CategoryTheory.ConcreteCategory A FA] [CategoryTheory.Limits.HasColimits A] [CategoryTheory.Limits.HasLimits A] [CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget A)] [CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.forget A)] [(CategoryTheory.forget A).ReflectsIsomorphisms] (f : X ⟶ Y) : CategoryTheory.Functor (Sheaf A Y) (Sheaf A X)

The pullback functor.

Equations

• TopCat.Sheaf.pullback A f = (TopologicalSpace.Opens.map f).sheafPullback A (Opens.grothendieckTopology ↑Y) (Opens.grothendieckTopology ↑X)

noncomputable def TopCat.Sheaf.pullbackIso {X Y : TopCat} (A : Type u_1) [CategoryTheory.Category.{w, u_1} A] {FA : A → A → Type u_2} {CA : A → Type w} [(X Y : A) → FunLike (FA X Y) (CA X) (CA Y)] [CategoryTheory.ConcreteCategory A FA] [CategoryTheory.Limits.HasColimits A] [CategoryTheory.Limits.HasLimits A] [CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget A)] [CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.forget A)] [(CategoryTheory.forget A).ReflectsIsomorphisms] (f : X ⟶ Y) : pullback A f ≅ (forget A Y).comp ((Presheaf.pullback A f).comp (CategoryTheory.presheafToSheaf (Opens.grothendieckTopology ↑X) A))

The pullback of a sheaf is isomorphic (actually definitionally equal) to the sheafification of the pullback as a presheaf.

Equations

• TopCat.Sheaf.pullbackIso A f = CategoryTheory.Functor.sheafPullbackConstruction.sheafPullbackIso (TopologicalSpace.Opens.map f) A (Opens.grothendieckTopology ↑Y) (Opens.grothendieckTopology ↑X)

noncomputable def TopCat.Sheaf.pullbackPushforwardAdjunction {X Y : TopCat} (A : Type u_1) [CategoryTheory.Category.{w, u_1} A] {FA : A → A → Type u_2} {CA : A → Type w} [(X Y : A) → FunLike (FA X Y) (CA X) (CA Y)] [CategoryTheory.ConcreteCategory A FA] [CategoryTheory.Limits.HasColimits A] [CategoryTheory.Limits.HasLimits A] [CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget A)] [CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.forget A)] [(CategoryTheory.forget A).ReflectsIsomorphisms] (f : X ⟶ Y) : pullback A f ⊣ pushforward A f

The adjunction between pullback and pushforward for sheaves on topological spaces.

Equations

• TopCat.Sheaf.pullbackPushforwardAdjunction A f = (TopologicalSpace.Opens.map f).sheafAdjunctionContinuous A (Opens.grothendieckTopology ↑Y) (Opens.grothendieckTopology ↑X)

instance TopCat.Sheaf.instIsLeftAdjointPullback {X Y : TopCat} (f : X ⟶ Y) (A : Type u_1) [CategoryTheory.Category.{w, u_1} A] {FA : A → A → Type u_2} {CA : A → Type w} [(X Y : A) → FunLike (FA X Y) (CA X) (CA Y)] [CategoryTheory.ConcreteCategory A FA] [CategoryTheory.Limits.HasColimits A] [CategoryTheory.Limits.HasLimits A] [CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget A)] [CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.forget A)] [(CategoryTheory.forget A).ReflectsIsomorphisms] : (pullback A f).IsLeftAdjoint

instance TopCat.Sheaf.instIsRightAdjointPushforward {X Y : TopCat} (f : X ⟶ Y) (A : Type u_1) [CategoryTheory.Category.{w, u_1} A] {FA : A → A → Type u_2} {CA : A → Type w} [(X Y : A) → FunLike (FA X Y) (CA X) (CA Y)] [CategoryTheory.ConcreteCategory A FA] [CategoryTheory.Limits.HasColimits A] [CategoryTheory.Limits.HasLimits A] [CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget A)] [CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.forget A)] [(CategoryTheory.forget A).ReflectsIsomorphisms] : (pushforward A f).IsRightAdjoint

def Topology.IsOpenEmbedding.sheafPullback {X Y : TopCat} (A : Type u_1) [CategoryTheory.Category.{w, u_1} A] {f : X ⟶ Y} (hf : IsOpenEmbedding ⇑(CategoryTheory.ConcreteCategory.hom f)) : CategoryTheory.Functor (TopCat.Sheaf A Y) (TopCat.Sheaf A X)

The "naive" sheaf pullback by an open embedding f: on the underlying presheaf, this is just composition by the functor IsOpenMap.functor f (sending an open U to f '' U).

Equations

• Topology.IsOpenEmbedding.sheafPullback A hf = hf.functor.sheafPushforwardContinuous A (Opens.grothendieckTopology ↑X) (Opens.grothendieckTopology ↑Y)

noncomputable def Topology.IsOpenEmbedding.sheafPullbackIso {X Y : TopCat} (A : Type u_1) [CategoryTheory.Category.{w, u_1} A] {f : X ⟶ Y} (hf : IsOpenEmbedding ⇑(CategoryTheory.ConcreteCategory.hom f)) {FA : A → A → Type u_2} {CA : A → Type w} [(X Y : A) → FunLike (FA X Y) (CA X) (CA Y)] [CategoryTheory.ConcreteCategory A FA] [CategoryTheory.Limits.HasColimits A] [CategoryTheory.Limits.HasLimits A] [CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget A)] [CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.forget A)] [(CategoryTheory.forget A).ReflectsIsomorphisms] : TopCat.Sheaf.pullback A f ≅ sheafPullback A hf

The pullback of a sheaf by an open embedding f is isomorphic to its naive pullback IsOpenEmbedding.sheafPullback, i.e. to the composition by the functor IsOpenMap.functor f. Also, this is an isomorphism of functors.

Equations

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