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Mathlib.CategoryTheory.Sites.SheafHom

Internal hom of sheaves

In this file, given two sheaves F and G on a site (C, J) with values in a category A, we define a sheaf of types sheafHom F G which sends X : C to the type of morphisms between the restrictions of F and G to the categories Over X.

We first define presheafHom F G when F and G are presheaves Cᵒᵖ ⥤ A and show that it is a sheaf when G is a sheaf.

TODO:

Given two presheaves F and G on a category C with values in a category A, this presheafHom F G is the presheaf of types which sends an object X : C to the type of morphisms between the "restrictions" of F and G to the category Over X.

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    Equational lemma for the presheaf structure on presheafHom. It is advisable to use this lemma rather than dsimp [presheafHom] which may result in the need to prove equalities of objects in an Over category.

    The sections of the presheaf presheafHom F G identify to morphisms F ⟶ G.

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      noncomputable def CategoryTheory.PresheafHom.IsSheafFor.app {C : Type u} [CategoryTheory.Category.{v, u} C] {A : Type u'} [CategoryTheory.Category.{v', u'} A] {F : CategoryTheory.Functor Cᵒᵖ A} {G : CategoryTheory.Functor Cᵒᵖ A} {X : C} {S : CategoryTheory.Sieve X} (hG : Y : C⦄ → (f : Y X) → CategoryTheory.Limits.IsLimit (G.mapCone (CategoryTheory.Sieve.pullback f S).arrows.cocone.op)) (x : CategoryTheory.Presieve.FamilyOfElements (CategoryTheory.presheafHom F G) S.arrows) {Y : C} (hx : x.Compatible) (g : Y X) :
      F.obj (Opposite.op Y) G.obj (Opposite.op Y)

      Auxiliary definition for presheafHom_isSheafFor.

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        The underlying presheaf of sheafHom F G. It is isomorphic to presheafHom F.1 G.1 (see sheafHom'Iso), but has better definitional properties.

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          The canonical isomorphism sheafHom' F G ≅ presheafHom F.1 G.1.

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            Given two sheaves F and G on a site (C, J) with values in a category A, this sheafHom F G is the sheaf of types which sends an object X : C to the type of morphisms between the "restrictions" of F and G to the category Over X.

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              The sections of the sheaf sheafHom F G identify to morphisms F ⟶ G.

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