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Mathlib.CategoryTheory.Whiskering

Whiskering #

Given a functor F : C ⥤ D and functors G H : D ⥤ E and a natural transformation α : G ⟶ H, we can construct a new natural transformation F ⋙ G ⟶ F ⋙ H, called whiskerLeft F α. This is the same as the horizontal composition of 𝟙 F with α.

This operation is functorial in F, and we package this as whiskeringLeft. Here (whiskeringLeft.obj F).obj G is F ⋙ G, and (whiskeringLeft.obj F).map α is whiskerLeft F α. (That is, we might have alternatively named this as the "left composition functor".)

We also provide analogues for composition on the right, and for these operations on isomorphisms.

At the end of the file, we provide the left and right unitors, and the associator, for functor composition. (In fact functor composition is definitionally associative, but very often relying on this causes extremely slow elaboration, so it is better to insert it explicitly.) We also show these natural isomorphisms satisfy the triangle and pentagon identities.

@[elab_without_expected_type]

If α : G ⟶ H then whiskerLeft F α : (F ⋙ G) ⟶ (F ⋙ H) has components α.app (F.obj X).

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    @[elab_without_expected_type]

    If α : G ⟶ H then whisker_right α F : (G ⋙ F) ⟶ (G ⋙ F) has components F.map (α.app X).

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      @[simp]
      theorem CategoryTheory.whiskeringLeft_map_app_app (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] (E : Type u₃) [CategoryTheory.Category.{v₃, u₃} E] :
      ∀ {X Y : CategoryTheory.Functor C D} (τ : X Y) (H : CategoryTheory.Functor D E) (c : C), (((CategoryTheory.whiskeringLeft C D E).map τ).app H).app c = H.map (τ.app c)

      Left-composition gives a functor (C ⥤ D) ⥤ ((D ⥤ E) ⥤ (C ⥤ E)).

      (whiskeringLeft.obj F).obj G is F ⋙ G, and (whiskeringLeft.obj F).map α is whiskerLeft F α.

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        @[simp]

        Right-composition gives a functor (D ⥤ E) ⥤ ((C ⥤ D) ⥤ (C ⥤ E)).

        (whiskeringRight.obj H).obj F is F ⋙ H, and (whiskeringRight.obj H).map α is whiskerRight α H.

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          If α : G ≅ H is a natural isomorphism then iso_whisker_left F α : (F ⋙ G) ≅ (F ⋙ H) has components α.app (F.obj X).

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            If α : G ≅ H then iso_whisker_right α F : (G ⋙ F) ≅ (H ⋙ F) has components F.map_iso (α.app X).

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              The left unitor, a natural isomorphism ((𝟭 _) ⋙ F) ≅ F.

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                The right unitor, a natural isomorphism (F ⋙ (𝟭 B)) ≅ F.

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                  The associator for functors, a natural isomorphism ((F ⋙ G) ⋙ H) ≅ (F ⋙ (G ⋙ H)).

                  (In fact, iso.refl _ will work here, but it tends to make Lean slow later, and it's usually best to insert explicit associators.)

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