Documentation

Mathlib.CategoryTheory.Adjunction.Triple

Adjoint triples #

This file concerns adjoint triples F ⊣ G ⊣ H of functors F H : C ⥤ D, G : D ⥤ C. We first prove that F is fully faithful iff H is, and then prove results about the two special cases where G is fully faithful or F and H are.

Main results #

All results are about an adjoint triple F ⊣ G ⊣ H where adj₁ : F ⊣ G and adj₂ : G ⊣ H. We bundle the adjunctions in a structure Triple F G H.

structure CategoryTheory.Adjunction.Triple {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] (F : Functor C D) (G : Functor D C) (H : Functor C D) :
Type (max (max (max u_1 u_2) u_3) u_4)

Structure containing the two adjunctions of an adjoint triple F ⊣ G ⊣ H.

  • adj₁ : F G

    Adjunction F ⊣ G of the adjoint triple F ⊣ G ⊣ H.

  • adj₂ : G H

    Adjunction G ⊣ H of the adjoint triple F ⊣ G ⊣ H.

Instances For
    noncomputable def CategoryTheory.Adjunction.Triple.fullyFaithfulEquiv {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] {F : Functor C D} {G : Functor D C} {H : Functor C D} (t : Triple F G H) :

    Given an adjoint triple F ⊣ G ⊣ H, the left adjoint F is fully faithful if and only if the right adjoint H is fully faithful.

    Equations
    • One or more equations did not get rendered due to their size.
    Instances For
      def CategoryTheory.Adjunction.Triple.op {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] {F : Functor C D} {G : Functor D C} {H : Functor C D} (t : Triple F G H) :
      Triple H.op G.op F.op

      The adjoint triple H.op ⊣ G.op ⊣ F.op dual to an adjoint triple F ⊣ G ⊣ H.

      Equations
      Instances For
        @[simp]
        theorem CategoryTheory.Adjunction.Triple.op_adj₂ {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] {F : Functor C D} {G : Functor D C} {H : Functor C D} (t : Triple F G H) :
        @[simp]
        theorem CategoryTheory.Adjunction.Triple.op_adj₁ {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] {F : Functor C D} {G : Functor D C} {H : Functor C D} (t : Triple F G H) :
        noncomputable def CategoryTheory.Adjunction.Triple.rightToLeft {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] {F : Functor C D} {G : Functor D C} {H : Functor C D} (t : Triple F G H) [G.Full] [G.Faithful] :
        H F

        The natural transformation H ⟶ F that exists for every adjoint triple F ⊣ G ⊣ H where G is fully faithful, given here as the preimage of adj₂.counit ≫ adj₁.unit : H ⋙ G ⟶ F ⋙ G under whiskering with G.

        Equations
        Instances For
          @[simp]

          For an adjoint triple F ⊣ G ⊣ H where G is fully faithful, whiskering the natural transformation H ⟶ F with G yields the composition of the counit of the second adjunction with the unit of the first adjunction.

          @[simp]
          theorem CategoryTheory.Adjunction.Triple.map_rightToLeft_app {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] {F : Functor C D} {G : Functor D C} {H : Functor C D} (t : Triple F G H) [G.Full] [G.Faithful] (X : C) :

          For an adjoint triple F ⊣ G ⊣ H where G is fully faithful, the images of the components of the natural transformation H ⟶ F under G are the components of the composition of counit of the second adjunction with the unit of the first adjunction.

          The natural transformation H ⟶ F for an adjoint triple F ⊣ G ⊣ H with G fully faithful is also equal to the whiskered unit H ⟶ F ⋙ G ⋙ H of the first adjunction followed by the inverse of the whiskered unit F ⟶ F ⋙ G ⋙ H of the second.

          The natural transformation H ⟶ F for an adjoint triple F ⊣ G ⊣ H with G fully faithful is also equal to the inverse of the whiskered counit H ⋙ G ⋙ F ⟶ H of the first adjunction followed by the whiskered counit H ⋙ G ⋙ F ⟶ F of the second.

          @[simp]
          @[simp]
          theorem CategoryTheory.Adjunction.Triple.rightToLeft_app_adj₂_unit_app_assoc {C : Type u_1} {D : Type u_2} [Category.{u_4, u_1} C] [Category.{u_3, u_2} D] {F : Functor C D} {G : Functor D C} {H : Functor C D} (t : Triple F G H) [G.Full] [G.Faithful] (X : C) {Z : D} (h : H.obj (G.obj (F.obj X)) Z) :
          @[simp]

          For an adjoint triple F ⊣ G ⊣ H where G is fully faithful, the natural transformation F.op ⟶ H.op obtained from the dual adjoint triple H.op ⊣ G.op ⊣ F.op is dual to the natural transformation H ⟶ F.

          For an adjoint triple F ⊣ G ⊣ H where G is fully faithful, the natural transformation H ⟶ F is epic at X iff the image of the unit of the adjunction F ⊣ G under H is.

          For an adjoint triple F ⊣ G ⊣ H where G is fully faithful, the natural transformation H ⟶ F is epic at X iff the image of the counit of the adjunction G ⊣ H under F is.

          For an adjoint triple F ⊣ G ⊣ H where G is fully faithful and H preserves epimorphisms (which is for example the case if H has a further right adjoint), the components of the natural transformation H ⟶ F are epic iff the respective components of the natural transformation H ⋙ G ⟶ F ⋙ G obtained from the units and counits of the adjunctions are.

          noncomputable def CategoryTheory.Adjunction.Triple.leftToRight {C : Type u_1} {D : Type u_2} [Category.{u_3, u_1} C] [Category.{u_4, u_2} D] {F : Functor C D} {G : Functor D C} {H : Functor C D} (t : Triple F G H) [F.Full] [F.Faithful] :
          F H

          The natural transformation F ⟶ H that exists for every adjoint triple F ⊣ G ⊣ H where F and H are fully faithful, given here as the whiskered unit F ⟶ F ⋙ G ⋙ H of the second adjunction followed by the inverse of the whiskered unit F ⋙ G ⋙ H ⟶ H of the first.

          Equations
          • One or more equations did not get rendered due to their size.
          Instances For
            theorem CategoryTheory.Adjunction.Triple.leftToRight_app {C : Type u_1} {D : Type u_2} [Category.{u_4, u_1} C] [Category.{u_3, u_2} D] {F : Functor C D} {G : Functor D C} {H : Functor C D} (t : Triple F G H) [F.Full] [F.Faithful] {X : C} :

            The natural transformation F ⟶ H for an adjoint triple F ⊣ G ⊣ H with F and H fully faithful is also equal to the inverse of the whiskered counit H ⋙ G ⋙ F ⟶ F of the second adjunction followed by the whiskered counit H ⋙ G ⋙ F ⟶ H of the first.

            @[simp]
            theorem CategoryTheory.Adjunction.Triple.leftToRight_app_obj {C : Type u_1} {D : Type u_2} [Category.{u_4, u_1} C] [Category.{u_3, u_2} D] {F : Functor C D} {G : Functor D C} {H : Functor C D} (t : Triple F G H) [F.Full] [F.Faithful] {X : D} :

            For an adjoint triple F ⊣ G ⊣ H where F and H are fully faithful, the components of the natural transformation F ⟶ H at G are precisely the components of the natural transformation G ⋙ F ⟶ G ⋙ H obtained from the units and counits of the adjunctions.

            @[simp]

            For an adjoint triple F ⊣ G ⊣ H where F and H are fully faithful, whiskering G with the natural transformation F ⟶ H yields the composition of the counit of the first adjunction with the unit of the second adjunction.

            @[simp]
            @[simp]

            For an adjoint triple F ⊣ G ⊣ H where F and H are fully faithful, the natural transformation H.op ⟶ F.op obtained from the dual adjoint triple H.op ⊣ G.op ⊣ F.op is dual to the natural transformation F ⟶ H.

            For an adjoint triple F ⊣ G ⊣ H where F and H are fully faithful, the natural transformation F ⟶ H is monic at X iff the unit of the adjunction G ⊣ H is monic at F.obj X.

            For an adjoint triple F ⊣ G ⊣ H where F and H are fully faithful, the natural transformation F ⟶ H is monic at X iff the counit of the adjunction F ⊣ G is monic at H.obj X.

            theorem CategoryTheory.Adjunction.Triple.mono_leftToRight_app_iff {C : Type u_1} {D : Type u_2} [Category.{u_4, u_1} C] [Category.{u_3, u_2} D] {F : Functor C D} {G : Functor D C} {H : Functor C D} (t : Triple F G H) [F.Full] [F.Faithful] :
            (∀ (X : C), Mono (t.leftToRight.app X)) ∀ (X : D), Mono (CategoryStruct.comp (t.adj₁.counit.app X) (t.adj₂.unit.app X))

            For an adjoint triple F ⊣ G ⊣ H where F and H are fully faithful, the natural transformation F ⟶ H is componentwise monic iff the natural transformation G ⋙ F ⟶ G ⋙ H obtained from the units and counits of the adjunctions is. Note that unlike epi_rightToLeft_app_iff, this equivalence does not make sense on a per-object basis because the components of the two natural transformations are indexed by different categories.