Documentation

Mathlib.CategoryTheory.Adjunction.Lifting

Adjoint lifting #

This file gives two constructions for building left adjoints: the adjoint triangle theorem and the adjoint lifting theorem. The adjoint triangle theorem concerns a functor U : B ⥤ C with a left adjoint F such that ε_X : FUX ⟶ X is a regular epi. Then for any category A with coequalizers of reflexive pairs, a functor R : A ⥤ B has a left adjoint if (and only if) the composite R ⋙ U does. Note that the condition on U regarding ε_X is automatically satisfied in the case when U is a monadic functor, giving the corollary: monadicAdjointTriangleLift, i.e. if U is monadic, A has reflexive coequalizers then R : A ⥤ B has a left adjoint provided R ⋙ U does.

The adjoint lifting theorem says that given a commutative square of functors (up to isomorphism):

  Q
A → B

U ↓ ↓ V C → D R

where U and V are monadic and A has reflexive coequalizers, then if R has a left adjoint then Q has a left adjoint.

Implementation #

It is more convenient to prove this theorem by assuming we are given the explicit adjunction rather than just a functor known to be a right adjoint. In docstrings, we write (η, ε) for the unit and counit of the adjunction adj₁ : F ⊣ U and (ι, δ) for the unit and counit of the adjunction adj₂ : F' ⊣ R ⋙ U.

TODO #

Dualise to lift right adjoints through comonads (by reversing 1-cells) and dualise to lift right adjoints through monads (by reversing 2-cells), and the combination.

References #

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def CategoryTheory.LiftAdjoint.counitCoequalises {B : Type u₂} {C : Type u₃} [CategoryTheory.Category.{v₂, u₂} B] [CategoryTheory.Category.{v₃, u₃} C] {U : CategoryTheory.Functor B C} {F : CategoryTheory.Functor C B} (adj₁ : F ⊣ U) [(X : B) → CategoryTheory.RegularEpi (adj₁.counit.app X)] (X : B) :
CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.Cofork.ofπ (adj₁.counit.app X) ⋯)

To show that ε_X is a coequalizer for (FUε_X, ε_FUX), it suffices to assume it's always a coequalizer of something (i.e. a regular epi).

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    def CategoryTheory.LiftAdjoint.otherMap {A : Type u₁} {B : Type u₂} {C : Type u₃} [CategoryTheory.Category.{v₁, u₁} A] [CategoryTheory.Category.{v₂, u₂} B] [CategoryTheory.Category.{v₃, u₃} C] {U : CategoryTheory.Functor B C} {F : CategoryTheory.Functor C B} (R : CategoryTheory.Functor A B) (F' : CategoryTheory.Functor C A) (adj₁ : F ⊣ U) (adj₂ : F' ⊣ R.comp U) (X : B) :
    F'.obj (U.obj (F.obj (U.obj X))) ⟶ F'.obj (U.obj X)

    (Implementation) To construct the left adjoint, we use the coequalizer of F' U ε_Y with the composite

    F' U F U X ⟶ F' U F U R F U' X ⟶ F' U R F' U X ⟶ F' U X

    where the first morphism is F' U F ι_UX, the second is F' U ε_RF'UX, and the third is δ_F'UX. We will show that this coequalizer exists and that it forms the object map for a left adjoint to R.

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      instance CategoryTheory.LiftAdjoint.instIsReflexivePairMapAppCounitOtherMap {A : Type u₁} {B : Type u₂} {C : Type u₃} [CategoryTheory.Category.{v₁, u₁} A] [CategoryTheory.Category.{v₂, u₂} B] [CategoryTheory.Category.{v₃, u₃} C] {U : CategoryTheory.Functor B C} {F : CategoryTheory.Functor C B} (R : CategoryTheory.Functor A B) (F' : CategoryTheory.Functor C A) (adj₁ : F ⊣ U) (adj₂ : F' ⊣ R.comp U) (X : B) :
      CategoryTheory.IsReflexivePair (F'.map (U.map (adj₁.counit.app X))) (CategoryTheory.LiftAdjoint.otherMap R F' adj₁ adj₂ X)

      (F'Uε_X, otherMap X) is a reflexive pair: in particular if A has reflexive coequalizers then it has a coequalizer.

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      noncomputable def CategoryTheory.LiftAdjoint.constructLeftAdjointObj {A : Type u₁} {B : Type u₂} {C : Type u₃} [CategoryTheory.Category.{v₁, u₁} A] [CategoryTheory.Category.{v₂, u₂} B] [CategoryTheory.Category.{v₃, u₃} C] {U : CategoryTheory.Functor B C} {F : CategoryTheory.Functor C B} (R : CategoryTheory.Functor A B) (F' : CategoryTheory.Functor C A) (adj₁ : F ⊣ U) (adj₂ : F' ⊣ R.comp U) [CategoryTheory.Limits.HasReflexiveCoequalizers A] (Y : B) :
      A

      Construct the object part of the desired left adjoint as the coequalizer of F'Uε_Y with otherMap.

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        @[simp]
        theorem CategoryTheory.LiftAdjoint.constructLeftAdjointEquiv_symm_apply {A : Type u₁} {B : Type u₂} {C : Type u₃} [CategoryTheory.Category.{v₁, u₁} A] [CategoryTheory.Category.{v₂, u₂} B] [CategoryTheory.Category.{v₃, u₃} C] {U : CategoryTheory.Functor B C} {F : CategoryTheory.Functor C B} (R : CategoryTheory.Functor A B) (F' : CategoryTheory.Functor C A) (adj₁ : F ⊣ U) (adj₂ : F' ⊣ R.comp U) [CategoryTheory.Limits.HasReflexiveCoequalizers A] [(X : B) → CategoryTheory.RegularEpi (adj₁.counit.app X)] (Y : A) (X : B) :
        ∀ (a : X ⟶ R.obj Y), (CategoryTheory.LiftAdjoint.constructLeftAdjointEquiv R F' adj₁ adj₂ Y X).symm a = (CategoryTheory.Limits.Cofork.IsColimit.homIso (CategoryTheory.Limits.colimit.isColimit (CategoryTheory.Limits.parallelPair (F'.map (U.map (adj₁.counit.app X))) (CategoryTheory.LiftAdjoint.otherMap R F' adj₁ adj₂ X))) Y).symm ⟨(adj₂.homEquiv (U.obj X) Y).symm (CategoryTheory.CategoryStruct.comp (adj₁.unit.app (U.obj X)) (U.map ↑((CategoryTheory.Limits.Cofork.IsColimit.homIso (CategoryTheory.LiftAdjoint.counitCoequalises adj₁ X) (R.obj Y)) a))), ⋯⟩
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        @[simp]
        theorem CategoryTheory.LiftAdjoint.constructLeftAdjointEquiv_apply {A : Type u₁} {B : Type u₂} {C : Type u₃} [CategoryTheory.Category.{v₁, u₁} A] [CategoryTheory.Category.{v₂, u₂} B] [CategoryTheory.Category.{v₃, u₃} C] {U : CategoryTheory.Functor B C} {F : CategoryTheory.Functor C B} (R : CategoryTheory.Functor A B) (F' : CategoryTheory.Functor C A) (adj₁ : F ⊣ U) (adj₂ : F' ⊣ R.comp U) [CategoryTheory.Limits.HasReflexiveCoequalizers A] [(X : B) → CategoryTheory.RegularEpi (adj₁.counit.app X)] (Y : A) (X : B) :
        ∀ (a : CategoryTheory.LiftAdjoint.constructLeftAdjointObj R F' adj₁ adj₂ X ⟶ Y), (CategoryTheory.LiftAdjoint.constructLeftAdjointEquiv R F' adj₁ adj₂ Y X) a = (CategoryTheory.Limits.Cofork.IsColimit.homIso (CategoryTheory.LiftAdjoint.counitCoequalises adj₁ X) (R.obj Y)).symm ⟨CategoryTheory.CategoryStruct.comp (F.map ((adj₂.homEquiv (U.obj X) Y) ↑((CategoryTheory.Limits.Cofork.IsColimit.homIso (CategoryTheory.Limits.colimit.isColimit (CategoryTheory.Limits.parallelPair (F'.map (U.map (adj₁.counit.app X))) (CategoryTheory.LiftAdjoint.otherMap R F' adj₁ adj₂ X))) Y) a))) (adj₁.counit.app (R.obj Y)), ⋯⟩
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        noncomputable def CategoryTheory.LiftAdjoint.constructLeftAdjointEquiv {A : Type u₁} {B : Type u₂} {C : Type u₃} [CategoryTheory.Category.{v₁, u₁} A] [CategoryTheory.Category.{v₂, u₂} B] [CategoryTheory.Category.{v₃, u₃} C] {U : CategoryTheory.Functor B C} {F : CategoryTheory.Functor C B} (R : CategoryTheory.Functor A B) (F' : CategoryTheory.Functor C A) (adj₁ : F ⊣ U) (adj₂ : F' ⊣ R.comp U) [CategoryTheory.Limits.HasReflexiveCoequalizers A] [(X : B) → CategoryTheory.RegularEpi (adj₁.counit.app X)] (Y : A) (X : B) :
        (CategoryTheory.LiftAdjoint.constructLeftAdjointObj R F' adj₁ adj₂ X ⟶ Y) ≃ (X ⟶ R.obj Y)

        The homset equivalence which helps show that R is a right adjoint.

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          noncomputable def CategoryTheory.LiftAdjoint.constructLeftAdjoint {A : Type u₁} {B : Type u₂} {C : Type u₃} [CategoryTheory.Category.{v₁, u₁} A] [CategoryTheory.Category.{v₂, u₂} B] [CategoryTheory.Category.{v₃, u₃} C] {U : CategoryTheory.Functor B C} {F : CategoryTheory.Functor C B} (R : CategoryTheory.Functor A B) (F' : CategoryTheory.Functor C A) (adj₁ : F ⊣ U) (adj₂ : F' ⊣ R.comp U) [CategoryTheory.Limits.HasReflexiveCoequalizers A] [(X : B) → CategoryTheory.RegularEpi (adj₁.counit.app X)] :
          CategoryTheory.Functor B A

          Construct the left adjoint to R, with object map constructLeftAdjointObj.

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            theorem CategoryTheory.adjointTriangleLift {A : Type u₁} {B : Type u₂} {C : Type u₃} [CategoryTheory.Category.{v₁, u₁} A] [CategoryTheory.Category.{v₂, u₂} B] [CategoryTheory.Category.{v₃, u₃} C] {U : CategoryTheory.Functor B C} {F : CategoryTheory.Functor C B} (R : CategoryTheory.Functor A B) (adj₁ : F ⊣ U) [(X : B) → CategoryTheory.RegularEpi (adj₁.counit.app X)] [CategoryTheory.Limits.HasReflexiveCoequalizers A] [(R.comp U).IsRightAdjoint] :
            R.IsRightAdjoint

            The adjoint triangle theorem: Suppose U : B ⥤ C has a left adjoint F such that each counit ε_X : FUX ⟶ X is a regular epimorphism. Then if a category A has coequalizers of reflexive pairs, then a functor R : A ⥤ B has a left adjoint if the composite R ⋙ U does.

            Note the converse is true (with weaker assumptions), by Adjunction.comp. See https://ncatlab.org/nlab/show/adjoint+triangle+theorem

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            theorem CategoryTheory.monadicAdjointTriangleLift {A : Type u₁} {B : Type u₂} {C : Type u₃} [CategoryTheory.Category.{v₁, u₁} A] [CategoryTheory.Category.{v₂, u₂} B] [CategoryTheory.Category.{v₃, u₃} C] (U : CategoryTheory.Functor B C) [CategoryTheory.MonadicRightAdjoint U] {R : CategoryTheory.Functor A B} [CategoryTheory.Limits.HasReflexiveCoequalizers A] [(R.comp U).IsRightAdjoint] :
            R.IsRightAdjoint

            If R ⋙ U has a left adjoint, the domain of R has reflexive coequalizers and U is a monadic functor, then R has a left adjoint. This is a special case of adjointTriangleLift which is often more useful in practice.

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            theorem CategoryTheory.adjointSquareLift {A : Type u₁} {B : Type u₂} {C : Type u₃} [CategoryTheory.Category.{v₁, u₁} A] [CategoryTheory.Category.{v₂, u₂} B] [CategoryTheory.Category.{v₃, u₃} C] {D : Type u₄} [CategoryTheory.Category.{v₄, u₄} D] (Q : CategoryTheory.Functor A B) (V : CategoryTheory.Functor B D) (U : CategoryTheory.Functor A C) (R : CategoryTheory.Functor C D) (comm : U.comp R ≅ Q.comp V) [U.IsRightAdjoint] [V.IsRightAdjoint] [R.IsRightAdjoint] [(X : B) → CategoryTheory.RegularEpi ((CategoryTheory.Adjunction.ofIsRightAdjoint V).counit.app X)] [CategoryTheory.Limits.HasReflexiveCoequalizers A] :
            Q.IsRightAdjoint

            Suppose we have a commutative square of functors

              Q
A → B

            U ↓ ↓ V C → D R

            where U has a left adjoint, A has reflexive coequalizers and V has a left adjoint such that each component of the counit is a regular epi. Then Q has a left adjoint if R has a left adjoint.

            See https://ncatlab.org/nlab/show/adjoint+lifting+theorem

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            theorem CategoryTheory.monadicAdjointSquareLift {A : Type u₁} {B : Type u₂} {C : Type u₃} [CategoryTheory.Category.{v₁, u₁} A] [CategoryTheory.Category.{v₂, u₂} B] [CategoryTheory.Category.{v₃, u₃} C] {D : Type u₄} [CategoryTheory.Category.{v₄, u₄} D] (Q : CategoryTheory.Functor A B) (V : CategoryTheory.Functor B D) (U : CategoryTheory.Functor A C) (R : CategoryTheory.Functor C D) (comm : U.comp R ≅ Q.comp V) [U.IsRightAdjoint] [CategoryTheory.MonadicRightAdjoint V] [R.IsRightAdjoint] [CategoryTheory.Limits.HasReflexiveCoequalizers A] :
            Q.IsRightAdjoint

            Suppose we have a commutative square of functors

              Q
A → B

            U ↓ ↓ V C → D R

            where U has a left adjoint, A has reflexive coequalizers and V is monadic. Then Q has a left adjoint if R has a left adjoint.

            See https://ncatlab.org/nlab/show/adjoint+lifting+theorem