Documentation

Mathlib.CategoryTheory.Adjunction.Lifting.Left

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: isRightAdjoint_triangle_lift_monadic, 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 V is monadic, U has a left adjoint, 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.

This file has been adapted to Mathlib.CategoryTheory.Adjunction.Lifting.Right. Please try to keep them in sync.

TODO #

References #

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

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

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

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

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            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

            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 isRightAdjoint_triangle_lift which is often more useful in practice.

            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

            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