# 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 #

• Adjoint Lifting Theorems for Categories of Algebras (PT Johnstone, 1975)
• A unified approach to the lifting of adjoints (AJ Power, 1988)
def CategoryTheory.LiftAdjoint.counitCoequalises {B : Type u₂} {C : Type u₃} [] [] {U : } {F : } (adj₁ : F U) [(X : B) → CategoryTheory.RegularEpi (adj₁.counit.app X)] (X : B) :
CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.Cofork.ofπ (adj₁.counit.app X) (_ : CategoryTheory.CategoryStruct.comp (F.map (U.map (adj₁.counit.app X))) (adj₁.counit.app X) = CategoryTheory.CategoryStruct.comp (adj₁.counit.app (F.obj (U.obj X))) (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).

Instances For
def CategoryTheory.LiftAdjoint.otherMap {A : Type u₁} {B : Type u₂} {C : Type u₃} [] [] [] {U : } {F : } (R : ) (F' : ) (adj₁ : F U) (adj₂ : ) (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.

Instances For
instance CategoryTheory.LiftAdjoint.instIsReflexivePairObjToQuiverToCategoryStructToQuiverToCategoryStructToPrefunctorObjToQuiverToCategoryStructToPrefunctorObjToPrefunctorCompIdMapMapAppCounitOtherMap {A : Type u₁} {B : Type u₂} {C : Type u₃} [] [] [] {U : } {F : } (R : ) (F' : ) (adj₁ : F U) (adj₂ : ) (X : B) :

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

noncomputable def CategoryTheory.LiftAdjoint.constructLeftAdjointObj {A : Type u₁} {B : Type u₂} {C : Type u₃} [] [] [] {U : } {F : } (R : ) (F' : ) (adj₁ : F U) (adj₂ : ) (Y : B) :
A

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

Instances For
@[simp]
theorem CategoryTheory.LiftAdjoint.constructLeftAdjointEquiv_apply {A : Type u₁} {B : Type u₂} {C : Type u₃} [] [] [] {U : } {F : } (R : ) (F' : ) (adj₁ : F U) (adj₂ : ) [(X : B) → CategoryTheory.RegularEpi (adj₁.counit.app X)] (Y : A) (X : B) :
@[simp]
theorem CategoryTheory.LiftAdjoint.constructLeftAdjointEquiv_symm_apply {A : Type u₁} {B : Type u₂} {C : Type u₃} [] [] [] {U : } {F : } (R : ) (F' : ) (adj₁ : F U) (adj₂ : ) [(X : B) → CategoryTheory.RegularEpi (adj₁.counit.app X)] (Y : A) (X : B) :
noncomputable def CategoryTheory.LiftAdjoint.constructLeftAdjointEquiv {A : Type u₁} {B : Type u₂} {C : Type u₃} [] [] [] {U : } {F : } (R : ) (F' : ) (adj₁ : F U) (adj₂ : ) [(X : B) → CategoryTheory.RegularEpi (adj₁.counit.app X)] (Y : A) (X : B) :
( Y) (X R.obj Y)

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

Instances For
noncomputable def CategoryTheory.LiftAdjoint.constructLeftAdjoint {A : Type u₁} {B : Type u₂} {C : Type u₃} [] [] [] {U : } {F : } (R : ) (F' : ) (adj₁ : F U) (adj₂ : ) [(X : B) → CategoryTheory.RegularEpi (adj₁.counit.app X)] :

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

Instances For
noncomputable def CategoryTheory.adjointTriangleLift {A : Type u₁} {B : Type u₂} {C : Type u₃} [] [] [] {U : } {F : } (R : ) (adj₁ : F U) [(X : B) → CategoryTheory.RegularEpi (adj₁.counit.app X)] :

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

Instances For
noncomputable def CategoryTheory.monadicAdjointTriangleLift {A : Type u₁} {B : Type u₂} {C : Type u₃} [] [] [] (U : ) {R : } :

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.

Instances For
noncomputable def CategoryTheory.adjointSquareLift {A : Type u₁} {B : Type u₂} {C : Type u₃} [] [] [] {D : Type u₄} [] (Q : ) (V : ) (U : ) (R : ) (comm : ) [(X : B) → CategoryTheory.RegularEpi (().counit.app X)] :

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.

Instances For
noncomputable def CategoryTheory.monadicAdjointSquareLift {A : Type u₁} {B : Type u₂} {C : Type u₃} [] [] [] {D : Type u₄} [] (Q : ) (V : ) (U : ) (R : ) (comm : ) :

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.