Adjoint functor theorem

This file proves the (general) adjoint functor theorem, in the form:

• If G : D ⥤ C preserves limits and D has limits, and satisfies the solution set condition, then it has a left adjoint: isRightAdjointOfPreservesLimitsOfIsCoseparating.

We show that the converse holds, i.e. that if G has a left adjoint then it satisfies the solution set condition, see solutionSetCondition_of_isRightAdjoint (the file CategoryTheory/Adjunction/Limits already shows it preserves limits).

We define the solution set condition for the functor G : D ⥤ C to mean, for every object A : C, there is a set-indexed family ${f_i : A ⟶ G (B_i)}$ such that any morphism A ⟶ G X factors through one of the f_i.

This file also proves the special adjoint functor theorem, in the form:

• If G : D ⥤ C preserves limits and D is complete, well-powered and has a small coseparating set, then G has a left adjoint: isRightAdjointOfPreservesLimitsOfIsCoseparating

Finally, we prove the following corollary of the special adjoint functor theorem:

• If C is complete, well-powered and has a small coseparating set, then it is cocomplete: hasColimits_of_hasLimits_of_isCoseparating

def CategoryTheory.SolutionSetCondition {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u} [CategoryTheory.Category.{v, u} D] (G : CategoryTheory.Functor D C) : Prop

The functor G : D ⥤ C satisfies the solution set condition if for every A : C, there is a family of morphisms {f_i : A ⟶ G (B_i) // i ∈ ι} such that given any morphism h : A ⟶ G X, there is some i ∈ ι such that h factors through f_i.

The key part of this definition is that the indexing set ι lives in Type v, where v is the universe of morphisms of the category: this is the "smallness" condition which allows the general adjoint functor theorem to go through.

Equations

• One or more equations did not get rendered due to their size.

theorem CategoryTheory.solutionSetCondition_of_isRightAdjoint {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u} [CategoryTheory.Category.{v, u} D] (G : CategoryTheory.Functor D C) [CategoryTheory.IsRightAdjoint G] : CategoryTheory.SolutionSetCondition G

If G : D ⥤ C is a right adjoint it satisfies the solution set condition.

noncomputable def CategoryTheory.isRightAdjointOfPreservesLimitsOfSolutionSetCondition {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u} [CategoryTheory.Category.{v, u} D] (G : CategoryTheory.Functor D C) [CategoryTheory.Limits.HasLimits D] [CategoryTheory.Limits.PreservesLimits G] (hG : CategoryTheory.SolutionSetCondition G) : CategoryTheory.IsRightAdjoint G

The general adjoint functor theorem says that if G : D ⥤ C preserves limits and D has them, if G satisfies the solution set condition then G is a right adjoint.

Equations

• CategoryTheory.isRightAdjointOfPreservesLimitsOfSolutionSetCondition G hG = CategoryTheory.isRightAdjointOfStructuredArrowInitials G

noncomputable def CategoryTheory.isRightAdjointOfPreservesLimitsOfIsCoseparating {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u'} [CategoryTheory.Category.{v, u'} D] [CategoryTheory.Limits.HasLimits D] [CategoryTheory.WellPowered D] {𝒢 : Set D} [Small.{v, u'} ↑𝒢] (h𝒢 : CategoryTheory.IsCoseparating 𝒢) (G : CategoryTheory.Functor D C) [CategoryTheory.Limits.PreservesLimits G] : CategoryTheory.IsRightAdjoint G

The special adjoint functor theorem: if G : D ⥤ C preserves limits and D is complete, well-powered and has a small coseparating set, then G has a left adjoint.

Equations

• CategoryTheory.isRightAdjointOfPreservesLimitsOfIsCoseparating h𝒢 G = let_fun this := ⋯; CategoryTheory.isRightAdjointOfStructuredArrowInitials G

noncomputable def CategoryTheory.isLeftAdjointOfPreservesColimitsOfIsSeparatig {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u'} [CategoryTheory.Category.{v, u'} D] [CategoryTheory.Limits.HasColimits C] [CategoryTheory.WellPowered Cᵒᵖ] {𝒢 : Set C} [Small.{v, u} ↑𝒢] (h𝒢 : CategoryTheory.IsSeparating 𝒢) (F : CategoryTheory.Functor C D) [CategoryTheory.Limits.PreservesColimits F] : CategoryTheory.IsLeftAdjoint F

The special adjoint functor theorem: if F : C ⥤ D preserves colimits and C is cocomplete, well-copowered and has a small separating set, then F has a right adjoint.

Equations

• CategoryTheory.isLeftAdjointOfPreservesColimitsOfIsSeparatig h𝒢 F = let_fun this := ⋯; CategoryTheory.isLeftAdjointOfCostructuredArrowTerminals F

theorem CategoryTheory.Limits.hasColimits_of_hasLimits_of_isCoseparating {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasLimits C] [CategoryTheory.WellPowered C] {𝒢 : Set C} [Small.{v, u} ↑𝒢] (h𝒢 : CategoryTheory.IsCoseparating 𝒢) : CategoryTheory.Limits.HasColimits C

A consequence of the special adjoint functor theorem: if C is complete, well-powered and has a small coseparating set, then it is cocomplete.

theorem CategoryTheory.Limits.hasLimits_of_hasColimits_of_isSeparating {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasColimits C] [CategoryTheory.WellPowered Cᵒᵖ] {𝒢 : Set C} [Small.{v, u} ↑𝒢] (h𝒢 : CategoryTheory.IsSeparating 𝒢) : CategoryTheory.Limits.HasLimits C

A consequence of the special adjoint functor theorem: if C is cocomplete, well-copowered and has a small separating set, then it is complete.