Theorems · Definition · category theory

CategoryTheory.Adjunction.derived

{C₁ : Type u_1} →
  {C₂ : Type u_2} →
    {D₁ : Type u_3} →
      {D₂ : Type u_4} →
        [inst : CategoryTheory.Category.{v_1, u_1} C₁] →
          [inst_1 : CategoryTheory.Category.{v_2, u_2} C₂] →
            [inst_2 : CategoryTheory.Category.{v_3, u_3} D₁] →
              [inst_3 : CategoryTheory.Category.{v_4, u_4} D₂] →
                {G : CategoryTheory.Functor C₁ C₂} →
                  {F : CategoryTheory.Functor C₂ C₁} →
                    (G ⊣ F) →
                      {L₁ : CategoryTheory.Functor C₁ D₁} →
                        {L₂ : CategoryTheory.Functor C₂ D₂} →
                          (W₁ : CategoryTheory.MorphismProperty C₁) →
                            (W₂ : CategoryTheory.MorphismProperty C₂) →
                              [inst_4 : L₁.IsLocalization W₁] →
                                [inst_5 : L₂.IsLocalization W₂] →
                                  {G' : CategoryTheory.Functor D₁ D₂} →
                                    {F' : CategoryTheory.Functor D₂ D₁} →
                                      (α : L₁.comp G' ⟶ G.comp L₂) →
                                        (β : F.comp L₁ ⟶ L₂.comp F') →
                                          [G'.IsLeftDerivedFunctor α W₁] →
                                            [F'.IsRightDerivedFunctor β W₂] →
                                              [(G'.comp F').IsLeftDerivedFunctor
                                                    (CategoryTheory.CategoryStruct.comp (L₁.associator G' F').inv
                                                      (CategoryTheory.Functor.whiskerRight α F'))
                                                    W₁] →
                                                [(F'.comp G').IsRightDerivedFunctor
                                                      (CategoryTheory.CategoryStruct.comp
                                                        (CategoryTheory.Functor.whiskerRight β G')
                                                        (L₂.associator F' G').hom)
                                                      W₂] →
                                                  G' ⊣ F'

An adjunction between functors induces an adjunction between the corresponding left/right derived functors, when these derived functors are absolute, i.e. they remain derived functors after the post-composition with any functor. (One actually only needs that G' ⋙ F' is the left derived functor of G ⋙ L₂ ⋙ F' and that F' ⋙ G' is the right derived functor of F ⋙ L₁ ⋙ G').

Defined in

Mathlib.CategoryTheory.Functor.Derived.Adjunction

Cited by

2 results in Mathlib

Foundations

Depth 37 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.IsLocalizationCategoryTheory.Functor.IsLocalizationCategoryTheory.Functor.IsLeftDerivedFunctorCategoryTheory.Functor.IsRightDerivedFunctorCategoryTheory.Functor.IsLeftDerivedFunctorCategoryTheory.Functor.IsRightDerivedFunctor