Documentation

Mathlib.CategoryTheory.Adjunction.Basic

Adjunctions between functors #

F ⊣ G represents the data of an adjunction between two functors F : C ⥤ D and G : D ⥤ C. F is the left adjoint and G is the right adjoint.

We provide various useful constructors:

There are also typeclasses IsLeftAdjoint / IsRightAdjoint, which asserts the existence of a adjoint functor. Given [F.IsLeftAdjoint], a chosen right adjoint can be obtained as F.rightAdjoint.

Adjunction.comp composes adjunctions.

toEquivalence upgrades an adjunction to an equivalence, given witnesses that the unit and counit are pointwise isomorphisms. Conversely Equivalence.toAdjunction recovers the underlying adjunction from an equivalence.

structure CategoryTheory.Adjunction {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) (G : CategoryTheory.Functor D C) :
Type (max (max (max u₁ u₂) v₁) v₂)

F ⊣ G represents the data of an adjunction between two functors F : C ⥤ D and G : D ⥤ C. F is the left adjoint and G is the right adjoint.

We use the unit-counit definition of an adjunction. There is a constructor Adjunction.mk' which constructs an adjunction from the data of a hom set equivalence, a unit, and a counit, together with proofs of the equalities homEquiv_unit and homEquiv_counit relating them to each other.

There is also a constructor Adjunction.mkOfHomEquiv which constructs an adjunction from a natural hom set equivalence.

To construct adjoints to a given functor, there are constructors leftAdjointOfEquiv and adjunctionOfEquivLeft (as well as their duals).

Uniqueness of adjoints is shown in CategoryTheory.Adjunction.Unique.

See https://stacks.math.columbia.edu/tag/0037.

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    Equality of the composition of the unit and counit with the identity F ⟶ FGF ⟶ F = 𝟙

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    Equality of the composition of the unit and counit with the identity G ⟶ GFG ⟶ G = 𝟙

    The notation F ⊣ G stands for Adjunction F G representing that F is left adjoint to G

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      A class asserting the existence of a right adjoint.

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        A class asserting the existence of a left adjoint.

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          A chosen left adjoint to a functor that is a right adjoint.

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          • R.leftAdjoint = .choose
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            A chosen right adjoint to a functor that is a left adjoint.

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            • L.rightAdjoint = .choose
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              noncomputable def CategoryTheory.Adjunction.ofIsLeftAdjoint {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (left : CategoryTheory.Functor C D) [left.IsLeftAdjoint] :
              left left.rightAdjoint

              The adjunction associated to a functor known to be a left adjoint.

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                noncomputable def CategoryTheory.Adjunction.ofIsRightAdjoint {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (right : CategoryTheory.Functor C D) [right.IsRightAdjoint] :
                right.leftAdjoint right

                The adjunction associated to a functor known to be a right adjoint.

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                  theorem CategoryTheory.Adjunction.left_triangle_components_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (self : F G) (X : C) {Z : D} (h : F.obj X Z) :
                  CategoryTheory.CategoryStruct.comp (F.map (self.unit.app X)) (CategoryTheory.CategoryStruct.comp (self.counit.app (F.obj X)) h) = h
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                  theorem CategoryTheory.Adjunction.right_triangle_components_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (self : F G) (Y : D) {Z : C} (h : G.obj Y Z) :
                  CategoryTheory.CategoryStruct.comp (self.unit.app (G.obj Y)) (CategoryTheory.CategoryStruct.comp (G.map (self.counit.app Y)) h) = h
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                  theorem CategoryTheory.Adjunction.homEquiv_apply {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) (X : C) (Y : D) (f : F.obj X Y) :
                  (adj.homEquiv X Y) f = CategoryTheory.CategoryStruct.comp (adj.unit.app X) (G.map f)
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                  theorem CategoryTheory.Adjunction.homEquiv_symm_apply {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) (X : C) (Y : D) (g : X G.obj Y) :
                  (adj.homEquiv X Y).symm g = CategoryTheory.CategoryStruct.comp (F.map g) (adj.counit.app Y)
                  def CategoryTheory.Adjunction.homEquiv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) (X : C) (Y : D) :
                  (F.obj X Y) (X G.obj Y)

                  The hom set equivalence associated to an adjunction.

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                    theorem CategoryTheory.Adjunction.homEquiv_unit {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) (X : C) (Y : D) (f : F.obj X Y) :
                    (adj.homEquiv X Y) f = CategoryTheory.CategoryStruct.comp (adj.unit.app X) (G.map f)

                    Alias of CategoryTheory.Adjunction.homEquiv_apply.

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                    theorem CategoryTheory.Adjunction.homEquiv_counit {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) (X : C) (Y : D) (g : X G.obj Y) :
                    (adj.homEquiv X Y).symm g = CategoryTheory.CategoryStruct.comp (F.map g) (adj.counit.app Y)

                    Alias of CategoryTheory.Adjunction.homEquiv_symm_apply.

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                    theorem CategoryTheory.Adjunction.homEquiv_id {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) (X : C) :
                    (adj.homEquiv X (F.obj X)) (CategoryTheory.CategoryStruct.id (F.obj X)) = adj.unit.app X
                    theorem CategoryTheory.Adjunction.homEquiv_symm_id {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) (X : D) :
                    (adj.homEquiv (G.obj X) X).symm (CategoryTheory.CategoryStruct.id (G.obj X)) = adj.counit.app X
                    theorem CategoryTheory.Adjunction.homEquiv_naturality_left_symm {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {X' : C} {X : C} {Y : D} (f : X' X) (g : X G.obj Y) :
                    (adj.homEquiv X' Y).symm (CategoryTheory.CategoryStruct.comp f g) = CategoryTheory.CategoryStruct.comp (F.map f) ((adj.homEquiv X Y).symm g)
                    theorem CategoryTheory.Adjunction.homEquiv_naturality_left {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {X' : C} {X : C} {Y : D} (f : X' X) (g : F.obj X Y) :
                    (adj.homEquiv X' Y) (CategoryTheory.CategoryStruct.comp (F.map f) g) = CategoryTheory.CategoryStruct.comp f ((adj.homEquiv X Y) g)
                    theorem CategoryTheory.Adjunction.homEquiv_naturality_right {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {X : C} {Y : D} {Y' : D} (f : F.obj X Y) (g : Y Y') :
                    (adj.homEquiv X Y') (CategoryTheory.CategoryStruct.comp f g) = CategoryTheory.CategoryStruct.comp ((adj.homEquiv X Y) f) (G.map g)
                    theorem CategoryTheory.Adjunction.homEquiv_naturality_right_symm {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {X : C} {Y : D} {Y' : D} (f : X G.obj Y) (g : Y Y') :
                    (adj.homEquiv X Y').symm (CategoryTheory.CategoryStruct.comp f (G.map g)) = CategoryTheory.CategoryStruct.comp ((adj.homEquiv X Y).symm f) g
                    theorem CategoryTheory.Adjunction.homEquiv_naturality_left_square_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {X' : C} {X : C} {Y : D} {Y' : D} (f : X' X) (g : F.obj X Y') (h : F.obj X' Y) (k : Y Y') (w : CategoryTheory.CategoryStruct.comp (F.map f) g = CategoryTheory.CategoryStruct.comp h✝ k) {Z : C} (h : G.obj Y' Z) :
                    theorem CategoryTheory.Adjunction.homEquiv_naturality_left_square {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {X' : C} {X : C} {Y : D} {Y' : D} (f : X' X) (g : F.obj X Y') (h : F.obj X' Y) (k : Y Y') (w : CategoryTheory.CategoryStruct.comp (F.map f) g = CategoryTheory.CategoryStruct.comp h k) :
                    CategoryTheory.CategoryStruct.comp f ((adj.homEquiv X Y') g) = CategoryTheory.CategoryStruct.comp ((adj.homEquiv X' Y) h) (G.map k)
                    theorem CategoryTheory.Adjunction.homEquiv_naturality_right_square_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {X' : C} {X : C} {Y : D} {Y' : D} (f : X' X) (g : X G.obj Y') (h : X' G.obj Y) (k : Y Y') (w : CategoryTheory.CategoryStruct.comp f g = CategoryTheory.CategoryStruct.comp h✝ (G.map k)) {Z : D} (h : Y' Z) :
                    theorem CategoryTheory.Adjunction.homEquiv_naturality_right_square {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {X' : C} {X : C} {Y : D} {Y' : D} (f : X' X) (g : X G.obj Y') (h : X' G.obj Y) (k : Y Y') (w : CategoryTheory.CategoryStruct.comp f g = CategoryTheory.CategoryStruct.comp h (G.map k)) :
                    CategoryTheory.CategoryStruct.comp (F.map f) ((adj.homEquiv X Y').symm g) = CategoryTheory.CategoryStruct.comp ((adj.homEquiv X' Y).symm h) k
                    theorem CategoryTheory.Adjunction.homEquiv_naturality_left_square_iff {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {X' : C} {X : C} {Y : D} {Y' : D} (f : X' X) (g : F.obj X Y') (h : F.obj X' Y) (k : Y Y') :
                    theorem CategoryTheory.Adjunction.homEquiv_naturality_right_square_iff {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {X' : C} {X : C} {Y : D} {Y' : D} (f : X' X) (g : X G.obj Y') (h : X' G.obj Y) (k : Y Y') :
                    @[simp]
                    theorem CategoryTheory.Adjunction.counit_naturality {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {X : D} {Y : D} (f : X Y) :
                    CategoryTheory.CategoryStruct.comp (F.map (G.map f)) (adj.counit.app Y) = CategoryTheory.CategoryStruct.comp (adj.counit.app X) f
                    @[simp]
                    theorem CategoryTheory.Adjunction.unit_naturality {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {X : C} {Y : C} (f : X Y) :
                    CategoryTheory.CategoryStruct.comp (adj.unit.app X) (G.map (F.map f)) = CategoryTheory.CategoryStruct.comp f (adj.unit.app Y)
                    theorem CategoryTheory.Adjunction.unit_comp_map_eq_iff {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {A : C} {B : D} (f : F.obj A B) (g : A G.obj B) :
                    CategoryTheory.CategoryStruct.comp (adj.unit.app A) (G.map f) = g f = CategoryTheory.CategoryStruct.comp (F.map g) (adj.counit.app B)
                    theorem CategoryTheory.Adjunction.eq_unit_comp_map_iff {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {A : C} {B : D} (f : F.obj A B) (g : A G.obj B) :
                    g = CategoryTheory.CategoryStruct.comp (adj.unit.app A) (G.map f) CategoryTheory.CategoryStruct.comp (F.map g) (adj.counit.app B) = f
                    theorem CategoryTheory.Adjunction.homEquiv_apply_eq {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {A : C} {B : D} (f : F.obj A B) (g : A G.obj B) :
                    (adj.homEquiv A B) f = g f = (adj.homEquiv A B).symm g
                    theorem CategoryTheory.Adjunction.eq_homEquiv_apply {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {A : C} {B : D} (f : F.obj A B) (g : A G.obj B) :
                    g = (adj.homEquiv A B) f (adj.homEquiv A B).symm g = f

                    This is an auxiliary data structure useful for constructing adjunctions. See Adjunction.mk'. This structure won't typically be used anywhere else.

                    • homEquiv : (X : C) → (Y : D) → (F.obj X Y) (X G.obj Y)

                      The equivalence between Hom (F X) Y and Hom X (G Y) coming from an adjunction

                    • unit : CategoryTheory.Functor.id C F.comp G

                      The unit of an adjunction

                    • counit : G.comp F CategoryTheory.Functor.id D

                      The counit of an adjunction

                    • homEquiv_unit : ∀ {X : C} {Y : D} {f : F.obj X Y}, (self.homEquiv X Y) f = CategoryTheory.CategoryStruct.comp (self.unit.app X) (G.map f)

                      The relationship between the unit and hom set equivalence of an adjunction

                    • homEquiv_counit : ∀ {X : C} {Y : D} {g : X G.obj Y}, (self.homEquiv X Y).symm g = CategoryTheory.CategoryStruct.comp (F.map g) (self.counit.app Y)

                      The relationship between the counit and hom set equivalence of an adjunction

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                      The relationship between the unit and hom set equivalence of an adjunction

                      @[simp]

                      The relationship between the counit and hom set equivalence of an adjunction

                      This is an auxiliary data structure useful for constructing adjunctions. See Adjunction.mkOfHomEquiv. This structure won't typically be used anywhere else.

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                        theorem CategoryTheory.Adjunction.CoreHomEquiv.homEquiv_naturality_left_symm {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (self : CategoryTheory.Adjunction.CoreHomEquiv F G) {X' : C} {X : C} {Y : D} (f : X' X) (g : X G.obj Y) :
                        (self.homEquiv X' Y).symm (CategoryTheory.CategoryStruct.comp f g) = CategoryTheory.CategoryStruct.comp (F.map f) ((self.homEquiv X Y).symm g)

                        The property that describes how homEquiv.symm transforms compositions X' ⟶ X ⟶ G Y

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                        The property that describes how homEquiv transforms compositions F X ⟶ Y ⟶ Y'

                        This is an auxiliary data structure useful for constructing adjunctions. See Adjunction.mkOfUnitCounit. This structure won't typically be used anywhere else.

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                          Equality of the composition of the unit, associator, and counit with the identity F ⟶ (F G) F ⟶ F (G F) ⟶ F = NatTrans.id F

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                          Equality of the composition of the unit, associator, and counit with the identity G ⟶ G (F G) ⟶ (F G) F ⟶ G = NatTrans.id G

                          Construct an adjunction from the data of a CoreHomEquivUnitCounit, i.e. a hom set equivalence, unit and counit natural transformations together with proofs of the equalities homEquiv_unit and homEquiv_counit relating them to each other.

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                            Construct an adjunction between F and G out of a natural bijection between each F.obj X ⟶ Y and X ⟶ G.obj Y.

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                              Construct an adjunction between functors F and G given a unit and counit for the adjunction satisfying the triangle identities.

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                                The adjunction between the identity functor on a category and itself.

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                                  • CategoryTheory.Adjunction.instInhabitedId = { default := CategoryTheory.Adjunction.id }

                                  If F and G are naturally isomorphic functors, establish an equivalence of hom-sets.

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                                    If G and H are naturally isomorphic functors, establish an equivalence of hom-sets.

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                                      Transport an adjunction along a natural isomorphism on the left.

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                                        Transport an adjunction along a natural isomorphism on the right.

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                                          theorem CategoryTheory.Adjunction.compYonedaIso_hom_app_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₁, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) (X : D) (X : Cᵒᵖ) :
                                          ∀ (a : ((G.comp CategoryTheory.yoneda).obj X✝).obj X), (adj.compYonedaIso.hom.app X✝).app X a = CategoryTheory.CategoryStruct.comp (F.map a) (adj.counit.app X✝)
                                          @[simp]
                                          theorem CategoryTheory.Adjunction.compYonedaIso_inv_app_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₁, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) (X : D) (X : Cᵒᵖ) :
                                          ∀ (a : ((CategoryTheory.yoneda.comp ((CategoryTheory.whiskeringLeft Cᵒᵖ Dᵒᵖ (Type v₁)).obj F.op)).obj X✝).obj X), (adj.compYonedaIso.inv.app X✝).app X a = CategoryTheory.CategoryStruct.comp (adj.unit.app (Opposite.unop X)) (G.map a)
                                          def CategoryTheory.Adjunction.compYonedaIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₁, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) :
                                          G.comp CategoryTheory.yoneda CategoryTheory.yoneda.comp ((CategoryTheory.whiskeringLeft Cᵒᵖ Dᵒᵖ (Type v₁)).obj F.op)

                                          The isomorpism which an adjunction F ⊣ G induces on G ⋙ yoneda. This states that Adjunction.homEquiv is natural in both arguments.

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                                            theorem CategoryTheory.Adjunction.compCoyonedaIso_hom_app_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₁, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) (X : Cᵒᵖ) (X : D) :
                                            ∀ (a : ((F.op.comp CategoryTheory.coyoneda).obj X✝).obj X), (adj.compCoyonedaIso.hom.app X✝).app X a = CategoryTheory.CategoryStruct.comp (adj.unit.app (Opposite.unop X✝)) (G.map a)
                                            @[simp]
                                            theorem CategoryTheory.Adjunction.compCoyonedaIso_inv_app_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₁, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) (X : Cᵒᵖ) (X : D) :
                                            ∀ (a : ((CategoryTheory.coyoneda.comp ((CategoryTheory.whiskeringLeft D C (Type v₁)).obj G)).obj X✝).obj X), (adj.compCoyonedaIso.inv.app X✝).app X a = CategoryTheory.CategoryStruct.comp (F.map a) (adj.counit.app X)
                                            def CategoryTheory.Adjunction.compCoyonedaIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₁, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) :
                                            F.op.comp CategoryTheory.coyoneda CategoryTheory.coyoneda.comp ((CategoryTheory.whiskeringLeft D C (Type v₁)).obj G)

                                            The isomorpism which an adjunction F ⊣ G induces on F.op ⋙ coyoneda. This states that Adjunction.homEquiv is natural in both arguments.

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                                              Composition of adjunctions.

                                              See https://stacks.math.columbia.edu/tag/0DV0.

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                                                theorem CategoryTheory.Adjunction.comp_unit_app_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} {E : Type u₃} [ℰ : CategoryTheory.Category.{v₃, u₃} E] {H : CategoryTheory.Functor D E} {I : CategoryTheory.Functor E D} (adj₁ : F G) (adj₂ : H I) (X : C) {Z : C} (h : G.obj (I.obj (H.obj (F.obj X))) Z) :
                                                CategoryTheory.CategoryStruct.comp ((adj₁.comp adj₂).unit.app X) h = CategoryTheory.CategoryStruct.comp (adj₁.unit.app X) (CategoryTheory.CategoryStruct.comp (G.map (adj₂.unit.app (F.obj X))) h)
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                                                theorem CategoryTheory.Adjunction.comp_unit_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} {E : Type u₃} [ℰ : CategoryTheory.Category.{v₃, u₃} E] {H : CategoryTheory.Functor D E} {I : CategoryTheory.Functor E D} (adj₁ : F G) (adj₂ : H I) (X : C) :
                                                (adj₁.comp adj₂).unit.app X = CategoryTheory.CategoryStruct.comp (adj₁.unit.app X) (G.map (adj₂.unit.app (F.obj X)))
                                                theorem CategoryTheory.Adjunction.comp_counit_app_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} {E : Type u₃} [ℰ : CategoryTheory.Category.{v₃, u₃} E] {H : CategoryTheory.Functor D E} {I : CategoryTheory.Functor E D} (adj₁ : F G) (adj₂ : H I) (X : E) {Z : E} (h : X Z) :
                                                CategoryTheory.CategoryStruct.comp ((adj₁.comp adj₂).counit.app X) h = CategoryTheory.CategoryStruct.comp (H.map (adj₁.counit.app (I.obj X))) (CategoryTheory.CategoryStruct.comp (adj₂.counit.app X) h)
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                                                theorem CategoryTheory.Adjunction.comp_counit_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} {E : Type u₃} [ℰ : CategoryTheory.Category.{v₃, u₃} E] {H : CategoryTheory.Functor D E} {I : CategoryTheory.Functor E D} (adj₁ : F G) (adj₂ : H I) (X : E) :
                                                (adj₁.comp adj₂).counit.app X = CategoryTheory.CategoryStruct.comp (H.map (adj₁.counit.app (I.obj X))) (adj₂.counit.app X)
                                                theorem CategoryTheory.Adjunction.comp_homEquiv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} {E : Type u₃} [ℰ : CategoryTheory.Category.{v₃, u₃} E] {H : CategoryTheory.Functor D E} {I : CategoryTheory.Functor E D} (adj₁ : F G) (adj₂ : H I) :
                                                (adj₁.comp adj₂).homEquiv = fun (x : C) (x_1 : E) => (adj₂.homEquiv (F.obj x) x_1).trans (adj₁.homEquiv x (I.obj x_1))
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                                                theorem CategoryTheory.Adjunction.leftAdjointOfEquiv_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {G : CategoryTheory.Functor D C} {F_obj : CD} (e : (X : C) → (Y : D) → (F_obj X Y) (X G.obj Y)) (he : ∀ (X : C) (Y Y' : D) (g : Y Y') (h : F_obj X Y), (e X Y') (CategoryTheory.CategoryStruct.comp h g) = CategoryTheory.CategoryStruct.comp ((e X Y) h) (G.map g)) {X : C} {X' : C} (f : X X') :
                                                (CategoryTheory.Adjunction.leftAdjointOfEquiv e he).map f = (e X (F_obj X')).symm (CategoryTheory.CategoryStruct.comp f ((e X' (F_obj X')) (CategoryTheory.CategoryStruct.id (F_obj X'))))
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                                                theorem CategoryTheory.Adjunction.leftAdjointOfEquiv_obj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {G : CategoryTheory.Functor D C} {F_obj : CD} (e : (X : C) → (Y : D) → (F_obj X Y) (X G.obj Y)) (he : ∀ (X : C) (Y Y' : D) (g : Y Y') (h : F_obj X Y), (e X Y') (CategoryTheory.CategoryStruct.comp h g) = CategoryTheory.CategoryStruct.comp ((e X Y) h) (G.map g)) :
                                                ∀ (a : C), (CategoryTheory.Adjunction.leftAdjointOfEquiv e he).obj a = F_obj a
                                                def CategoryTheory.Adjunction.leftAdjointOfEquiv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {G : CategoryTheory.Functor D C} {F_obj : CD} (e : (X : C) → (Y : D) → (F_obj X Y) (X G.obj Y)) (he : ∀ (X : C) (Y Y' : D) (g : Y Y') (h : F_obj X Y), (e X Y') (CategoryTheory.CategoryStruct.comp h g) = CategoryTheory.CategoryStruct.comp ((e X Y) h) (G.map g)) :

                                                Construct a left adjoint functor to G, given the functor's value on objects F_obj and a bijection e between F_obj X ⟶ Y and X ⟶ G.obj Y satisfying a naturality law he : ∀ X Y Y' g h, e X Y' (h ≫ g) = e X Y h ≫ G.map g. Dual to rightAdjointOfEquiv.

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                                                  theorem CategoryTheory.Adjunction.adjunctionOfEquivLeft_counit_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {G : CategoryTheory.Functor D C} {F_obj : CD} (e : (X : C) → (Y : D) → (F_obj X Y) (X G.obj Y)) (he : ∀ (X : C) (Y Y' : D) (g : Y Y') (h : F_obj X Y), (e X Y') (CategoryTheory.CategoryStruct.comp h g) = CategoryTheory.CategoryStruct.comp ((e X Y) h) (G.map g)) (Y : D) :
                                                  (CategoryTheory.Adjunction.adjunctionOfEquivLeft e he).counit.app Y = (e (G.obj Y) Y).symm (CategoryTheory.CategoryStruct.id (G.obj Y))
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                                                  theorem CategoryTheory.Adjunction.adjunctionOfEquivLeft_unit_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {G : CategoryTheory.Functor D C} {F_obj : CD} (e : (X : C) → (Y : D) → (F_obj X Y) (X G.obj Y)) (he : ∀ (X : C) (Y Y' : D) (g : Y Y') (h : F_obj X Y), (e X Y') (CategoryTheory.CategoryStruct.comp h g) = CategoryTheory.CategoryStruct.comp ((e X Y) h) (G.map g)) (X : C) :
                                                  def CategoryTheory.Adjunction.adjunctionOfEquivLeft {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {G : CategoryTheory.Functor D C} {F_obj : CD} (e : (X : C) → (Y : D) → (F_obj X Y) (X G.obj Y)) (he : ∀ (X : C) (Y Y' : D) (g : Y Y') (h : F_obj X Y), (e X Y') (CategoryTheory.CategoryStruct.comp h g) = CategoryTheory.CategoryStruct.comp ((e X Y) h) (G.map g)) :

                                                  Show that the functor given by leftAdjointOfEquiv is indeed left adjoint to G. Dual to adjunctionOfRightEquiv.

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                                                    theorem CategoryTheory.Adjunction.rightAdjointOfEquiv_obj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G_obj : DC} (e : (X : C) → (Y : D) → (F.obj X Y) (X G_obj Y)) (he : ∀ (X' X : C) (Y : D) (f : X' X) (g : F.obj X Y), (e X' Y) (CategoryTheory.CategoryStruct.comp (F.map f) g) = CategoryTheory.CategoryStruct.comp f ((e X Y) g)) :
                                                    ∀ (a : D), (CategoryTheory.Adjunction.rightAdjointOfEquiv e he).obj a = G_obj a
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                                                    theorem CategoryTheory.Adjunction.rightAdjointOfEquiv_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G_obj : DC} (e : (X : C) → (Y : D) → (F.obj X Y) (X G_obj Y)) (he : ∀ (X' X : C) (Y : D) (f : X' X) (g : F.obj X Y), (e X' Y) (CategoryTheory.CategoryStruct.comp (F.map f) g) = CategoryTheory.CategoryStruct.comp f ((e X Y) g)) {Y : D} {Y' : D} (g : Y Y') :
                                                    def CategoryTheory.Adjunction.rightAdjointOfEquiv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G_obj : DC} (e : (X : C) → (Y : D) → (F.obj X Y) (X G_obj Y)) (he : ∀ (X' X : C) (Y : D) (f : X' X) (g : F.obj X Y), (e X' Y) (CategoryTheory.CategoryStruct.comp (F.map f) g) = CategoryTheory.CategoryStruct.comp f ((e X Y) g)) :

                                                    Construct a right adjoint functor to F, given the functor's value on objects G_obj and a bijection e between F.obj X ⟶ Y and X ⟶ G_obj Y satisfying a naturality law he : ∀ X Y Y' g h, e X' Y (F.map f ≫ g) = f ≫ e X Y g. Dual to leftAdjointOfEquiv.

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                                                      theorem CategoryTheory.Adjunction.adjunctionOfEquivRight_counit_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G_obj : DC} (e : (X : C) → (Y : D) → (F.obj X Y) (X G_obj Y)) (he : ∀ (X' X : C) (Y : D) (f : X' X) (g : F.obj X Y), (e X' Y) (CategoryTheory.CategoryStruct.comp (F.map f) g) = CategoryTheory.CategoryStruct.comp f ((e X Y) g)) (Y : D) :
                                                      (CategoryTheory.Adjunction.adjunctionOfEquivRight e he).counit.app Y = (e (G_obj Y) Y).symm (CategoryTheory.CategoryStruct.id (G_obj Y))
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                                                      theorem CategoryTheory.Adjunction.adjunctionOfEquivRight_unit_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G_obj : DC} (e : (X : C) → (Y : D) → (F.obj X Y) (X G_obj Y)) (he : ∀ (X' X : C) (Y : D) (f : X' X) (g : F.obj X Y), (e X' Y) (CategoryTheory.CategoryStruct.comp (F.map f) g) = CategoryTheory.CategoryStruct.comp f ((e X Y) g)) (X : C) :
                                                      def CategoryTheory.Adjunction.adjunctionOfEquivRight {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G_obj : DC} (e : (X : C) → (Y : D) → (F.obj X Y) (X G_obj Y)) (he : ∀ (X' X : C) (Y : D) (f : X' X) (g : F.obj X Y), (e X' Y) (CategoryTheory.CategoryStruct.comp (F.map f) g) = CategoryTheory.CategoryStruct.comp f ((e X Y) g)) :

                                                      Show that the functor given by rightAdjointOfEquiv is indeed right adjoint to F. Dual to adjunctionOfEquivRight.

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                                                        theorem CategoryTheory.Adjunction.toEquivalence_counitIso_inv_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) [∀ (X : C), CategoryTheory.IsIso (adj.unit.app X)] [∀ (Y : D), CategoryTheory.IsIso (adj.counit.app Y)] (X : D) :
                                                        adj.toEquivalence.counitIso.inv.app X = CategoryTheory.inv (adj.counit.app X)
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                                                        theorem CategoryTheory.Adjunction.toEquivalence_inverse {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) [∀ (X : C), CategoryTheory.IsIso (adj.unit.app X)] [∀ (Y : D), CategoryTheory.IsIso (adj.counit.app Y)] :
                                                        adj.toEquivalence.inverse = G
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                                                        theorem CategoryTheory.Adjunction.toEquivalence_counitIso_hom_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) [∀ (X : C), CategoryTheory.IsIso (adj.unit.app X)] [∀ (Y : D), CategoryTheory.IsIso (adj.counit.app Y)] (X : D) :
                                                        adj.toEquivalence.counitIso.hom.app X = adj.counit.app X
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                                                        theorem CategoryTheory.Adjunction.toEquivalence_unitIso_hom_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) [∀ (X : C), CategoryTheory.IsIso (adj.unit.app X)] [∀ (Y : D), CategoryTheory.IsIso (adj.counit.app Y)] (X : C) :
                                                        adj.toEquivalence.unitIso.hom.app X = adj.unit.app X
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                                                        theorem CategoryTheory.Adjunction.toEquivalence_functor {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) [∀ (X : C), CategoryTheory.IsIso (adj.unit.app X)] [∀ (Y : D), CategoryTheory.IsIso (adj.counit.app Y)] :
                                                        adj.toEquivalence.functor = F
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                                                        theorem CategoryTheory.Adjunction.toEquivalence_unitIso_inv_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) [∀ (X : C), CategoryTheory.IsIso (adj.unit.app X)] [∀ (Y : D), CategoryTheory.IsIso (adj.counit.app Y)] (X : C) :
                                                        adj.toEquivalence.unitIso.inv.app X = CategoryTheory.inv (adj.unit.app X)
                                                        noncomputable def CategoryTheory.Adjunction.toEquivalence {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) [∀ (X : C), CategoryTheory.IsIso (adj.unit.app X)] [∀ (Y : D), CategoryTheory.IsIso (adj.counit.app Y)] :
                                                        C D

                                                        If the unit and counit of a given adjunction are (pointwise) isomorphisms, then we can upgrade the adjunction to an equivalence.

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                                                          If the unit and counit for the adjunction corresponding to a right adjoint functor are (pointwise) isomorphisms, then the functor is an equivalence of categories.

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                                                          The adjunction given by an equivalence of categories. (To obtain the opposite adjunction, simply use e.symm.toAdjunction.

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                                                          • e.toAdjunction = { unit := e.unit, counit := e.counit, left_triangle_components := , right_triangle_components := }
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                                                            instance CategoryTheory.Functor.isLeftAdjoint_comp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] (F : CategoryTheory.Functor C D) (G : CategoryTheory.Functor D E) [F.IsLeftAdjoint] [G.IsLeftAdjoint] :
                                                            (F.comp G).IsLeftAdjoint

                                                            If F and G are left adjoints then F ⋙ G is a left adjoint too.

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                                                            instance CategoryTheory.Functor.isRightAdjoint_comp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D E} [F.IsRightAdjoint] [G.IsRightAdjoint] :
                                                            (F.comp G).IsRightAdjoint

                                                            If F and G are right adjoints then F ⋙ G is a right adjoint too.

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                                                            Transport being a right adjoint along a natural isomorphism.

                                                            Transport being a left adjoint along a natural isomorphism.

                                                            An equivalence E is left adjoint to its inverse.

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                                                            • E.adjunction = E.asEquivalence.toAdjunction
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                                                              If F is an equivalence, it's a left adjoint.

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                                                              If F is an equivalence, it's a right adjoint.

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