Opposite adjunctions #

This file contains constructions to relate adjunctions of functors to adjunctions of their opposites.

Tags #

adjunction, opposite, uniqueness

def CategoryTheory.Adjunction.unop {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {F : Functor Cᵒᵖ Dᵒᵖ} {G : Functor Dᵒᵖ Cᵒᵖ} (h : G ⊣ F) :

F.unop ⊣ G.unop

If G is adjoint to F then F.unop is adjoint to G.unop.

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@[simp]

theorem CategoryTheory.Adjunction.unop_counit {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {F : Functor Cᵒᵖ Dᵒᵖ} {G : Functor Dᵒᵖ Cᵒᵖ} (h : G ⊣ F) :

h.unop.counit = NatTrans.unop h.unit

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@[simp]

theorem CategoryTheory.Adjunction.unop_unit {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {F : Functor Cᵒᵖ Dᵒᵖ} {G : Functor Dᵒᵖ Cᵒᵖ} (h : G ⊣ F) :

h.unop.unit = NatTrans.unop h.counit

source

def CategoryTheory.Adjunction.op {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {F : Functor C D} {G : Functor D C} (h : G ⊣ F) :

F.op ⊣ G.op

If G is adjoint to F then F.op is adjoint to G.op.

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@[simp]

theorem CategoryTheory.Adjunction.op_unit {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {F : Functor C D} {G : Functor D C} (h : G ⊣ F) :

h.op.unit = NatTrans.op h.counit

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@[simp]

theorem CategoryTheory.Adjunction.op_counit {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {F : Functor C D} {G : Functor D C} (h : G ⊣ F) :

h.op.counit = NatTrans.op h.unit

source

def CategoryTheory.Adjunction.leftOp {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {F : Functor C Dᵒᵖ} {G : Functor D Cᵒᵖ} (a : F ⊣ G.leftOp) :

G ⊣ F.leftOp

If F is adjoint to G.leftOp then G is adjoint to F.leftOp.

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@[simp]

theorem CategoryTheory.Adjunction.leftOp_counit {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {F : Functor C Dᵒᵖ} {G : Functor D Cᵒᵖ} (a : F ⊣ G.leftOp) :

a.leftOp.counit = NatTrans.op a.unit

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@[simp]

theorem CategoryTheory.Adjunction.leftOp_unit {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {F : Functor C Dᵒᵖ} {G : Functor D Cᵒᵖ} (a : F ⊣ G.leftOp) :

a.leftOp.unit = NatTrans.unop a.counit

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def CategoryTheory.Adjunction.rightOp {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {F : Functor Cᵒᵖ D} {G : Functor Dᵒᵖ C} (a : F.rightOp ⊣ G) :

G.rightOp ⊣ F

If F.rightOp is adjoint to G then G.rightOp is adjoint to F.

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@[simp]

theorem CategoryTheory.Adjunction.rightOp_counit {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {F : Functor Cᵒᵖ D} {G : Functor Dᵒᵖ C} (a : F.rightOp ⊣ G) :

a.rightOp.counit = NatTrans.op a.unit

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@[simp]

theorem CategoryTheory.Adjunction.rightOp_unit {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {F : Functor Cᵒᵖ D} {G : Functor Dᵒᵖ C} (a : F.rightOp ⊣ G) :

a.rightOp.unit = NatTrans.unop a.counit

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theorem CategoryTheory.Adjunction.leftOp_eq {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {F : Functor C Dᵒᵖ} {G : Functor D Cᵒᵖ} (a : F ⊣ G.leftOp) :

a.leftOp = (opOpEquivalence D).symm.toAdjunction.comp a.op

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theorem CategoryTheory.Adjunction.rightOp_eq {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {F : Functor Cᵒᵖ D} {G : Functor Dᵒᵖ C} (a : F.rightOp ⊣ G) :

a.rightOp = (opOpEquivalence D).symm.toAdjunction.comp a.op

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def CategoryTheory.Adjunction.leftAdjointsCoyonedaEquiv {C : Type u₁} [Category.{v₁, u₁} C] {D : Type u₂} [Category.{v₂, u₂} D] {F F' : Functor C D} {G : Functor D C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G) :

F.op.comp coyoneda ≅ F'.op.comp coyoneda

If F and F' are both adjoint to G, there is a natural isomorphism F.op ⋙ coyoneda ≅ F'.op ⋙ coyoneda. We use this in combination with fullyFaithfulCancelRight to show left adjoints are unique.

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@[deprecated "Use `(Adjunction.conjugateIsoEquiv adj1 adj2).symm` (requires `import Mathlib.CategoryTheory.Adjunction.Mates`)." (since := "2026-01-31")]