Documentation

Mathlib.CategoryTheory.Opposites

Opposite categories #

We provide a category instance on Cᵒᵖ. The morphisms X ⟶ Y are defined to be the morphisms unop Y ⟶ unop X in C.

Here Cᵒᵖ is an irreducible typeclass synonym for C (it is the same one used in the algebra library).

We also provide various mechanisms for constructing opposite morphisms, functors, and natural transformations.

Unfortunately, because we do not have a definitional equality op (op X) = X, there are quite a few variations that are needed in practice.

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theorem Quiver.Hom.op_inj {C : Type u₁} [Quiver C] {X : C} {Y : C} :
Function.Injective Quiver.Hom.op
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theorem Quiver.Hom.unop_inj {C : Type u₁} [Quiver C] {X : Cᵒᵖ} {Y : Cᵒᵖ} :
Function.Injective Quiver.Hom.unop
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@[simp]
theorem Quiver.Hom.unop_op {C : Type u₁} [Quiver C] {X : C} {Y : C} (f : X ⟶ Y) :
f.op.unop = f
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@[simp]
theorem Quiver.Hom.op_unop {C : Type u₁} [Quiver C] {X : Cᵒᵖ} {Y : Cᵒᵖ} (f : X ⟶ Y) :
f.unop.op = f
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instance CategoryTheory.Category.opposite {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] :
CategoryTheory.Category.{v₁, u₁} Cᵒᵖ

The opposite category.

See .

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theorem CategoryTheory.op_comp_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {Y : C} {Z : C} {f : X ⟶ Y} {g : Y ⟶ Z✝} {Z : Cᵒᵖ} (h : Opposite.op X ⟶ Z) :
CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.comp f g).op h = CategoryTheory.CategoryStruct.comp g.op (CategoryTheory.CategoryStruct.comp f.op h)
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@[simp]
theorem CategoryTheory.op_comp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {Y : C} {Z : C} {f : X ⟶ Y} {g : Y ⟶ Z} :
(CategoryTheory.CategoryStruct.comp f g).op = CategoryTheory.CategoryStruct.comp g.op f.op
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@[simp]
theorem CategoryTheory.op_id {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} :
(CategoryTheory.CategoryStruct.id X).op = CategoryTheory.CategoryStruct.id (Opposite.op X)
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theorem CategoryTheory.unop_comp_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : Cᵒᵖ} {Y : Cᵒᵖ} {Z : Cᵒᵖ} {f : X ⟶ Y} {g : Y ⟶ Z✝} {Z : C} (h : X.unop ⟶ Z) :
CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.comp f g).unop h = CategoryTheory.CategoryStruct.comp g.unop (CategoryTheory.CategoryStruct.comp f.unop h)
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@[simp]
theorem CategoryTheory.unop_comp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : Cᵒᵖ} {Y : Cᵒᵖ} {Z : Cᵒᵖ} {f : X ⟶ Y} {g : Y ⟶ Z} :
(CategoryTheory.CategoryStruct.comp f g).unop = CategoryTheory.CategoryStruct.comp g.unop f.unop
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@[simp]
theorem CategoryTheory.unop_id {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : Cᵒᵖ} :
(CategoryTheory.CategoryStruct.id X).unop = CategoryTheory.CategoryStruct.id X.unop
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@[simp]
theorem CategoryTheory.unop_id_op {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} :
(CategoryTheory.CategoryStruct.id (Opposite.op X)).unop = CategoryTheory.CategoryStruct.id X
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@[simp]
theorem CategoryTheory.op_id_unop {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : Cᵒᵖ} :
(CategoryTheory.CategoryStruct.id X.unop).op = CategoryTheory.CategoryStruct.id X
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@[simp]
theorem CategoryTheory.unopUnop_map (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] :
∀ {X Y : Cᵒᵖᵒᵖ} (f : X ⟶ Y), (CategoryTheory.unopUnop C).map f = f.unop.unop
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@[simp]
theorem CategoryTheory.unopUnop_obj (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (X : Cᵒᵖᵒᵖ) :
(CategoryTheory.unopUnop C).obj X = X.unop.unop
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def CategoryTheory.unopUnop (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] :
CategoryTheory.Functor Cᵒᵖᵒᵖ C

The functor from the double-opposite of a category to the underlying category.

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@[simp]
theorem CategoryTheory.opOp_map (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] :
∀ {X Y : C} (f : X ⟶ Y), (CategoryTheory.opOp C).map f = f.op.op
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@[simp]
theorem CategoryTheory.opOp_obj (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (X : C) :
(CategoryTheory.opOp C).obj X = Opposite.op (Opposite.op X)
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def CategoryTheory.opOp (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] :
CategoryTheory.Functor C Cᵒᵖᵒᵖ

The functor from a category to its double-opposite.

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@[simp]
theorem CategoryTheory.opOpEquivalence_counitIso (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] :
(CategoryTheory.opOpEquivalence C).counitIso = CategoryTheory.Iso.refl (CategoryTheory.Functor.comp (CategoryTheory.opOp C) (CategoryTheory.unopUnop C))
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@[simp]
theorem CategoryTheory.opOpEquivalence_inverse (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] :
(CategoryTheory.opOpEquivalence C).inverse = CategoryTheory.opOp C
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@[simp]
theorem CategoryTheory.opOpEquivalence_unitIso (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] :
(CategoryTheory.opOpEquivalence C).unitIso = CategoryTheory.Iso.refl (CategoryTheory.Functor.id Cᵒᵖᵒᵖ)
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@[simp]
theorem CategoryTheory.opOpEquivalence_functor (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] :
(CategoryTheory.opOpEquivalence C).functor = CategoryTheory.unopUnop C
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def CategoryTheory.opOpEquivalence (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] :
Cᵒᵖᵒᵖ ≌ C

The double opposite category is equivalent to the original.

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  • One or more equations did not get rendered due to their size.
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instance CategoryTheory.isIso_op {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {Y : C} (f : X ⟶ Y) [CategoryTheory.IsIso f] :
CategoryTheory.IsIso f.op

If f is an isomorphism, so is f.op

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  • ⋯ = ⋯
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theorem CategoryTheory.isIso_of_op {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {Y : C} (f : X ⟶ Y) [CategoryTheory.IsIso f.op] :
CategoryTheory.IsIso f

If f.op is an isomorphism f must be too. (This cannot be an instance as it would immediately loop!)

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theorem CategoryTheory.isIso_op_iff {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {Y : C} (f : X ⟶ Y) :
CategoryTheory.IsIso f.op ↔ CategoryTheory.IsIso f
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theorem CategoryTheory.isIso_unop_iff {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : Cᵒᵖ} {Y : Cᵒᵖ} (f : X ⟶ Y) :
CategoryTheory.IsIso f.unop ↔ CategoryTheory.IsIso f
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instance CategoryTheory.isIso_unop {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : Cᵒᵖ} {Y : Cᵒᵖ} (f : X ⟶ Y) [CategoryTheory.IsIso f] :
CategoryTheory.IsIso f.unop
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  • ⋯ = ⋯
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@[simp]
theorem CategoryTheory.op_inv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {Y : C} (f : X ⟶ Y) [CategoryTheory.IsIso f] :
(CategoryTheory.inv f).op = CategoryTheory.inv f.op
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@[simp]
theorem CategoryTheory.unop_inv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : Cᵒᵖ} {Y : Cᵒᵖ} (f : X ⟶ Y) [CategoryTheory.IsIso f] :
(CategoryTheory.inv f).unop = CategoryTheory.inv f.unop
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@[simp]
theorem CategoryTheory.Functor.op_obj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) (X : Cᵒᵖ) :
F.op.obj X = Opposite.op (F.obj X.unop)
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@[simp]
theorem CategoryTheory.Functor.op_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) :
∀ {X Y : Cᵒᵖ} (f : X ⟶ Y), F.op.map f = (F.map f.unop).op
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def CategoryTheory.Functor.op {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) :
CategoryTheory.Functor Cᵒᵖ Dᵒᵖ

The opposite of a functor, i.e. considering a functor F : C ⥤ D as a functor Cᵒᵖ ⥤ Dᵒᵖ. In informal mathematics no distinction is made between these.

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  • F.op = { toPrefunctor := { obj := fun (X : Cᵒᵖ) => Opposite.op (F.obj X.unop), map := fun {X Y : Cᵒᵖ} (f : X ⟶ Y) => (F.map f.unop).op }, map_id := ⋯, map_comp := ⋯ }
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@[simp]
theorem CategoryTheory.Functor.unop_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ) :
∀ {X Y : C} (f : X ⟶ Y), (CategoryTheory.Functor.unop F).map f = (F.map f.op).unop
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@[simp]
theorem CategoryTheory.Functor.unop_obj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ) (X : C) :
(CategoryTheory.Functor.unop F).obj X = (F.obj (Opposite.op X)).unop
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def CategoryTheory.Functor.unop {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ) :
CategoryTheory.Functor C D

Given a functor F : Cᵒᵖ ⥤ Dᵒᵖ we can take the "unopposite" functor F : C ⥤ D. In informal mathematics no distinction is made between these.

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  • One or more equations did not get rendered due to their size.
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@[simp]
theorem CategoryTheory.Functor.opUnopIso_inv_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) (X : C) :
(CategoryTheory.Functor.opUnopIso F).inv.app X = CategoryTheory.CategoryStruct.id (F.obj X)
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@[simp]
theorem CategoryTheory.Functor.opUnopIso_hom_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) (X : C) :
(CategoryTheory.Functor.opUnopIso F).hom.app X = CategoryTheory.CategoryStruct.id (F.obj X)
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def CategoryTheory.Functor.opUnopIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) :
CategoryTheory.Functor.unop F.op ≅ F

The isomorphism between F.op.unop and F.

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theorem CategoryTheory.Functor.unopOpIso_hom_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ) (X : Cᵒᵖ) :
(CategoryTheory.Functor.unopOpIso F).hom.app X = CategoryTheory.CategoryStruct.id (F.obj X)
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@[simp]
theorem CategoryTheory.Functor.unopOpIso_inv_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ) (X : Cᵒᵖ) :
(CategoryTheory.Functor.unopOpIso F).inv.app X = CategoryTheory.CategoryStruct.id (F.obj X)
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def CategoryTheory.Functor.unopOpIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ) :
(CategoryTheory.Functor.unop F).op ≅ F

The isomorphism between F.unop.op and F.

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@[simp]
theorem CategoryTheory.Functor.opHom_map_app (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] :
∀ {X Y : (CategoryTheory.Functor C D)ᵒᵖ} (α : X ⟶ Y) (X_1 : Cᵒᵖ), ((CategoryTheory.Functor.opHom C D).map α).app X_1 = (α.unop.app X_1.unop).op
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@[simp]
theorem CategoryTheory.Functor.opHom_obj (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] (F : (CategoryTheory.Functor C D)ᵒᵖ) :
(CategoryTheory.Functor.opHom C D).obj F = F.unop.op
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def CategoryTheory.Functor.opHom (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] :
CategoryTheory.Functor (CategoryTheory.Functor C D)ᵒᵖ (CategoryTheory.Functor Cᵒᵖ Dᵒᵖ)

Taking the opposite of a functor is functorial.

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  • One or more equations did not get rendered due to their size.
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@[simp]
theorem CategoryTheory.Functor.opInv_map (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] :
∀ {X Y : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ} (α : X ⟶ Y), (CategoryTheory.Functor.opInv C D).map α = { app := fun (X_1 : C) => (α.app (Opposite.op X_1)).unop, naturality := ⋯ }.op
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@[simp]
theorem CategoryTheory.Functor.opInv_obj (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ) :
(CategoryTheory.Functor.opInv C D).obj F = Opposite.op (CategoryTheory.Functor.unop F)
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def CategoryTheory.Functor.opInv (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] :
CategoryTheory.Functor (CategoryTheory.Functor Cᵒᵖ Dᵒᵖ) (CategoryTheory.Functor C D)ᵒᵖ

Take the "unopposite" of a functor is functorial.

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  • One or more equations did not get rendered due to their size.
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@[simp]
theorem CategoryTheory.Functor.leftOp_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C Dᵒᵖ) :
∀ {X Y : Cᵒᵖ} (f : X ⟶ Y), F.leftOp.map f = (F.map f.unop).unop
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@[simp]
theorem CategoryTheory.Functor.leftOp_obj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C Dᵒᵖ) (X : Cᵒᵖ) :
F.leftOp.obj X = (F.obj X.unop).unop
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def CategoryTheory.Functor.leftOp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C Dᵒᵖ) :
CategoryTheory.Functor Cᵒᵖ D

Another variant of the opposite of functor, turning a functor C ⥤ Dᵒᵖ into a functor Cᵒᵖ ⥤ D. In informal mathematics no distinction is made.

Equations
  • F.leftOp = { toPrefunctor := { obj := fun (X : Cᵒᵖ) => (F.obj X.unop).unop, map := fun {X Y : Cᵒᵖ} (f : X ⟶ Y) => (F.map f.unop).unop }, map_id := ⋯, map_comp := ⋯ }
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@[simp]
theorem CategoryTheory.Functor.rightOp_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor Cᵒᵖ D) :
∀ {X Y : C} (f : X ⟶ Y), F.rightOp.map f = (F.map f.op).op
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@[simp]
theorem CategoryTheory.Functor.rightOp_obj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor Cᵒᵖ D) (X : C) :
F.rightOp.obj X = Opposite.op (F.obj (Opposite.op X))
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def CategoryTheory.Functor.rightOp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor Cᵒᵖ D) :
CategoryTheory.Functor C Dᵒᵖ

Another variant of the opposite of functor, turning a functor Cᵒᵖ ⥤ D into a functor C ⥤ Dᵒᵖ. In informal mathematics no distinction is made.

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  • One or more equations did not get rendered due to their size.
Instances For
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instance CategoryTheory.Functor.instFullOppositeOppositeOppositeOppositeOp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} [CategoryTheory.Full F] :
CategoryTheory.Full F.op
Equations
  • CategoryTheory.Functor.instFullOppositeOppositeOppositeOppositeOp = { preimage := fun (X Y : Cᵒᵖ) (f : F.op.obj X ⟶ F.op.obj Y) => (F.preimage f.unop).op, witness := ⋯ }
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instance CategoryTheory.Functor.instFaithfulOppositeOppositeOppositeOppositeOp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} [CategoryTheory.Faithful F] :
CategoryTheory.Faithful F.op
Equations
  • ⋯ = ⋯
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instance CategoryTheory.Functor.rightOp_faithful {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor Cᵒᵖ D} [CategoryTheory.Faithful F] :
CategoryTheory.Faithful F.rightOp

If F is faithful then the right_op of F is also faithful.

Equations
  • ⋯ = ⋯
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instance CategoryTheory.Functor.leftOp_faithful {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C Dᵒᵖ} [CategoryTheory.Faithful F] :
CategoryTheory.Faithful F.leftOp

If F is faithful then the left_op of F is also faithful.

Equations
  • ⋯ = ⋯
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@[simp]
theorem CategoryTheory.Functor.leftOpRightOpIso_inv_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C Dᵒᵖ) (X : C) :
(CategoryTheory.Functor.leftOpRightOpIso F).inv.app X = CategoryTheory.CategoryStruct.id (F.obj X)
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@[simp]
theorem CategoryTheory.Functor.leftOpRightOpIso_hom_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C Dᵒᵖ) (X : C) :
(CategoryTheory.Functor.leftOpRightOpIso F).hom.app X = CategoryTheory.CategoryStruct.id (F.obj X)
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def CategoryTheory.Functor.leftOpRightOpIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C Dᵒᵖ) :
F.leftOp.rightOp ≅ F

The isomorphism between F.leftOp.rightOp and F.

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@[simp]
theorem CategoryTheory.Functor.rightOpLeftOpIso_hom_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor Cᵒᵖ D) (X : Cᵒᵖ) :
(CategoryTheory.Functor.rightOpLeftOpIso F).hom.app X = CategoryTheory.CategoryStruct.id (F.obj X)
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@[simp]
theorem CategoryTheory.Functor.rightOpLeftOpIso_inv_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor Cᵒᵖ D) (X : Cᵒᵖ) :
(CategoryTheory.Functor.rightOpLeftOpIso F).inv.app X = CategoryTheory.CategoryStruct.id (F.obj X)
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def CategoryTheory.Functor.rightOpLeftOpIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor Cᵒᵖ D) :
F.rightOp.leftOp ≅ F

The isomorphism between F.rightOp.leftOp and F.

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theorem CategoryTheory.Functor.rightOp_leftOp_eq {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor Cᵒᵖ D) :
F.rightOp.leftOp = F

Whenever possible, it is advisable to use the isomorphism rightOpLeftOpIso instead of this equality of functors.

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@[simp]
theorem CategoryTheory.NatTrans.op_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 C D} (α : F ⟶ G) (X : Cᵒᵖ) :
(CategoryTheory.NatTrans.op α).app X = (α.app X.unop).op
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def CategoryTheory.NatTrans.op {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor C D} (α : F ⟶ G) :
G.op ⟶ F.op

The opposite of a natural transformation.

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@[simp]
theorem CategoryTheory.NatTrans.op_id {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) :
CategoryTheory.NatTrans.op (CategoryTheory.CategoryStruct.id F) = CategoryTheory.CategoryStruct.id F.op
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@[simp]
theorem CategoryTheory.NatTrans.unop_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ} {G : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ} (α : F ⟶ G) (X : C) :
(CategoryTheory.NatTrans.unop α).app X = (α.app (Opposite.op X)).unop
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def CategoryTheory.NatTrans.unop {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ} {G : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ} (α : F ⟶ G) :
CategoryTheory.Functor.unop G ⟶ CategoryTheory.Functor.unop F

The "unopposite" of a natural transformation.

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@[simp]
theorem CategoryTheory.NatTrans.unop_id {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ) :
CategoryTheory.NatTrans.unop (CategoryTheory.CategoryStruct.id F) = CategoryTheory.CategoryStruct.id (CategoryTheory.Functor.unop F)
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@[simp]
theorem CategoryTheory.NatTrans.removeOp_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 C D} (α : F.op ⟶ G.op) (X : C) :
(CategoryTheory.NatTrans.removeOp α).app X = (α.app (Opposite.op X)).unop
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def CategoryTheory.NatTrans.removeOp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor C D} (α : F.op ⟶ G.op) :
G ⟶ F

Given a natural transformation α : F.op ⟶ G.op, we can take the "unopposite" of each component obtaining a natural transformation G ⟶ F.

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@[simp]
theorem CategoryTheory.NatTrans.removeOp_id {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) :
CategoryTheory.NatTrans.removeOp (CategoryTheory.CategoryStruct.id F.op) = CategoryTheory.CategoryStruct.id F
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@[simp]
theorem CategoryTheory.NatTrans.removeUnop_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ} {G : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ} (α : CategoryTheory.Functor.unop F ⟶ CategoryTheory.Functor.unop G) (X : Cᵒᵖ) :
(CategoryTheory.NatTrans.removeUnop α).app X = (α.app X.unop).op
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def CategoryTheory.NatTrans.removeUnop {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ} {G : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ} (α : CategoryTheory.Functor.unop F ⟶ CategoryTheory.Functor.unop G) :
G ⟶ F

Given a natural transformation α : F.unop ⟶ G.unop, we can take the opposite of each component obtaining a natural transformation G ⟶ F.

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@[simp]
theorem CategoryTheory.NatTrans.removeUnop_id {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ) :
CategoryTheory.NatTrans.removeUnop (CategoryTheory.CategoryStruct.id (CategoryTheory.Functor.unop F)) = CategoryTheory.CategoryStruct.id F
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@[simp]
theorem CategoryTheory.NatTrans.leftOp_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C Dᵒᵖ} {G : CategoryTheory.Functor C Dᵒᵖ} (α : F ⟶ G) (X : Cᵒᵖ) :
(CategoryTheory.NatTrans.leftOp α).app X = (α.app X.unop).unop
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def CategoryTheory.NatTrans.leftOp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C Dᵒᵖ} {G : CategoryTheory.Functor C Dᵒᵖ} (α : F ⟶ G) :
G.leftOp ⟶ F.leftOp

Given a natural transformation α : F ⟶ G, for F G : C ⥤ Dᵒᵖ, taking unop of each component gives a natural transformation G.leftOp ⟶ F.leftOp.

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@[simp]
theorem CategoryTheory.NatTrans.leftOp_id {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C Dᵒᵖ} :
CategoryTheory.NatTrans.leftOp (CategoryTheory.CategoryStruct.id F) = CategoryTheory.CategoryStruct.id F.leftOp
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@[simp]
theorem CategoryTheory.NatTrans.leftOp_comp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C Dᵒᵖ} {G : CategoryTheory.Functor C Dᵒᵖ} {H : CategoryTheory.Functor C Dᵒᵖ} (α : F ⟶ G) (β : G ⟶ H) :
CategoryTheory.NatTrans.leftOp (CategoryTheory.CategoryStruct.comp α β) = CategoryTheory.CategoryStruct.comp (CategoryTheory.NatTrans.leftOp β) (CategoryTheory.NatTrans.leftOp α)
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@[simp]
theorem CategoryTheory.NatTrans.removeLeftOp_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C Dᵒᵖ} {G : CategoryTheory.Functor C Dᵒᵖ} (α : F.leftOp ⟶ G.leftOp) (X : C) :
(CategoryTheory.NatTrans.removeLeftOp α).app X = (α.app (Opposite.op X)).op
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def CategoryTheory.NatTrans.removeLeftOp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C Dᵒᵖ} {G : CategoryTheory.Functor C Dᵒᵖ} (α : F.leftOp ⟶ G.leftOp) :
G ⟶ F

Given a natural transformation α : F.leftOp ⟶ G.leftOp, for F G : C ⥤ Dᵒᵖ, taking op of each component gives a natural transformation G ⟶ F.

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@[simp]
theorem CategoryTheory.NatTrans.removeLeftOp_id {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C Dᵒᵖ} :
CategoryTheory.NatTrans.removeLeftOp (CategoryTheory.CategoryStruct.id F.leftOp) = CategoryTheory.CategoryStruct.id F
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@[simp]
theorem CategoryTheory.NatTrans.rightOp_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor Cᵒᵖ D} {G : CategoryTheory.Functor Cᵒᵖ D} (α : F ⟶ G) (X : C) :
(CategoryTheory.NatTrans.rightOp α).app X = (α.app (Opposite.op X)).op
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def CategoryTheory.NatTrans.rightOp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor Cᵒᵖ D} {G : CategoryTheory.Functor Cᵒᵖ D} (α : F ⟶ G) :
G.rightOp ⟶ F.rightOp

Given a natural transformation α : F ⟶ G, for F G : Cᵒᵖ ⥤ D, taking op of each component gives a natural transformation G.rightOp ⟶ F.rightOp.

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@[simp]
theorem CategoryTheory.NatTrans.rightOp_id {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor Cᵒᵖ D} :
CategoryTheory.NatTrans.rightOp (CategoryTheory.CategoryStruct.id F) = CategoryTheory.CategoryStruct.id F.rightOp
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@[simp]
theorem CategoryTheory.NatTrans.rightOp_comp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor Cᵒᵖ D} {G : CategoryTheory.Functor Cᵒᵖ D} {H : CategoryTheory.Functor Cᵒᵖ D} (α : F ⟶ G) (β : G ⟶ H) :
CategoryTheory.NatTrans.rightOp (CategoryTheory.CategoryStruct.comp α β) = CategoryTheory.CategoryStruct.comp (CategoryTheory.NatTrans.rightOp β) (CategoryTheory.NatTrans.rightOp α)
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@[simp]
theorem CategoryTheory.NatTrans.removeRightOp_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor Cᵒᵖ D} {G : CategoryTheory.Functor Cᵒᵖ D} (α : F.rightOp ⟶ G.rightOp) (X : Cᵒᵖ) :
(CategoryTheory.NatTrans.removeRightOp α).app X = (α.app X.unop).unop
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def CategoryTheory.NatTrans.removeRightOp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor Cᵒᵖ D} {G : CategoryTheory.Functor Cᵒᵖ D} (α : F.rightOp ⟶ G.rightOp) :
G ⟶ F

Given a natural transformation α : F.rightOp ⟶ G.rightOp, for F G : Cᵒᵖ ⥤ D, taking unop of each component gives a natural transformation G ⟶ F.

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@[simp]
theorem CategoryTheory.NatTrans.removeRightOp_id {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor Cᵒᵖ D} :
CategoryTheory.NatTrans.removeRightOp (CategoryTheory.CategoryStruct.id F.rightOp) = CategoryTheory.CategoryStruct.id F
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@[simp]
theorem CategoryTheory.Iso.op_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {Y : C} (α : X ≅ Y) :
(CategoryTheory.Iso.op α).hom = α.hom.op
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@[simp]
theorem CategoryTheory.Iso.op_inv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {Y : C} (α : X ≅ Y) :
(CategoryTheory.Iso.op α).inv = α.inv.op
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def CategoryTheory.Iso.op {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {Y : C} (α : X ≅ Y) :
Opposite.op Y ≅ Opposite.op X

The opposite isomorphism.

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@[simp]
theorem CategoryTheory.Iso.unop_inv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : Cᵒᵖ} {Y : Cᵒᵖ} (f : X ≅ Y) :
(CategoryTheory.Iso.unop f).inv = f.inv.unop
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@[simp]
theorem CategoryTheory.Iso.unop_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : Cᵒᵖ} {Y : Cᵒᵖ} (f : X ≅ Y) :
(CategoryTheory.Iso.unop f).hom = f.hom.unop
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def CategoryTheory.Iso.unop {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : Cᵒᵖ} {Y : Cᵒᵖ} (f : X ≅ Y) :
Y.unop ≅ X.unop

The isomorphism obtained from an isomorphism in the opposite category.

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@[simp]
theorem CategoryTheory.Iso.unop_op {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : Cᵒᵖ} {Y : Cᵒᵖ} (f : X ≅ Y) :
CategoryTheory.Iso.op (CategoryTheory.Iso.unop f) = f
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@[simp]
theorem CategoryTheory.Iso.op_unop {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {X : C} {Y : C} (f : X ≅ Y) :
CategoryTheory.Iso.unop (CategoryTheory.Iso.op f) = f
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@[simp]
theorem CategoryTheory.Iso.unop_hom_inv_id_app_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u_1} [CategoryTheory.Category.{u_2, u_1} D] {F : CategoryTheory.Functor C Dᵒᵖ} {G : CategoryTheory.Functor C Dᵒᵖ} (e : F ≅ G) (X : C) {Z : D} (h : (G.obj X).unop ⟶ Z) :
CategoryTheory.CategoryStruct.comp (e.hom.app X).unop (CategoryTheory.CategoryStruct.comp (e.inv.app X).unop h) = h
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@[simp]
theorem CategoryTheory.Iso.unop_hom_inv_id_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u_1} [CategoryTheory.Category.{u_2, u_1} D] {F : CategoryTheory.Functor C Dᵒᵖ} {G : CategoryTheory.Functor C Dᵒᵖ} (e : F ≅ G) (X : C) :
CategoryTheory.CategoryStruct.comp (e.hom.app X).unop (e.inv.app X).unop = CategoryTheory.CategoryStruct.id (G.obj X).unop
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@[simp]
theorem CategoryTheory.Iso.unop_inv_hom_id_app_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u_1} [CategoryTheory.Category.{u_2, u_1} D] {F : CategoryTheory.Functor C Dᵒᵖ} {G : CategoryTheory.Functor C Dᵒᵖ} (e : F ≅ G) (X : C) {Z : D} (h : (F.obj X).unop ⟶ Z) :
CategoryTheory.CategoryStruct.comp (e.inv.app X).unop (CategoryTheory.CategoryStruct.comp (e.hom.app X).unop h) = h
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@[simp]
theorem CategoryTheory.Iso.unop_inv_hom_id_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u_1} [CategoryTheory.Category.{u_2, u_1} D] {F : CategoryTheory.Functor C Dᵒᵖ} {G : CategoryTheory.Functor C Dᵒᵖ} (e : F ≅ G) (X : C) :
CategoryTheory.CategoryStruct.comp (e.inv.app X).unop (e.hom.app X).unop = CategoryTheory.CategoryStruct.id (F.obj X).unop
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@[simp]
theorem CategoryTheory.NatIso.op_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor C D} (α : F ≅ G) :
(CategoryTheory.NatIso.op α).hom = CategoryTheory.NatTrans.op α.hom
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@[simp]
theorem CategoryTheory.NatIso.op_inv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor C D} (α : F ≅ G) :
(CategoryTheory.NatIso.op α).inv = CategoryTheory.NatTrans.op α.inv
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def CategoryTheory.NatIso.op {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor C D} (α : F ≅ G) :
G.op ≅ F.op

The natural isomorphism between opposite functors G.op ≅ F.op induced by a natural isomorphism between the original functors F ≅ G.

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@[simp]
theorem CategoryTheory.NatIso.removeOp_inv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor C D} (α : F.op ≅ G.op) :
(CategoryTheory.NatIso.removeOp α).inv = CategoryTheory.NatTrans.removeOp α.inv
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@[simp]
theorem CategoryTheory.NatIso.removeOp_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor C D} (α : F.op ≅ G.op) :
(CategoryTheory.NatIso.removeOp α).hom = CategoryTheory.NatTrans.removeOp α.hom
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def CategoryTheory.NatIso.removeOp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor C D} (α : F.op ≅ G.op) :
G ≅ F

The natural isomorphism between functors G ≅ F induced by a natural isomorphism between the opposite functors F.op ≅ G.op.

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@[simp]
theorem CategoryTheory.NatIso.unop_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ} {G : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ} (α : F ≅ G) :
(CategoryTheory.NatIso.unop α).hom = CategoryTheory.NatTrans.unop α.hom
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@[simp]
theorem CategoryTheory.NatIso.unop_inv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ} {G : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ} (α : F ≅ G) :
(CategoryTheory.NatIso.unop α).inv = CategoryTheory.NatTrans.unop α.inv
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def CategoryTheory.NatIso.unop {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ} {G : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ} (α : F ≅ G) :
CategoryTheory.Functor.unop G ≅ CategoryTheory.Functor.unop F

The natural isomorphism between functors G.unop ≅ F.unop induced by a natural isomorphism between the original functors F ≅ G.

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@[simp]
theorem CategoryTheory.Equivalence.op_unitIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (e : C ≌ D) :
(CategoryTheory.Equivalence.op e).unitIso = (CategoryTheory.NatIso.op e.unitIso).symm
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@[simp]
theorem CategoryTheory.Equivalence.op_counitIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (e : C ≌ D) :
(CategoryTheory.Equivalence.op e).counitIso = (CategoryTheory.NatIso.op e.counitIso).symm
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@[simp]
theorem CategoryTheory.Equivalence.op_functor {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (e : C ≌ D) :
(CategoryTheory.Equivalence.op e).functor = e.functor.op
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@[simp]
theorem CategoryTheory.Equivalence.op_inverse {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (e : C ≌ D) :
(CategoryTheory.Equivalence.op e).inverse = e.inverse.op
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def CategoryTheory.Equivalence.op {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (e : C ≌ D) :
Cᵒᵖ ≌ Dᵒᵖ

An equivalence between categories gives an equivalence between the opposite categories.

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  • One or more equations did not get rendered due to their size.
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@[simp]
theorem CategoryTheory.Equivalence.unop_counitIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (e : Cᵒᵖ ≌ Dᵒᵖ) :
(CategoryTheory.Equivalence.unop e).counitIso = (CategoryTheory.NatIso.unop e.counitIso).symm
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@[simp]
theorem CategoryTheory.Equivalence.unop_unitIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (e : Cᵒᵖ ≌ Dᵒᵖ) :
(CategoryTheory.Equivalence.unop e).unitIso = (CategoryTheory.NatIso.unop e.unitIso).symm
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@[simp]
theorem CategoryTheory.Equivalence.unop_inverse {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (e : Cᵒᵖ ≌ Dᵒᵖ) :
(CategoryTheory.Equivalence.unop e).inverse = CategoryTheory.Functor.unop e.inverse
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@[simp]
theorem CategoryTheory.Equivalence.unop_functor {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (e : Cᵒᵖ ≌ Dᵒᵖ) :
(CategoryTheory.Equivalence.unop e).functor = CategoryTheory.Functor.unop e.functor
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def CategoryTheory.Equivalence.unop {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (e : Cᵒᵖ ≌ Dᵒᵖ) :
C ≌ D

An equivalence between opposite categories gives an equivalence between the original categories.

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@[simp]
theorem CategoryTheory.opEquiv_symm_apply {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (A : Cᵒᵖ) (B : Cᵒᵖ) (g : B.unop ⟶ A.unop) :
(CategoryTheory.opEquiv A B).symm g = g.op
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@[simp]
theorem CategoryTheory.opEquiv_apply {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (A : Cᵒᵖ) (B : Cᵒᵖ) (f : A ⟶ B) :
(CategoryTheory.opEquiv A B) f = f.unop
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def CategoryTheory.opEquiv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (A : Cᵒᵖ) (B : Cᵒᵖ) :
(A ⟶ B) ≃ (B.unop ⟶ A.unop)

The equivalence between arrows of the form A ⟶ B and B.unop ⟶ A.unop. Useful for building adjunctions. Note that this (definitionally) gives variants

def opEquiv' (A : C) (B : Cᵒᵖ) : (Opposite.op A ⟶ B) ≃ (B.unop ⟶ A) :=
  opEquiv _ _

def opEquiv'' (A : Cᵒᵖ) (B : C) : (A ⟶ Opposite.op B) ≃ (B ⟶ A.unop) :=
  opEquiv _ _

def opEquiv''' (A B : C) : (Opposite.op A ⟶ Opposite.op B) ≃ (B ⟶ A) :=
  opEquiv _ _
Equations
  • CategoryTheory.opEquiv A B = { toFun := fun (f : A ⟶ B) => f.unop, invFun := fun (g : B.unop ⟶ A.unop) => g.op, left_inv := ⋯, right_inv := ⋯ }
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instance CategoryTheory.subsingleton_of_unop {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (A : Cᵒᵖ) (B : Cᵒᵖ) [Subsingleton (B.unop ⟶ A.unop)] :
Subsingleton (A ⟶ B)
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  • ⋯ = ⋯
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instance CategoryTheory.decidableEqOfUnop {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (A : Cᵒᵖ) (B : Cᵒᵖ) [DecidableEq (B.unop ⟶ A.unop)] :
DecidableEq (A ⟶ B)
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@[simp]
theorem CategoryTheory.isoOpEquiv_apply {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (A : Cᵒᵖ) (B : Cᵒᵖ) (f : A ≅ B) :
(CategoryTheory.isoOpEquiv A B) f = CategoryTheory.Iso.unop f
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@[simp]
theorem CategoryTheory.isoOpEquiv_symm_apply {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (A : Cᵒᵖ) (B : Cᵒᵖ) (g : B.unop ≅ A.unop) :
(CategoryTheory.isoOpEquiv A B).symm g = CategoryTheory.Iso.op g
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def CategoryTheory.isoOpEquiv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (A : Cᵒᵖ) (B : Cᵒᵖ) :
(A ≅ B) ≃ (B.unop ≅ A.unop)

The equivalence between isomorphisms of the form A ≅ B and B.unop ≅ A.unop.

Note this is definitionally the same as the other three variants:

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@[simp]
theorem CategoryTheory.Functor.opUnopEquiv_inverse (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] :
(CategoryTheory.Functor.opUnopEquiv C D).inverse = CategoryTheory.Functor.opInv C D
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@[simp]
theorem CategoryTheory.Functor.opUnopEquiv_functor (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] :
(CategoryTheory.Functor.opUnopEquiv C D).functor = CategoryTheory.Functor.opHom C D
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@[simp]
theorem CategoryTheory.Functor.opUnopEquiv_counitIso (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] :
(CategoryTheory.Functor.opUnopEquiv C D).counitIso = CategoryTheory.NatIso.ofComponents (fun (F : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ) => CategoryTheory.Functor.unopOpIso F) ⋯
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@[simp]
theorem CategoryTheory.Functor.opUnopEquiv_unitIso (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] :
(CategoryTheory.Functor.opUnopEquiv C D).unitIso = CategoryTheory.NatIso.ofComponents (fun (F : (CategoryTheory.Functor C D)ᵒᵖ) => CategoryTheory.Iso.op (CategoryTheory.Functor.opUnopIso F.unop)) ⋯
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def CategoryTheory.Functor.opUnopEquiv (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] :
(CategoryTheory.Functor C D)ᵒᵖ ≌ CategoryTheory.Functor Cᵒᵖ Dᵒᵖ

The equivalence of functor categories induced by op and unop.

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  • One or more equations did not get rendered due to their size.
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@[simp]
theorem CategoryTheory.Functor.leftOpRightOpEquiv_counitIso_inv_app_app (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] (X : CategoryTheory.Functor C Dᵒᵖ) (X : C) :
((CategoryTheory.Functor.leftOpRightOpEquiv C D).counitIso.inv.app X✝).app X = CategoryTheory.CategoryStruct.id (X✝.obj X)
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@[simp]
theorem CategoryTheory.Functor.leftOpRightOpEquiv_functor_obj_obj (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] (F : (CategoryTheory.Functor Cᵒᵖ D)ᵒᵖ) (X : C) :
((CategoryTheory.Functor.leftOpRightOpEquiv C D).functor.obj F).obj X = Opposite.op (F.unop.obj (Opposite.op X))
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@[simp]
theorem CategoryTheory.Functor.leftOpRightOpEquiv_unitIso_hom_app (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] (X : (CategoryTheory.Functor Cᵒᵖ D)ᵒᵖ) :
(CategoryTheory.Functor.leftOpRightOpEquiv C D).unitIso.hom.app X = (CategoryTheory.Functor.rightOpLeftOpIso X.unop).hom.op
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@[simp]
theorem CategoryTheory.Functor.leftOpRightOpEquiv_functor_map_app (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] :
∀ {X Y : (CategoryTheory.Functor Cᵒᵖ D)ᵒᵖ} (η : X ⟶ Y) (X_1 : C), ((CategoryTheory.Functor.leftOpRightOpEquiv C D).functor.map η).app X_1 = (η.unop.app (Opposite.op X_1)).op
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@[simp]
theorem CategoryTheory.Functor.leftOpRightOpEquiv_unitIso_inv_app (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] (X : (CategoryTheory.Functor Cᵒᵖ D)ᵒᵖ) :
(CategoryTheory.Functor.leftOpRightOpEquiv C D).unitIso.inv.app X = (CategoryTheory.Functor.rightOpLeftOpIso X.unop).inv.op
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@[simp]
theorem CategoryTheory.Functor.leftOpRightOpEquiv_inverse_obj (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C Dᵒᵖ) :
(CategoryTheory.Functor.leftOpRightOpEquiv C D).inverse.obj F = Opposite.op F.leftOp
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@[simp]
theorem CategoryTheory.Functor.leftOpRightOpEquiv_counitIso_hom_app_app (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] (X : CategoryTheory.Functor C Dᵒᵖ) (X : C) :
((CategoryTheory.Functor.leftOpRightOpEquiv C D).counitIso.hom.app X✝).app X = CategoryTheory.CategoryStruct.id (X✝.obj X)
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@[simp]
theorem CategoryTheory.Functor.leftOpRightOpEquiv_functor_obj_map (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] (F : (CategoryTheory.Functor Cᵒᵖ D)ᵒᵖ) :
∀ {X Y : C} (f : X ⟶ Y), ((CategoryTheory.Functor.leftOpRightOpEquiv C D).functor.obj F).map f = (F.unop.map f.op).op
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@[simp]
theorem CategoryTheory.Functor.leftOpRightOpEquiv_inverse_map (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] :
∀ {X Y : CategoryTheory.Functor C Dᵒᵖ} (η : X ⟶ Y), (CategoryTheory.Functor.leftOpRightOpEquiv C D).inverse.map η = (CategoryTheory.NatTrans.leftOp η).op
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def CategoryTheory.Functor.leftOpRightOpEquiv (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] :
(CategoryTheory.Functor Cᵒᵖ D)ᵒᵖ ≌ CategoryTheory.Functor C Dᵒᵖ

The equivalence of functor categories induced by leftOp and rightOp.

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