mathlib documentation

category_theory.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.

theorem quiver.hom.op_inj {C : Type u₁} [quiver C] {X Y : C} :
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theorem quiver.hom.unop_op {C : Type u₁} [quiver C] {X Y : C} (f : X Y) :
f.op.unop = f
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theorem quiver.hom.op_unop {C : Type u₁} [quiver C] {X Y : Cᵒᵖ} (f : X Y) :
f.unop.op = f
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theorem category_theory.op_comp {C : Type u₁} [category_theory.category C] {X Y Z : C} {f : X Y} {g : Y Z} :
(f g).op = g.op f.op
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theorem category_theory.op_id {C : Type u₁} [category_theory.category C] {X : C} :
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theorem category_theory.unop_comp {C : Type u₁} [category_theory.category C] {X Y Z : Cᵒᵖ} {f : X Y} {g : Y Z} :
(f g).unop = g.unop f.unop
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The functor from the double-opposite of a category to the underlying category.

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theorem category_theory.unop_unop_map (C : Type u₁) [category_theory.category C] (X Y : C) (f : X Y) :

The functor from a category to its double-opposite.

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If f is an isomorphism, so is f.op

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 category_theory.functor.op_map {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : C D) (X Y : Cᵒᵖ) (f : X Y) :
F.op.map f = (F.map f.unop).op
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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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Instances for category_theory.functor.op
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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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Instances for category_theory.functor.unop
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theorem category_theory.functor.unop_map {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : Cᵒᵖ Dᵒᵖ) (X Y : C) (f : X Y) :
F.unop.map f = (F.map f.op).unop
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The isomorphism between F.op.unop and F.

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The isomorphism between F.unop.op and F.

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Taking the opposite of a functor is functorial.

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Take the "unopposite" of a functor is functorial.

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theorem category_theory.functor.left_op_map {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : C Dᵒᵖ) (X Y : Cᵒᵖ) (f : X Y) :
F.left_op.map f = (F.map f.unop).unop
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Another variant of the opposite of functor, turning a functor C ⥤ Dᵒᵖ into a functor Cᵒᵖ ⥤ D. In informal mathematics no distinction is made.

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Instances for category_theory.functor.left_op
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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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Instances for category_theory.functor.right_op
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theorem category_theory.functor.right_op_map {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : Cᵒᵖ D) (X Y : C) (f : X Y) :
F.right_op.map f = (F.map f.op).op
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If F is faithful then the right_op of F is also faithful.

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

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

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theorem category_theory.nat_trans.op_app {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {F G : C D} (α : F G) (X : Cᵒᵖ) :
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def category_theory.nat_trans.op {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {F G : C D} (α : F G) :
G.op F.op

The opposite of a natural transformation.

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The "unopposite" of a natural transformation.

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def category_theory.nat_trans.remove_op {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {F G : 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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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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Given a natural transformation α : F ⟶ G, for F G : C ⥤ Dᵒᵖ, taking unop of each component gives a natural transformation G.left_op ⟶ F.left_op.

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

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

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

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theorem category_theory.iso.op_hom {C : Type u₁} [category_theory.category C] {X Y : C} (α : X Y) :
α.op.hom = α.hom.op
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def category_theory.iso.op {C : Type u₁} [category_theory.category C] {X Y : C} (α : X Y) :

The opposite isomorphism.

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theorem category_theory.iso.op_inv {C : Type u₁} [category_theory.category C] {X Y : C} (α : X Y) :
α.op.inv = α.inv.op
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theorem category_theory.iso.unop_inv {C : Type u₁} [category_theory.category C] {X Y : Cᵒᵖ} (f : X Y) :
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theorem category_theory.iso.unop_hom {C : Type u₁} [category_theory.category C] {X Y : Cᵒᵖ} (f : X Y) :

The isomorphism obtained from an isomorphism in the opposite category.

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theorem category_theory.iso.unop_op {C : Type u₁} [category_theory.category C] {X Y : Cᵒᵖ} (f : X Y) :
f.unop.op = f
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theorem category_theory.iso.op_unop {C : Type u₁} [category_theory.category C] {X Y : C} (f : X Y) :
f.op.unop = f
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def category_theory.nat_iso.op {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {F G : 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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def category_theory.nat_iso.remove_op {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {F G : 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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The natural isomorphism between functors G.unop ≅ F.unop induced by a natural isomorphism between the original functors F ≅ G.

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

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

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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 op_equiv' (A : C) (B : Cᵒᵖ) : (opposite.op A  B)  (B.unop  A) :=
op_equiv _ _

def op_equiv'' (A : Cᵒᵖ) (B : C) : (A  opposite.op B)  (B  A.unop) :=
op_equiv _ _

def op_equiv''' (A B : C) : (opposite.op A  opposite.op B)  (B  A) :=
op_equiv _ _
```
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@[simp]

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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