mathlib documentation

category_theory.opposites

@[instance]

The hom types of the opposite of a category (or graph).

As with the objects, we'll make this irreducible below. Use f.op and f.unop to convert between morphisms of C and morphisms of Cᵒᵖ.

Equations
def category_theory.has_hom.hom.op {C : Type u₁} [category_theory.has_hom C] {X Y : C} :
(X Y)(opposite.op Y opposite.op X)

The opposite of a morphism in C.

Equations

Given a morphism in Cᵒᵖ, we can take the "unopposite" back in C.

Equations
@[simp]
theorem category_theory.has_hom.hom.unop_op {C : Type u₁} [category_theory.has_hom C] {X Y : C} {f : X Y} :
f.op.unop = f

@[simp]
theorem category_theory.has_hom.hom.op_unop {C : Type u₁} [category_theory.has_hom C] {X Y : Cᵒᵖ} {f : X Y} :
f.unop.op = f

@[instance]

The opposite category.

See https://stacks.math.columbia.edu/tag/001M.

Equations
@[simp]
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

@[simp]
theorem category_theory.op_id {C : Type u₁} [category_theory.category C] {X : C} :

@[simp]
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

@[simp]
theorem category_theory.unop_id {C : Type u₁} [category_theory.category C] {X : Cᵒᵖ} :

@[simp]
theorem category_theory.unop_id_op {C : Type u₁} [category_theory.category C] {X : C} :

@[simp]
theorem category_theory.op_id_unop {C : Type u₁} [category_theory.category C] {X : Cᵒᵖ} :

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

Equations
@[simp]
theorem category_theory.op_op_map (C : Type u₁) [category_theory.category C] (X Y : Cᵒᵖᵒᵖ) (f : X Y) :

@[simp]
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.

Equations

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

Equations
@[simp]
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

def category_theory.functor.op {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] :
C DCᵒᵖ 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.

Equations
@[simp]
theorem category_theory.functor.op_obj {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : C D) (X : Cᵒᵖ) :

def category_theory.functor.unop {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] :
Cᵒᵖ DᵒᵖC 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.

Equations
@[simp]
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

@[simp]
theorem category_theory.functor.unop_obj {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : Cᵒᵖ Dᵒᵖ) (X : C) :

def category_theory.functor.op_unop_iso {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : C D) :
F.op.unop F

The isomorphism between F.op.unop and F.

Equations

The isomorphism between F.unop.op and F.

Equations

Taking the opposite of a functor is functorial.

Equations
@[simp]
theorem category_theory.functor.op_hom_map_app (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (F G : (C D)ᵒᵖ) (α : F G) (X : Cᵒᵖ) :

@[simp]

Take the "unopposite" of a functor is functorial.

Equations
@[simp]
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

def category_theory.functor.left_op {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] :
C DᵒᵖCᵒᵖ 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
@[simp]
theorem category_theory.functor.left_op_obj {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : C Dᵒᵖ) (X : Cᵒᵖ) :

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

Equations
@[simp]
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

@[simp]
theorem category_theory.functor.right_op_obj {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (F : Cᵒᵖ D) (X : C) :

@[instance]

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

@[instance]

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

@[simp]
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) :

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

The opposite of a natural transformation.

Equations
@[simp]

def category_theory.nat_trans.unop {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {F G : Cᵒᵖ Dᵒᵖ} :
(F G)(G.unop F.unop)

The "unopposite" of a natural transformation.

Equations
@[simp]
theorem category_theory.nat_trans.unop_app {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {F G : Cᵒᵖ Dᵒᵖ} (α : F G) (X : C) :

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} :
(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.

Equations
@[simp]
theorem category_theory.nat_trans.remove_op_app {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {F G : C D} (α : F.op G.op) (X : C) :

def category_theory.nat_trans.left_op {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {F G : C Dᵒᵖ} :
(F G)(G.left_op F.left_op)

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.

Equations
@[simp]
theorem category_theory.nat_trans.left_op_app {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {F G : C Dᵒᵖ} (α : F G) (X : Cᵒᵖ) :

def category_theory.nat_trans.remove_left_op {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {F G : C Dᵒᵖ} :
(F.left_op G.left_op)(G F)

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.

Equations
def category_theory.iso.op {C : Type u₁} [category_theory.category C] {X Y : C} :
(X Y)(opposite.op Y opposite.op X)

The opposite isomorphism.

Equations
@[simp]
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

@[simp]
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

def category_theory.nat_iso.op {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {F G : C D} :
(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.

Equations
@[simp]

@[simp]

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} :
(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.

Equations
def category_theory.nat_iso.unop {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {F G : Cᵒᵖ Dᵒᵖ} :
(F G)(G.unop F.unop)

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

Equations
@[simp]
theorem category_theory.equivalence.op_inverse {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (e : C D) :

@[simp]
theorem category_theory.equivalence.op_functor {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (e : C D) :

def category_theory.equivalence.op {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] :
(C D)(Cᵒᵖ Dᵒᵖ)

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

Equations
def category_theory.equivalence.unop {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] :
(Cᵒᵖ Dᵒᵖ)(C D)

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

Equations

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 _ _
Equations
@[simp]
theorem category_theory.op_equiv_apply {C : Type u₁} [category_theory.category C] (A B : Cᵒᵖ) (f : A B) :

def category_theory.op_hom_of_le {α : Type v} [preorder α] {U V : αᵒᵖ} :

Construct a morphism in the opposite of a preorder category from an inequality.

Equations
theorem category_theory.le_of_op_hom {α : Type v} [preorder α] {U V : αᵒᵖ} :