Documentation

Mathlib.Data.Opposite

Opposites #

In this file we define a type synonym Opposite α := α, denoted by αᵒᵖ and two synonyms for the identity map, op : α → αᵒᵖ and unop : αᵒᵖ → α. If α is a category, then αᵒᵖ is the opposite category, with all arrows reversed.

def Opposite (α : Sort u) :

The type of objects of the opposite of α; used to define the opposite category.

In order to avoid confusion between α and its opposite type, we set up the type of objects Opposite α using the following pattern, which will be repeated later for the morphisms.

Define Opposite α := α.
Define the isomorphisms op : α → Opposite α, unop : Opposite α → α.
Make the definition Opposite irreducible.

This has the following consequences.

Opposite α and α are distinct types in the elaborator, so you must use op and unop explicitly to convert between them.
Both unop (op X) = X and op (unop X) = X are definitional equalities. Notably, every object of the opposite category is definitionally of the form op X, which greatly simplifies the definition of the structure of the opposite category, for example.

(Now that Lean 4 supports definitional eta equality for records, we could achieve the same goals using a structure with one field.)

Equations

αᵒᵖ = α

def «term_ᵒᵖ» :

Lean.TrailingParserDescr

The type of objects of the opposite of α; used to define the opposite category.

In order to avoid confusion between α and its opposite type, we set up the type of objects Opposite α using the following pattern, which will be repeated later for the morphisms.

Define Opposite α := α.
Define the isomorphisms op : α → Opposite α, unop : Opposite α → α.
Make the definition Opposite irreducible.

This has the following consequences.

Opposite α and α are distinct types in the elaborator, so you must use op and unop explicitly to convert between them.
Both unop (op X) = X and op (unop X) = X are definitional equalities. Notably, every object of the opposite category is definitionally of the form op X, which greatly simplifies the definition of the structure of the opposite category, for example.

(Now that Lean 4 supports definitional eta equality for records, we could achieve the same goals using a structure with one field.)

Equations

«term_ᵒᵖ» = Lean.ParserDescr.trailingNode `term_ᵒᵖ 1024 0 (Lean.ParserDescr.symbol "ᵒᵖ")

def Opposite.op {α : Sort u} :

α → αᵒᵖ

The canonical map α → αᵒᵖ.

Equations

Opposite.op = id

def Opposite.unop {α : Sort u} :

αᵒᵖ → α

The canonical map αᵒᵖ → α.

Equations

Opposite.unop = id

theorem Opposite.op_injective {α : Sort u} :

Function.Injective Opposite.op

theorem Opposite.unop_injective {α : Sort u} :

Function.Injective Opposite.unop

@[simp]

theorem Opposite.op_unop {α : Sort u} (x : αᵒᵖ) :

Opposite.op (Opposite.unop x) = x

@[simp]

theorem Opposite.unop_op {α : Sort u} (x : α) :

Opposite.unop (Opposite.op x) = x

@[simp]

theorem Opposite.op_inj_iff {α : Sort u} (x : α) (y : α) :

Opposite.op x = Opposite.op y ↔ x = y

@[simp]

theorem Opposite.unop_inj_iff {α : Sort u} (x : αᵒᵖ) (y : αᵒᵖ) :

Opposite.unop x = Opposite.unop y ↔ x = y

def Opposite.equivToOpposite {α : Sort u} :

α ≃ αᵒᵖ

The type-level equivalence between a type and its opposite.

Equations

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

@[simp]

theorem Opposite.equivToOpposite_coe {α : Sort u} :

↑Opposite.equivToOpposite = Opposite.op

@[simp]

theorem Opposite.equivToOpposite_symm_coe {α : Sort u} :

↑(Equiv.symm Opposite.equivToOpposite) = Opposite.unop

theorem Opposite.op_eq_iff_eq_unop {α : Sort u} {x : α} {y : αᵒᵖ} :

Opposite.op x = y ↔ x = Opposite.unop y

theorem Opposite.unop_eq_iff_eq_op {α : Sort u} {x : αᵒᵖ} {y : α} :

Opposite.unop x = y ↔ x = Opposite.op y

instance Opposite.instInhabitedOpposite {α : Sort u} [inst : Inhabited α] :

Inhabited αᵒᵖ

Equations

Opposite.instInhabitedOpposite = { default := Opposite.op default }

def Opposite.rec {α : Sort u} {F : αᵒᵖ → Sort v} (h : (X : α) → F (Opposite.op X)) (X : αᵒᵖ) :

F X

A recursor for Opposite. Use as induction x using Opposite.rec.

Equations

Opposite.rec h X = h (Opposite.unop X)