Opposites

In this file we define a structure Opposite α containing a single field of type α and two bijections op : α → αᵒᵖ and unop : αᵒᵖ → α. If α is a category, then αᵒᵖ is the opposite category, with all arrows reversed.

structure Opposite (α : Sort u) : Sort (max 1 u)

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

Now that Lean 4 supports definitional eta equality for records, both unop (op X) = X and op (unop X) = X are definitional equalities.

• unop : α

  The canonical map αᵒᵖ → α.

def «term_ᵒᵖ» : Lean.TrailingParserDescr

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

Now that Lean 4 supports definitional eta equality for records, both unop (op X) = X and op (unop X) = X are definitional equalities.

Equations

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

def Opposite.op {α : Sort u} (x : α) : αᵒᵖ

The canonical map α → αᵒᵖ.

Equations

• Opposite.op x = { unop := x }

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 x.unop = x

@[simp]

theorem Opposite.unop_op {α : Sort u} (x : α) : (Opposite.op x).unop = 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) : x.unop = y.unop ↔ x = y

def Opposite.equivToOpposite {α : Sort u} : α ≃ αᵒᵖ

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

Equations

• Opposite.equivToOpposite = { toFun := Opposite.op, invFun := Opposite.unop, left_inv := (_ : ∀ (x : α), (Opposite.op x).unop = x), right_inv := (_ : ∀ (x : αᵒᵖ), Opposite.op x.unop = x) }

@[simp]

theorem Opposite.equivToOpposite_coe {α : Sort u} : ⇑Opposite.equivToOpposite = Opposite.op

@[simp]

theorem Opposite.equivToOpposite_symm_coe {α : Sort u} : ⇑Opposite.equivToOpposite.symm = Opposite.unop

theorem Opposite.op_eq_iff_eq_unop {α : Sort u} {x : α} {y : αᵒᵖ} : Opposite.op x = y ↔ x = y.unop

theorem Opposite.unop_eq_iff_eq_op {α : Sort u} {x : αᵒᵖ} {y : α} : x.unop = y ↔ x = Opposite.op y

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

Equations

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

instance Opposite.instNonemptyOpposite {α : Sort u} [Nonempty α] : Nonempty αᵒᵖ

Equations

• (_ : Nonempty αᵒᵖ) = (_ : Nonempty αᵒᵖ)

instance Opposite.instSubsingletonOpposite {α : Sort u} [Subsingleton α] : Subsingleton αᵒᵖ

Equations

• (_ : Subsingleton αᵒᵖ) = (_ : Subsingleton αᵒᵖ)

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. The @[eliminator] attribute makes it the default induction principle for Opposite so you don't need to use induction x using Opposite.rec'.

Equations

• Opposite.rec' h X = h X.unop