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.

• op :: (
• unop : α

  The canonical map αᵒᵖ → α.

• )

def Opposite.unexpander_op : Lean.PrettyPrinter.Unexpander

Make sure that Opposite.op a is pretty-printed as op a instead of { unop := a } or ⟨a⟩.

Equations

• Opposite.unexpander_op x = pure x

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")

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

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

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

• Opposite.equivToOpposite = { toFun := Opposite.op, invFun := Opposite.unop, left_inv := ⋯, right_inv := ⋯ }

theorem Opposite.op_surjective {α : Sort u} : Function.Surjective Opposite.op

theorem Opposite.unop_surjective {α : Sort u} : Function.Surjective Opposite.unop

@[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 = 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.instInhabited {α : Sort u} [Inhabited α] : Inhabited αᵒᵖ

Equations

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

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

Equations

• ⋯ = ⋯

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

Equations

• ⋯ = ⋯

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 @[induction_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 (Opposite.unop X)