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 op

theorem Opposite.unop_injective {α : Sort u} : Function.Injective unop

@[simp]

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

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

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

@[simp]

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

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

@[simp]

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

@[simp]

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

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

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

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

@[deprecated Opposite.rec (since := "2025-04-04")]

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

A deprecated alias for Opposite.rec.

Equations

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

theorem Opposite.small {X : Type v} [Small.{u, v} X] : Small.{u, v} Xᵒᵖ

If X is u-small, also Xᵒᵖ is u-small. Note: This is not an instance, because it tends to mislead typeclass search.