# 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.

• )
Instances For

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

Equations
Instances For

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
Instances For
theorem Opposite.unop_injective {α : Sort u} :
Function.Injective Opposite.unop
@[simp]
theorem Opposite.op_unop {α : Sort u} (x : αᵒᵖ) :
theorem Opposite.unop_op {α : Sort u} (x : α) :
theorem Opposite.op_inj_iff {α : Sort u} (x : α) (y : α) :
x = y
@[simp]
theorem Opposite.unop_inj_iff {α : Sort u} (x : αᵒᵖ) (y : αᵒᵖ) :
x = y

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

Equations
• Opposite.equivToOpposite = { toFun := Opposite.op, invFun := Opposite.unop, left_inv := , right_inv := }
Instances For
@[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} :
= y
theorem Opposite.unop_eq_iff_eq_op {α : Sort u} {x : αᵒᵖ} {y : α} :
x =
instance Opposite.instInhabited {α : Sort u} [] :
Equations
instance Opposite.instNonempty {α : Sort u} [] :
Equations
• =
instance Opposite.instSubsingleton {α : Sort u} [] :
Equations
• =
def Opposite.rec' {α : Sort u} {F : αᵒᵖSort 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
• = h
Instances For