Documentation

Mathlib.Data.Opposite

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) :
        @[simp]
        theorem Opposite.unop_inj_iff {α : Sort u} (x : αᵒᵖ) (y : αᵒᵖ) :

        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} :
          theorem Opposite.unop_eq_iff_eq_op {α : Sort u} {x : αᵒᵖ} {y : α} :
          Equations
          Equations
          • =
          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
          Instances For