Duality.agda

module Duality (X : Set)
              (_~_ : (x0 x1 : X) -> Set)
              (sym : {x0 x1 : X} -> x0 ~ x1 -> x1 ~ x0) where

  Object : Set1
  Object = X -> Set

  Dual : Object -> Object
  Dual A x0 = {x1 : X} -> A x1 -> x0 ~ x1

  _<=_ : Object -> Object -> Set
  A0 <= A1 = {x : X} -> A0 x -> A1 x

  contra : {A0 A1 : Object} -> A0 <= A1 -> Dual A1 <= Dual A0
  contra i f x = f (i x)

  inject : {A : Object} -> A <= Dual (Dual A)
  inject x f = sym (f x)

  cancel : {A : Object} -> Dual (Dual (Dual A)) <= Dual A
  cancel = contra inject