Use long name
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@ -10,8 +10,8 @@ open import Cat.Category.NaturalTransformation
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open import Cat.Category.Yoneda
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open import Cat.Category.Monoid
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open import Cat.Category.Monad
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open Cat.Category.Monad.Monoidal
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open Cat.Category.Monad.Kleisli
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open import Cat.Category.Monad.Monoidal
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open import Cat.Category.Monad.Kleisli
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open import Cat.Category.Monad.Voevodsky
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open import Cat.Categories.Sets
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@ -28,186 +28,205 @@ import Cat.Category.Monad.Monoidal
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import Cat.Category.Monad.Kleisli
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open import Cat.Categories.Fun
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module Monoidal = Cat.Category.Monad.Monoidal
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module Kleisli = Cat.Category.Monad.Kleisli
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-- | The monoidal- and kleisli presentation of monads are equivalent.
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module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
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open Cat.Category.NaturalTransformation ℂ ℂ using (NaturalTransformation ; propIsNatural)
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private
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module ℂ = Category ℂ
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open ℂ using (Object ; Arrow ; identity ; _<<<_ ; _>>>_)
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module M = Monoidal ℂ
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module K = Kleisli ℂ
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module _ (m : M.RawMonad) where
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open M.RawMonad m
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module Monoidal = Cat.Category.Monad.Monoidal ℂ
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module Kleisli = Cat.Category.Monad.Kleisli ℂ
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forthRaw : K.RawMonad
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K.RawMonad.omap forthRaw = Romap
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K.RawMonad.pure forthRaw = pureT _
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K.RawMonad.bind forthRaw = bind
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module _ (m : Monoidal.RawMonad) where
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open Monoidal.RawMonad m
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module _ {raw : M.RawMonad} (m : M.IsMonad raw) where
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private
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module MI = M.IsMonad m
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forthIsMonad : K.IsMonad (forthRaw raw)
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K.IsMonad.isIdentity forthIsMonad = snd MI.isInverse
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K.IsMonad.isNatural forthIsMonad = MI.isNatural
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K.IsMonad.isDistributive forthIsMonad = MI.isDistributive
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toKleisliRaw : Kleisli.RawMonad
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Kleisli.RawMonad.omap toKleisliRaw = Romap
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Kleisli.RawMonad.pure toKleisliRaw = pure
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Kleisli.RawMonad.bind toKleisliRaw = bind
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forth : M.Monad → K.Monad
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Kleisli.Monad.raw (forth m) = forthRaw (M.Monad.raw m)
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Kleisli.Monad.isMonad (forth m) = forthIsMonad (M.Monad.isMonad m)
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module _ {raw : Monoidal.RawMonad} (m : Monoidal.IsMonad raw) where
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open Monoidal.IsMonad m
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module _ (m : K.Monad) where
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open K.Monad m
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open Kleisli.RawMonad (toKleisliRaw raw) using (_>=>_)
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toKleisliIsMonad : Kleisli.IsMonad (toKleisliRaw raw)
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Kleisli.IsMonad.isIdentity toKleisliIsMonad = begin
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bind pure ≡⟨⟩
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join <<< (fmap pure) ≡⟨ snd isInverse ⟩
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identity ∎
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Kleisli.IsMonad.isNatural toKleisliIsMonad f = begin
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pure >=> f ≡⟨⟩
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pure >>> bind f ≡⟨⟩
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bind f <<< pure ≡⟨⟩
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(join <<< fmap f) <<< pure ≡⟨ isNatural f ⟩
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f ∎
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Kleisli.IsMonad.isDistributive toKleisliIsMonad f g = begin
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bind g >>> bind f ≡⟨⟩
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(join <<< fmap g) >>> (join <<< fmap f) ≡⟨ isDistributive f g ⟩
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bind (g >=> f) ∎
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-- Kleisli.IsMonad.isDistributive toKleisliIsMonad = isDistributive
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backRaw : M.RawMonad
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M.RawMonad.R backRaw = R
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M.RawMonad.pureNT backRaw = pureNT
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M.RawMonad.joinNT backRaw = joinNT
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toKleisli : Monoidal.Monad → Kleisli.Monad
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Kleisli.Monad.raw (toKleisli m) = toKleisliRaw (Monoidal.Monad.raw m)
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Kleisli.Monad.isMonad (toKleisli m) = toKleisliIsMonad (Monoidal.Monad.isMonad m)
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private
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open M.RawMonad backRaw renaming
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( join to join*
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; pure to pure*
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; bind to bind*
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; fmap to fmap*
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)
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module R = Functor (M.RawMonad.R backRaw)
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module _ (m : Kleisli.Monad) where
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open Kleisli.Monad m
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backIsMonad : M.IsMonad backRaw
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M.IsMonad.isAssociative backIsMonad = begin
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join* <<< R.fmap join* ≡⟨⟩
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toMonoidalRaw : Monoidal.RawMonad
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Monoidal.RawMonad.R toMonoidalRaw = R
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Monoidal.RawMonad.pureNT toMonoidalRaw = pureNT
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Monoidal.RawMonad.joinNT toMonoidalRaw = joinNT
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open Monoidal.RawMonad toMonoidalRaw renaming
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( join to join*
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; pure to pure*
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; bind to bind*
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; fmap to fmap*
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) using ()
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toMonoidalIsMonad : Monoidal.IsMonad toMonoidalRaw
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Monoidal.IsMonad.isAssociative toMonoidalIsMonad = begin
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join* <<< fmap join* ≡⟨⟩
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join <<< fmap join ≡⟨ isNaturalForeign ⟩
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join <<< join ∎
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M.IsMonad.isInverse backIsMonad {X} = inv-l , inv-r
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Monoidal.IsMonad.isInverse toMonoidalIsMonad {X} = inv-l , inv-r
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where
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inv-l = begin
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join <<< pure ≡⟨ fst isInverse ⟩
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identity ∎
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inv-r = begin
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joinT X <<< R.fmap (pureT X) ≡⟨⟩
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join* <<< fmap* pure* ≡⟨⟩
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join <<< fmap pure ≡⟨ snd isInverse ⟩
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identity ∎
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back : K.Monad → M.Monad
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Monoidal.Monad.raw (back m) = backRaw m
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Monoidal.Monad.isMonad (back m) = backIsMonad m
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toMonoidal : Kleisli.Monad → Monoidal.Monad
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Monoidal.Monad.raw (toMonoidal m) = toMonoidalRaw m
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Monoidal.Monad.isMonad (toMonoidal m) = toMonoidalIsMonad m
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module _ (m : K.Monad) where
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module _ (m : Kleisli.Monad) where
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private
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open K.Monad m
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open Kleisli.Monad m
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bindEq : ∀ {X Y}
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→ K.RawMonad.bind (forthRaw (backRaw m)) {X} {Y}
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≡ K.RawMonad.bind (K.Monad.raw m)
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bindEq {X} {Y} = begin
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K.RawMonad.bind (forthRaw (backRaw m)) ≡⟨⟩
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(λ f → join <<< fmap f) ≡⟨⟩
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(λ f → bind (f >>> pure) >>> bind identity) ≡⟨ funExt lem ⟩
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(λ f → bind f) ≡⟨⟩
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bind ∎
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→ Kleisli.RawMonad.bind (toKleisliRaw (toMonoidalRaw m)) {X} {Y}
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≡ bind
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bindEq {X} {Y} = funExt lem
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where
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lem : (f : Arrow X (omap Y))
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→ bind (f >>> pure) >>> bind identity
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≡ bind f
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lem f = begin
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join <<< fmap f
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≡⟨⟩
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bind (f >>> pure) >>> bind identity
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≡⟨ isDistributive _ _ ⟩
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bind ((f >>> pure) >=> identity)
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≡⟨⟩
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bind ((f >>> pure) >>> bind identity)
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≡⟨ cong bind ℂ.isAssociative ⟩
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bind (f >>> (pure >>> bind identity))
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≡⟨⟩
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bind (f >>> (pure >=> identity))
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≡⟨ cong (λ φ → bind (f >>> φ)) (isNatural _) ⟩
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bind (f >>> identity)
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≡⟨ cong bind ℂ.leftIdentity ⟩
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bind f ∎
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forthRawEq : forthRaw (backRaw m) ≡ K.Monad.raw m
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K.RawMonad.omap (forthRawEq _) = omap
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K.RawMonad.pure (forthRawEq _) = pure
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K.RawMonad.bind (forthRawEq i) = bindEq i
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toKleisliRawEq : toKleisliRaw (toMonoidalRaw m) ≡ Kleisli.Monad.raw m
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Kleisli.RawMonad.omap (toKleisliRawEq i) = (begin
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Kleisli.RawMonad.omap (toKleisliRaw (toMonoidalRaw m)) ≡⟨⟩
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Monoidal.RawMonad.Romap (toMonoidalRaw m) ≡⟨⟩
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omap ∎
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) i
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Kleisli.RawMonad.pure (toKleisliRawEq i) = (begin
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Kleisli.RawMonad.pure (toKleisliRaw (toMonoidalRaw m)) ≡⟨⟩
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Monoidal.RawMonad.pure (toMonoidalRaw m) ≡⟨⟩
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pure ∎
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) i
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Kleisli.RawMonad.bind (toKleisliRawEq i) = bindEq i
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fortheq : (m : K.Monad) → forth (back m) ≡ m
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fortheq m = K.Monad≡ (forthRawEq m)
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toKleislieq : (m : Kleisli.Monad) → toKleisli (toMonoidal m) ≡ m
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toKleislieq m = Kleisli.Monad≡ (toKleisliRawEq m)
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module _ (m : M.Monad) where
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module _ (m : Monoidal.Monad) where
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private
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open M.Monad m
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module KM = K.Monad (forth m)
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open Monoidal.Monad m
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-- module KM = Kleisli.Monad (toKleisli m)
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open Kleisli.Monad (toKleisli m) renaming
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( bind to bind* ; omap to omap* ; join to join*
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; fmap to fmap* ; pure to pure* ; R to R*)
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using ()
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module R = Functor R
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omapEq : KM.omap ≡ Romap
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omapEq : omap* ≡ Romap
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omapEq = refl
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bindEq : ∀ {X Y} {f : Arrow X (Romap Y)} → KM.bind f ≡ bind f
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bindEq : ∀ {X Y} {f : Arrow X (Romap Y)} → bind* f ≡ bind f
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bindEq {X} {Y} {f} = begin
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KM.bind f ≡⟨⟩
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joinT Y <<< fmap f ≡⟨⟩
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bind f ∎
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bind* f ≡⟨⟩
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join <<< fmap f ≡⟨⟩
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bind f ∎
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joinEq : ∀ {X} → KM.join ≡ joinT X
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joinEq : ∀ {X} → join* ≡ joinT X
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joinEq {X} = begin
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KM.join ≡⟨⟩
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KM.bind identity ≡⟨⟩
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bind identity ≡⟨⟩
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joinT X <<< fmap identity ≡⟨ cong (λ φ → _ <<< φ) R.isIdentity ⟩
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joinT X <<< identity ≡⟨ ℂ.rightIdentity ⟩
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joinT X ∎
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join* ≡⟨⟩
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bind* identity ≡⟨⟩
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bind identity ≡⟨⟩
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join <<< fmap identity ≡⟨ cong (λ φ → _ <<< φ) R.isIdentity ⟩
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join <<< identity ≡⟨ ℂ.rightIdentity ⟩
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join ∎
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fmapEq : ∀ {A B} → KM.fmap {A} {B} ≡ fmap
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fmapEq : ∀ {A B} → fmap* {A} {B} ≡ fmap
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fmapEq {A} {B} = funExt (λ f → begin
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KM.fmap f ≡⟨⟩
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KM.bind (f >>> KM.pure) ≡⟨⟩
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bind (f >>> pureT _) ≡⟨⟩
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fmap (f >>> pureT B) >>> joinT B ≡⟨⟩
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fmap (f >>> pureT B) >>> joinT B ≡⟨ cong (λ φ → φ >>> joinT B) R.isDistributive ⟩
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fmap f >>> fmap (pureT B) >>> joinT B ≡⟨ ℂ.isAssociative ⟩
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joinT B <<< fmap (pureT B) <<< fmap f ≡⟨ cong (λ φ → φ <<< fmap f) (snd isInverse) ⟩
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identity <<< fmap f ≡⟨ ℂ.leftIdentity ⟩
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fmap f ∎
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fmap* f ≡⟨⟩
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bind* (f >>> pure*) ≡⟨⟩
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bind (f >>> pure) ≡⟨⟩
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fmap (f >>> pure) >>> join ≡⟨⟩
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fmap (f >>> pure) >>> join ≡⟨ cong (λ φ → φ >>> joinT B) R.isDistributive ⟩
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fmap f >>> fmap pure >>> join ≡⟨ ℂ.isAssociative ⟩
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join <<< fmap pure <<< fmap f ≡⟨ cong (λ φ → φ <<< fmap f) (snd isInverse) ⟩
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identity <<< fmap f ≡⟨ ℂ.leftIdentity ⟩
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fmap f ∎
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)
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rawEq : Functor.raw KM.R ≡ Functor.raw R
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rawEq : Functor.raw R* ≡ Functor.raw R
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RawFunctor.omap (rawEq i) = omapEq i
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RawFunctor.fmap (rawEq i) = fmapEq i
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Req : M.RawMonad.R (backRaw (forth m)) ≡ R
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Req : Monoidal.RawMonad.R (toMonoidalRaw (toKleisli m)) ≡ R
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Req = Functor≡ rawEq
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pureTEq : M.RawMonad.pureT (backRaw (forth m)) ≡ pureT
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pureTEq : Monoidal.RawMonad.pureT (toMonoidalRaw (toKleisli m)) ≡ pureT
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pureTEq = funExt (λ X → refl)
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pureNTEq : (λ i → NaturalTransformation Functors.identity (Req i))
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[ M.RawMonad.pureNT (backRaw (forth m)) ≡ pureNT ]
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[ Monoidal.RawMonad.pureNT (toMonoidalRaw (toKleisli m)) ≡ pureNT ]
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pureNTEq = lemSigP (λ i → propIsNatural Functors.identity (Req i)) _ _ pureTEq
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joinTEq : M.RawMonad.joinT (backRaw (forth m)) ≡ joinT
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joinTEq : Monoidal.RawMonad.joinT (toMonoidalRaw (toKleisli m)) ≡ joinT
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joinTEq = funExt (λ X → begin
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M.RawMonad.joinT (backRaw (forth m)) X ≡⟨⟩
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KM.join ≡⟨⟩
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joinT X <<< fmap identity ≡⟨ cong (λ φ → joinT X <<< φ) R.isIdentity ⟩
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joinT X <<< identity ≡⟨ ℂ.rightIdentity ⟩
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joinT X ∎)
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Monoidal.RawMonad.joinT (toMonoidalRaw (toKleisli m)) X ≡⟨⟩
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join* ≡⟨⟩
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join <<< fmap identity ≡⟨ cong (λ φ → join <<< φ) R.isIdentity ⟩
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join <<< identity ≡⟨ ℂ.rightIdentity ⟩
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join ∎)
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joinNTEq : (λ i → NaturalTransformation F[ Req i ∘ Req i ] (Req i))
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[ M.RawMonad.joinNT (backRaw (forth m)) ≡ joinNT ]
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[ Monoidal.RawMonad.joinNT (toMonoidalRaw (toKleisli m)) ≡ joinNT ]
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joinNTEq = lemSigP (λ i → propIsNatural F[ Req i ∘ Req i ] (Req i)) _ _ joinTEq
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backRawEq : backRaw (forth m) ≡ M.Monad.raw m
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M.RawMonad.R (backRawEq i) = Req i
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M.RawMonad.pureNT (backRawEq i) = pureNTEq i
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M.RawMonad.joinNT (backRawEq i) = joinNTEq i
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toMonoidalRawEq : toMonoidalRaw (toKleisli m) ≡ Monoidal.Monad.raw m
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Monoidal.RawMonad.R (toMonoidalRawEq i) = Req i
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Monoidal.RawMonad.pureNT (toMonoidalRawEq i) = pureNTEq i
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Monoidal.RawMonad.joinNT (toMonoidalRawEq i) = joinNTEq i
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backeq : (m : M.Monad) → back (forth m) ≡ m
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backeq m = M.Monad≡ (backRawEq m)
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eqv : isEquiv M.Monad K.Monad forth
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eqv = gradLemma forth back fortheq backeq
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toMonoidaleq : (m : Monoidal.Monad) → toMonoidal (toKleisli m) ≡ m
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toMonoidaleq m = Monoidal.Monad≡ (toMonoidalRawEq m)
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open import Cat.Equivalence
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Monoidal≊Kleisli : M.Monad ≅ K.Monad
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Monoidal≊Kleisli = forth , back , funExt backeq , funExt fortheq
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Monoidal≊Kleisli : Monoidal.Monad ≅ Kleisli.Monad
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Monoidal≊Kleisli = toKleisli , toMonoidal , funExt toMonoidaleq , funExt toKleislieq
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Monoidal≡Kleisli : M.Monad ≡ K.Monad
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Monoidal≡Kleisli : Monoidal.Monad ≡ Kleisli.Monad
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Monoidal≡Kleisli = isoToPath Monoidal≊Kleisli
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@ -18,7 +18,7 @@ private
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open Category ℂ using (Object ; Arrow ; identity ; _<<<_)
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open import Cat.Category.NaturalTransformation ℂ ℂ
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using (NaturalTransformation ; Transformation ; Natural)
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using (NaturalTransformation ; Transformation ; Natural ; NaturalTransformation≡)
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record RawMonad : Set ℓ where
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field
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@ -78,15 +78,39 @@ record IsMonad (raw : RawMonad) : Set ℓ where
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isNatural : IsNatural
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isNatural {X} {Y} f = begin
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joinT Y <<< R.fmap f <<< pureT X ≡⟨ sym ℂ.isAssociative ⟩
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joinT Y <<< (R.fmap f <<< pureT X) ≡⟨ cong (λ φ → joinT Y <<< φ) (sym (pureN f)) ⟩
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joinT Y <<< (pureT (R.omap Y) <<< f) ≡⟨ ℂ.isAssociative ⟩
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joinT Y <<< pureT (R.omap Y) <<< f ≡⟨ cong (λ φ → φ <<< f) (fst isInverse) ⟩
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identity <<< f ≡⟨ ℂ.leftIdentity ⟩
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f ∎
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join <<< fmap f <<< pure ≡⟨ sym ℂ.isAssociative ⟩
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join <<< (fmap f <<< pure) ≡⟨ cong (λ φ → join <<< φ) (sym (pureN f)) ⟩
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join <<< (pure <<< f) ≡⟨ ℂ.isAssociative ⟩
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join <<< pure <<< f ≡⟨ cong (λ φ → φ <<< f) (fst isInverse) ⟩
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identity <<< f ≡⟨ ℂ.leftIdentity ⟩
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f ∎
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isDistributive : IsDistributive
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isDistributive {X} {Y} {Z} g f = sym aux
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isDistributive {X} {Y} {Z} g f = begin
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join <<< fmap g <<< (join <<< fmap f)
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≡⟨ Category.isAssociative ℂ ⟩
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join <<< fmap g <<< join <<< fmap f
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≡⟨ cong (_<<< fmap f) (sym ℂ.isAssociative) ⟩
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(join <<< (fmap g <<< join)) <<< fmap f
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≡⟨ cong (λ φ → φ <<< fmap f) (cong (_<<<_ join) (sym (joinN g))) ⟩
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||||
(join <<< (join <<< R².fmap g)) <<< fmap f
|
||||
≡⟨ cong (_<<< fmap f) ℂ.isAssociative ⟩
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((join <<< join) <<< R².fmap g) <<< fmap f
|
||||
≡⟨⟩
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||||
join <<< join <<< R².fmap g <<< fmap f
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||||
≡⟨ sym ℂ.isAssociative ⟩
|
||||
(join <<< join) <<< (R².fmap g <<< fmap f)
|
||||
≡⟨ cong (λ φ → φ <<< (R².fmap g <<< fmap f)) (sym isAssociative) ⟩
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(join <<< fmap join) <<< (R².fmap g <<< fmap f)
|
||||
≡⟨ sym ℂ.isAssociative ⟩
|
||||
join <<< (fmap join <<< (R².fmap g <<< fmap f))
|
||||
≡⟨ cong (_<<<_ join) ℂ.isAssociative ⟩
|
||||
join <<< (fmap join <<< R².fmap g <<< fmap f)
|
||||
≡⟨⟩
|
||||
join <<< (fmap join <<< fmap (fmap g) <<< fmap f)
|
||||
≡⟨ cong (λ φ → join <<< φ) (sym distrib3) ⟩
|
||||
join <<< fmap (join <<< fmap g <<< f)
|
||||
∎
|
||||
where
|
||||
module R² = Functor F[ R ∘ R ]
|
||||
distrib3 : ∀ {A B C D} {a : Arrow C D} {b : Arrow B C} {c : Arrow A B}
|
||||
|
|
@ -96,31 +120,6 @@ record IsMonad (raw : RawMonad) : Set ℓ where
|
|||
R.fmap (a <<< b <<< c) ≡⟨ R.isDistributive ⟩
|
||||
R.fmap (a <<< b) <<< R.fmap c ≡⟨ cong (_<<< _) R.isDistributive ⟩
|
||||
R.fmap a <<< R.fmap b <<< R.fmap c ∎
|
||||
aux = begin
|
||||
joinT Z <<< R.fmap (joinT Z <<< R.fmap g <<< f)
|
||||
≡⟨ cong (λ φ → joinT Z <<< φ) distrib3 ⟩
|
||||
joinT Z <<< (R.fmap (joinT Z) <<< R.fmap (R.fmap g) <<< R.fmap f)
|
||||
≡⟨⟩
|
||||
joinT Z <<< (R.fmap (joinT Z) <<< R².fmap g <<< R.fmap f)
|
||||
≡⟨ cong (_<<<_ (joinT Z)) (sym ℂ.isAssociative) ⟩
|
||||
joinT Z <<< (R.fmap (joinT Z) <<< (R².fmap g <<< R.fmap f))
|
||||
≡⟨ ℂ.isAssociative ⟩
|
||||
(joinT Z <<< R.fmap (joinT Z)) <<< (R².fmap g <<< R.fmap f)
|
||||
≡⟨ cong (λ φ → φ <<< (R².fmap g <<< R.fmap f)) isAssociative ⟩
|
||||
(joinT Z <<< joinT (R.omap Z)) <<< (R².fmap g <<< R.fmap f)
|
||||
≡⟨ ℂ.isAssociative ⟩
|
||||
joinT Z <<< joinT (R.omap Z) <<< R².fmap g <<< R.fmap f
|
||||
≡⟨⟩
|
||||
((joinT Z <<< joinT (R.omap Z)) <<< R².fmap g) <<< R.fmap f
|
||||
≡⟨ cong (_<<< R.fmap f) (sym ℂ.isAssociative) ⟩
|
||||
(joinT Z <<< (joinT (R.omap Z) <<< R².fmap g)) <<< R.fmap f
|
||||
≡⟨ cong (λ φ → φ <<< R.fmap f) (cong (_<<<_ (joinT Z)) (joinN g)) ⟩
|
||||
(joinT Z <<< (R.fmap g <<< joinT Y)) <<< R.fmap f
|
||||
≡⟨ cong (_<<< R.fmap f) ℂ.isAssociative ⟩
|
||||
joinT Z <<< R.fmap g <<< joinT Y <<< R.fmap f
|
||||
≡⟨ sym (Category.isAssociative ℂ) ⟩
|
||||
joinT Z <<< R.fmap g <<< (joinT Y <<< R.fmap f)
|
||||
∎
|
||||
|
||||
record Monad : Set ℓ where
|
||||
field
|
||||
|
|
|
|||
|
|
@ -10,6 +10,8 @@ open import Cat.Category
|
|||
open import Cat.Category.Functor as F
|
||||
import Cat.Category.NaturalTransformation
|
||||
open import Cat.Category.Monad
|
||||
import Cat.Category.Monad.Monoidal as Monoidal
|
||||
import Cat.Category.Monad.Kleisli as Kleisli
|
||||
open import Cat.Categories.Fun
|
||||
open import Cat.Equivalence
|
||||
|
||||
|
|
|
|||
Loading…
Reference in a new issue