Move voevodsky's construction to own module
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@ -9,6 +9,7 @@ open import Cat.Category.CartesianClosed
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open import Cat.Category.NaturalTransformation
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open import Cat.Category.NaturalTransformation
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open import Cat.Category.Yoneda
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open import Cat.Category.Yoneda
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open import Cat.Category.Monad
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open import Cat.Category.Monad
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open import Cat.Category.Monad.Voevodsky
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open import Cat.Categories.Sets
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open import Cat.Categories.Sets
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open import Cat.Categories.Cat
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open import Cat.Categories.Cat
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@ -389,9 +389,6 @@ module Kleisli {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
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Monad.isMonad (Monad≡ i) = eqIsMonad i
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Monad.isMonad (Monad≡ i) = eqIsMonad i
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-- | The monoidal- and kleisli presentation of monads are equivalent.
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-- | The monoidal- and kleisli presentation of monads are equivalent.
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--
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-- This is *not* problem 2.3 in [voe].
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-- This is problem 2.3 in [voe].
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module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
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module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
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private
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private
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module ℂ = Category ℂ
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module ℂ = Category ℂ
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@ -565,208 +562,3 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
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Monoidal≃Kleisli : M.Monad ≃ K.Monad
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Monoidal≃Kleisli : M.Monad ≃ K.Monad
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Monoidal≃Kleisli = forth , eqv
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Monoidal≃Kleisli = forth , eqv
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module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
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private
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ℓ = ℓa ⊔ ℓb
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module ℂ = Category ℂ
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open ℂ using (Object ; Arrow ; _∘_)
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open NaturalTransformation ℂ ℂ
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module M = Monoidal ℂ
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module K = Kleisli ℂ
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module voe-2-3 (omap : Omap ℂ ℂ) (pure : {X : Object} → Arrow X (omap X)) where
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record voe-2-3-1 : Set ℓ where
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open M
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field
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fmap : Fmap ℂ ℂ omap
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join : {A : Object} → ℂ [ omap (omap A) , omap A ]
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Rraw : RawFunctor ℂ ℂ
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Rraw = record
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{ omap = omap
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; fmap = fmap
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}
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field
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RisFunctor : IsFunctor ℂ ℂ Rraw
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R : EndoFunctor ℂ
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R = record
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{ raw = Rraw
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; isFunctor = RisFunctor
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}
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pureT : (X : Object) → Arrow X (omap X)
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pureT X = pure {X}
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field
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pureN : Natural F.identity R pureT
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pureNT : NaturalTransformation F.identity R
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pureNT = pureT , pureN
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joinT : (A : Object) → ℂ [ omap (omap A) , omap A ]
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joinT A = join {A}
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field
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joinN : Natural F[ R ∘ R ] R joinT
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joinNT : NaturalTransformation F[ R ∘ R ] R
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joinNT = joinT , joinN
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rawMnd : RawMonad
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rawMnd = record
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{ R = R
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; pureNT = pureNT
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; joinNT = joinNT
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}
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field
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isMnd : IsMonad rawMnd
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toMonad : Monad
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toMonad = record
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{ raw = rawMnd
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; isMonad = isMnd
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}
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record voe-2-3-2 : Set ℓ where
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open K
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field
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bind : {X Y : Object} → ℂ [ X , omap Y ] → ℂ [ omap X , omap Y ]
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rawMnd : RawMonad
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rawMnd = record
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{ omap = omap
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; pure = pure
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; bind = bind
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}
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field
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isMnd : IsMonad rawMnd
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toMonad : Monad
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toMonad = record
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{ raw = rawMnd
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; isMonad = isMnd
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}
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module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
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private
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module M = Monoidal ℂ
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module K = Kleisli ℂ
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open voe-2-3 ℂ
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voe-2-3-1-fromMonad : (m : M.Monad) → voe-2-3-1 (M.Monad.Romap m) (λ {X} → M.Monad.pureT m X)
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voe-2-3-1-fromMonad m = record
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{ fmap = Functor.fmap R
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; RisFunctor = Functor.isFunctor R
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; pureN = pureN
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; join = λ {X} → joinT X
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; joinN = joinN
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; isMnd = M.Monad.isMonad m
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}
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where
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raw = M.Monad.raw m
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R = M.RawMonad.R raw
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pureT = M.RawMonad.pureT raw
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pureN = M.RawMonad.pureN raw
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joinT = M.RawMonad.joinT raw
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joinN = M.RawMonad.joinN raw
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voe-2-3-2-fromMonad : (m : K.Monad) → voe-2-3-2 (K.Monad.omap m) (K.Monad.pure m)
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voe-2-3-2-fromMonad m = record
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{ bind = K.Monad.bind m
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; isMnd = K.Monad.isMonad m
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}
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module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
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private
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ℓ = ℓa ⊔ ℓb
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module ℂ = Category ℂ
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open ℂ using (Object ; Arrow)
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open NaturalTransformation ℂ ℂ
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module M = Monoidal ℂ
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module K = Kleisli ℂ
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open import Function using (_∘_ ; _$_)
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module _ (omap : Omap ℂ ℂ) (pure : {X : Object} → Arrow X (omap X)) where
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open voe-2-3 ℂ
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private
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Monoidal→Kleisli : M.Monad → K.Monad
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Monoidal→Kleisli = proj₁ Monoidal≃Kleisli
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Kleisli→Monoidal : K.Monad → M.Monad
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Kleisli→Monoidal = inverse Monoidal≃Kleisli
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forth : voe-2-3-1 omap pure → voe-2-3-2 omap pure
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forth = voe-2-3-2-fromMonad ∘ Monoidal→Kleisli ∘ voe-2-3.voe-2-3-1.toMonad
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back : voe-2-3-2 omap pure → voe-2-3-1 omap pure
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back = voe-2-3-1-fromMonad ∘ Kleisli→Monoidal ∘ voe-2-3.voe-2-3-2.toMonad
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forthEq : ∀ m → _ ≡ _
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forthEq m = begin
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(forth ∘ back) m ≡⟨⟩
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-- In full gory detail:
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( voe-2-3-2-fromMonad
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∘ Monoidal→Kleisli
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∘ voe-2-3.voe-2-3-1.toMonad
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∘ voe-2-3-1-fromMonad
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∘ Kleisli→Monoidal
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∘ voe-2-3.voe-2-3-2.toMonad
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) m ≡⟨⟩ -- fromMonad and toMonad are inverses
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( voe-2-3-2-fromMonad
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∘ Monoidal→Kleisli
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∘ Kleisli→Monoidal
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∘ voe-2-3.voe-2-3-2.toMonad
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) m ≡⟨ u ⟩
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-- Monoidal→Kleisli and Kleisli→Monoidal are inverses
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-- I should be able to prove this using congruence and `lem` below.
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( voe-2-3-2-fromMonad
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∘ voe-2-3.voe-2-3-2.toMonad
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) m ≡⟨⟩
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( voe-2-3-2-fromMonad
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∘ voe-2-3.voe-2-3-2.toMonad
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) m ≡⟨⟩ -- fromMonad and toMonad are inverses
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m ∎
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where
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lem : Monoidal→Kleisli ∘ Kleisli→Monoidal ≡ Function.id
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lem = {!!} -- verso-recto Monoidal≃Kleisli
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t : {ℓ : Level} {A B : Set ℓ} {a : _ → A} {b : B → _}
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→ a ∘ (Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ b ≡ a ∘ b
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t {a = a} {b} = cong (λ φ → a ∘ φ ∘ b) lem
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u : {ℓ : Level} {A B : Set ℓ} {a : _ → A} {b : B → _}
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→ {m : _} → (a ∘ (Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ b) m ≡ (a ∘ b) m
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u {m = m} = cong (λ φ → φ m) t
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backEq : ∀ m → (back ∘ forth) m ≡ m
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backEq m = begin
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(back ∘ forth) m ≡⟨⟩
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( voe-2-3-1-fromMonad
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∘ Kleisli→Monoidal
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∘ voe-2-3.voe-2-3-2.toMonad
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∘ voe-2-3-2-fromMonad
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∘ Monoidal→Kleisli
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∘ voe-2-3.voe-2-3-1.toMonad
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) m ≡⟨⟩ -- fromMonad and toMonad are inverses
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( voe-2-3-1-fromMonad
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∘ Kleisli→Monoidal
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∘ Monoidal→Kleisli
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∘ voe-2-3.voe-2-3-1.toMonad
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) m ≡⟨ cong (λ φ → φ m) t ⟩ -- Monoidal→Kleisli and Kleisli→Monoidal are inverses
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( voe-2-3-1-fromMonad
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∘ voe-2-3.voe-2-3-1.toMonad
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) m ≡⟨⟩ -- fromMonad and toMonad are inverses
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m ∎
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where
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t = {!!} -- cong (λ φ → voe-2-3-1-fromMonad ∘ φ ∘ voe-2-3.voe-2-3-1.toMonad) (recto-verso Monoidal≃Kleisli)
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voe-isEquiv : isEquiv (voe-2-3-1 omap pure) (voe-2-3-2 omap pure) forth
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voe-isEquiv = gradLemma forth back forthEq backEq
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equiv-2-3 : voe-2-3-1 omap pure ≃ voe-2-3-2 omap pure
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equiv-2-3 = forth , voe-isEquiv
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221
src/Cat/Category/Monad/Voevodsky.agda
Normal file
221
src/Cat/Category/Monad/Voevodsky.agda
Normal file
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@ -0,0 +1,221 @@
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{-# OPTIONS --cubical --allow-unsolved-metas #-}
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module Cat.Category.Monad.Voevodsky where
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open import Agda.Primitive
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open import Data.Product
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open import Cubical
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open import Cubical.NType.Properties using (lemPropF ; lemSig ; lemSigP)
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open import Cubical.GradLemma using (gradLemma)
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open import Cat.Category
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open import Cat.Category.Functor as F
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open import Cat.Category.NaturalTransformation
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open import Cat.Category.Monad
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open import Cat.Categories.Fun
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module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
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private
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ℓ = ℓa ⊔ ℓb
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module ℂ = Category ℂ
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open ℂ using (Object ; Arrow ; _∘_)
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open NaturalTransformation ℂ ℂ
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module M = Monoidal ℂ
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module K = Kleisli ℂ
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module voe-2-3 (omap : Omap ℂ ℂ) (pure : {X : Object} → Arrow X (omap X)) where
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record voe-2-3-1 : Set ℓ where
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open M
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field
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fmap : Fmap ℂ ℂ omap
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join : {A : Object} → ℂ [ omap (omap A) , omap A ]
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Rraw : RawFunctor ℂ ℂ
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Rraw = record
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{ omap = omap
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; fmap = fmap
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}
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field
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RisFunctor : IsFunctor ℂ ℂ Rraw
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R : EndoFunctor ℂ
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R = record
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{ raw = Rraw
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; isFunctor = RisFunctor
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}
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pureT : (X : Object) → Arrow X (omap X)
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pureT X = pure {X}
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field
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pureN : Natural F.identity R pureT
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pureNT : NaturalTransformation F.identity R
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pureNT = pureT , pureN
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joinT : (A : Object) → ℂ [ omap (omap A) , omap A ]
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joinT A = join {A}
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field
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joinN : Natural F[ R ∘ R ] R joinT
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joinNT : NaturalTransformation F[ R ∘ R ] R
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joinNT = joinT , joinN
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rawMnd : RawMonad
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rawMnd = record
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{ R = R
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; pureNT = pureNT
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; joinNT = joinNT
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}
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field
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isMnd : IsMonad rawMnd
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toMonad : Monad
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toMonad = record
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{ raw = rawMnd
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; isMonad = isMnd
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}
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record voe-2-3-2 : Set ℓ where
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open K
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field
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bind : {X Y : Object} → ℂ [ X , omap Y ] → ℂ [ omap X , omap Y ]
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rawMnd : RawMonad
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rawMnd = record
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{ omap = omap
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; pure = pure
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; bind = bind
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}
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field
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isMnd : IsMonad rawMnd
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toMonad : Monad
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toMonad = record
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{ raw = rawMnd
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; isMonad = isMnd
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}
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module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
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private
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module M = Monoidal ℂ
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module K = Kleisli ℂ
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open voe-2-3 ℂ
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voe-2-3-1-fromMonad : (m : M.Monad) → voe-2-3-1 (M.Monad.Romap m) (λ {X} → M.Monad.pureT m X)
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voe-2-3-1-fromMonad m = record
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{ fmap = Functor.fmap R
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; RisFunctor = Functor.isFunctor R
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; pureN = pureN
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; join = λ {X} → joinT X
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; joinN = joinN
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; isMnd = M.Monad.isMonad m
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}
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where
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raw = M.Monad.raw m
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R = M.RawMonad.R raw
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pureT = M.RawMonad.pureT raw
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pureN = M.RawMonad.pureN raw
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joinT = M.RawMonad.joinT raw
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joinN = M.RawMonad.joinN raw
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voe-2-3-2-fromMonad : (m : K.Monad) → voe-2-3-2 (K.Monad.omap m) (K.Monad.pure m)
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voe-2-3-2-fromMonad m = record
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{ bind = K.Monad.bind m
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; isMnd = K.Monad.isMonad m
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}
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module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
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private
|
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ℓ = ℓa ⊔ ℓb
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module ℂ = Category ℂ
|
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open ℂ using (Object ; Arrow)
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open NaturalTransformation ℂ ℂ
|
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module M = Monoidal ℂ
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|
module K = Kleisli ℂ
|
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|
open import Function using (_∘_ ; _$_)
|
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|
|
||||||
|
module _ (omap : Omap ℂ ℂ) (pure : {X : Object} → Arrow X (omap X)) where
|
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|
open voe-2-3 ℂ
|
||||||
|
private
|
||||||
|
Monoidal→Kleisli : M.Monad → K.Monad
|
||||||
|
Monoidal→Kleisli = proj₁ Monoidal≃Kleisli
|
||||||
|
|
||||||
|
Kleisli→Monoidal : K.Monad → M.Monad
|
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|
Kleisli→Monoidal = inverse Monoidal≃Kleisli
|
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|
|
||||||
|
forth : voe-2-3-1 omap pure → voe-2-3-2 omap pure
|
||||||
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forth = voe-2-3-2-fromMonad ∘ Monoidal→Kleisli ∘ voe-2-3.voe-2-3-1.toMonad
|
||||||
|
|
||||||
|
back : voe-2-3-2 omap pure → voe-2-3-1 omap pure
|
||||||
|
back = voe-2-3-1-fromMonad ∘ Kleisli→Monoidal ∘ voe-2-3.voe-2-3-2.toMonad
|
||||||
|
|
||||||
|
forthEq : ∀ m → _ ≡ _
|
||||||
|
forthEq m = begin
|
||||||
|
(forth ∘ back) m ≡⟨⟩
|
||||||
|
-- In full gory detail:
|
||||||
|
( voe-2-3-2-fromMonad
|
||||||
|
∘ Monoidal→Kleisli
|
||||||
|
∘ voe-2-3.voe-2-3-1.toMonad
|
||||||
|
∘ voe-2-3-1-fromMonad
|
||||||
|
∘ Kleisli→Monoidal
|
||||||
|
∘ voe-2-3.voe-2-3-2.toMonad
|
||||||
|
) m ≡⟨⟩ -- fromMonad and toMonad are inverses
|
||||||
|
( voe-2-3-2-fromMonad
|
||||||
|
∘ Monoidal→Kleisli
|
||||||
|
∘ Kleisli→Monoidal
|
||||||
|
∘ voe-2-3.voe-2-3-2.toMonad
|
||||||
|
) m ≡⟨ u ⟩
|
||||||
|
-- Monoidal→Kleisli and Kleisli→Monoidal are inverses
|
||||||
|
-- I should be able to prove this using congruence and `lem` below.
|
||||||
|
( voe-2-3-2-fromMonad
|
||||||
|
∘ voe-2-3.voe-2-3-2.toMonad
|
||||||
|
) m ≡⟨⟩
|
||||||
|
( voe-2-3-2-fromMonad
|
||||||
|
∘ voe-2-3.voe-2-3-2.toMonad
|
||||||
|
) m ≡⟨⟩ -- fromMonad and toMonad are inverses
|
||||||
|
m ∎
|
||||||
|
where
|
||||||
|
lem : Monoidal→Kleisli ∘ Kleisli→Monoidal ≡ Function.id
|
||||||
|
lem = {!!} -- verso-recto Monoidal≃Kleisli
|
||||||
|
t : {ℓ : Level} {A B : Set ℓ} {a : _ → A} {b : B → _}
|
||||||
|
→ a ∘ (Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ b ≡ a ∘ b
|
||||||
|
t {a = a} {b} = cong (λ φ → a ∘ φ ∘ b) lem
|
||||||
|
u : {ℓ : Level} {A B : Set ℓ} {a : _ → A} {b : B → _}
|
||||||
|
→ {m : _} → (a ∘ (Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ b) m ≡ (a ∘ b) m
|
||||||
|
u {m = m} = cong (λ φ → φ m) t
|
||||||
|
|
||||||
|
backEq : ∀ m → (back ∘ forth) m ≡ m
|
||||||
|
backEq m = begin
|
||||||
|
(back ∘ forth) m ≡⟨⟩
|
||||||
|
( voe-2-3-1-fromMonad
|
||||||
|
∘ Kleisli→Monoidal
|
||||||
|
∘ voe-2-3.voe-2-3-2.toMonad
|
||||||
|
∘ voe-2-3-2-fromMonad
|
||||||
|
∘ Monoidal→Kleisli
|
||||||
|
∘ voe-2-3.voe-2-3-1.toMonad
|
||||||
|
) m ≡⟨⟩ -- fromMonad and toMonad are inverses
|
||||||
|
( voe-2-3-1-fromMonad
|
||||||
|
∘ Kleisli→Monoidal
|
||||||
|
∘ Monoidal→Kleisli
|
||||||
|
∘ voe-2-3.voe-2-3-1.toMonad
|
||||||
|
) m ≡⟨ cong (λ φ → φ m) t ⟩ -- Monoidal→Kleisli and Kleisli→Monoidal are inverses
|
||||||
|
( voe-2-3-1-fromMonad
|
||||||
|
∘ voe-2-3.voe-2-3-1.toMonad
|
||||||
|
) m ≡⟨⟩ -- fromMonad and toMonad are inverses
|
||||||
|
m ∎
|
||||||
|
where
|
||||||
|
t = {!!} -- cong (λ φ → voe-2-3-1-fromMonad ∘ φ ∘ voe-2-3.voe-2-3-1.toMonad) (recto-verso Monoidal≃Kleisli)
|
||||||
|
|
||||||
|
voe-isEquiv : isEquiv (voe-2-3-1 omap pure) (voe-2-3-2 omap pure) forth
|
||||||
|
voe-isEquiv = gradLemma forth back forthEq backEq
|
||||||
|
|
||||||
|
equiv-2-3 : voe-2-3-1 omap pure ≃ voe-2-3-2 omap pure
|
||||||
|
equiv-2-3 = forth , voe-isEquiv
|
||||||
Loading…
Reference in a new issue