diff --git a/src/Cat/Category/Monad/Voevodsky.agda b/src/Cat/Category/Monad/Voevodsky.agda index 27bf449..3c0554f 100644 --- a/src/Cat/Category/Monad/Voevodsky.agda +++ b/src/Cat/Category/Monad/Voevodsky.agda @@ -109,9 +109,9 @@ module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where ; isMonad = isMnd } - voe-2-3-1-fromMonad : (m : M.Monad) → §2-3.§1 (M.Monad.Romap m) (λ {X} → M.Monad.pureT m X) + §1-fromMonad : (m : M.Monad) → §2-3.§1 (M.Monad.Romap m) (λ {X} → M.Monad.pureT m X) -- voe-2-3-1-fromMonad : (m : M.Monad) → voe.§2-3.§1 (M.Monad.Romap m) (λ {X} → M.Monad.pureT m X) - voe-2-3-1-fromMonad m = record + §1-fromMonad m = record { fmap = Functor.fmap R ; RisFunctor = Functor.isFunctor R ; pureN = pureN @@ -127,8 +127,8 @@ module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where joinT = M.RawMonad.joinT raw joinN = M.RawMonad.joinN raw - voe-2-3-2-fromMonad : (m : K.Monad) → §2-3.§2 (K.Monad.omap m) (K.Monad.pure m) - voe-2-3-2-fromMonad m = record + §2-fromMonad : (m : K.Monad) → §2-3.§2 (K.Monad.omap m) (K.Monad.pure m) + §2-fromMonad m = record { bind = K.Monad.bind m ; isMnd = K.Monad.isMonad m } @@ -142,33 +142,33 @@ module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where Kleisli→Monoidal = inverse Monoidal≃Kleisli forth : §2-3.§1 omap pure → §2-3.§2 omap pure - forth = voe-2-3-2-fromMonad ∘ Monoidal→Kleisli ∘ §2-3.§1.toMonad + forth = §2-fromMonad ∘ Monoidal→Kleisli ∘ §2-3.§1.toMonad back : §2-3.§2 omap pure → §2-3.§1 omap pure - back = voe-2-3-1-fromMonad ∘ Kleisli→Monoidal ∘ §2-3.§2.toMonad + back = §1-fromMonad ∘ Kleisli→Monoidal ∘ §2-3.§2.toMonad forthEq : ∀ m → _ ≡ _ forthEq m = begin (forth ∘ back) m ≡⟨⟩ -- In full gory detail: - ( voe-2-3-2-fromMonad + ( §2-fromMonad ∘ Monoidal→Kleisli ∘ §2-3.§1.toMonad - ∘ voe-2-3-1-fromMonad + ∘ §1-fromMonad ∘ Kleisli→Monoidal ∘ §2-3.§2.toMonad ) m ≡⟨⟩ -- fromMonad and toMonad are inverses - ( voe-2-3-2-fromMonad + ( §2-fromMonad ∘ Monoidal→Kleisli ∘ Kleisli→Monoidal ∘ §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 + ( §2-fromMonad ∘ §2-3.§2.toMonad ) m ≡⟨⟩ - ( voe-2-3-2-fromMonad + ( §2-fromMonad ∘ §2-3.§2.toMonad ) m ≡⟨⟩ -- fromMonad and toMonad are inverses m ∎ @@ -185,19 +185,19 @@ module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where backEq : ∀ m → (back ∘ forth) m ≡ m backEq m = begin (back ∘ forth) m ≡⟨⟩ - ( voe-2-3-1-fromMonad + ( §1-fromMonad ∘ Kleisli→Monoidal ∘ §2-3.§2.toMonad - ∘ voe-2-3-2-fromMonad + ∘ §2-fromMonad ∘ Monoidal→Kleisli ∘ §2-3.§1.toMonad ) m ≡⟨⟩ -- fromMonad and toMonad are inverses - ( voe-2-3-1-fromMonad + ( §1-fromMonad ∘ Kleisli→Monoidal ∘ Monoidal→Kleisli ∘ §2-3.§1.toMonad ) m ≡⟨ cong (λ φ → φ m) t ⟩ -- Monoidal→Kleisli and Kleisli→Monoidal are inverses - ( voe-2-3-1-fromMonad + ( §1-fromMonad ∘ §2-3.§1.toMonad ) m ≡⟨⟩ -- fromMonad and toMonad are inverses m ∎