More readable goal for voevodsky's construction
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BACKLOG.md
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BACKLOG.md
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@ -1,13 +1,14 @@
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Backlog
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=======
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Prove postulates in `Cat.Wishlist`
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`ntypeCommulative` might be there as well.
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Prove that the opposite category is a category.
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Prove postulates in `Cat.Wishlist`:
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* `ntypeCommulative : n ≤ m → HasLevel ⟨ n ⟩₋₂ A → HasLevel ⟨ m ⟩₋₂ A`
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Prove univalence for the category of
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* the opposite category
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* sets
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This does not follow trivially from `Cubical.Univalence.univalence` because
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objects are not `Set` but `hSet`
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* functors and natural transformations
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Prove:
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@ -24,7 +25,10 @@ Prove:
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* Monad
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* Monoidal monad ✓
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* Kleisli monad ✓
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* Problem 2.3 in voe
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* Kleisli ≃ Monoidal ✓
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* Problem 2.3 in [voe]
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* 1st contruction ~ monoidal ✓
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* 2nd contruction ~ klesli ✓
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* 1st ≃ 2nd ✗
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I've managed to set this up so I should be able to reuse the proof that
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Kleisli ≃ Monoidal, but I don't know why it doesn't typecheck.
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@ -7,6 +7,7 @@ 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 Function using (_∘_ ; _$_)
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open import Cubical
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open import Cubical.NType.Properties using (lemPropF ; lemSig ; lemSigP)
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@ -20,13 +21,26 @@ import Cat.Category.Monad.Monoidal as Monoidal
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import Cat.Category.Monad.Kleisli as Kleisli
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open import Cat.Categories.Fun
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-- Utilities
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module _ {ℓa ℓb : Level} {A : Set ℓa} {B : Set ℓb} where
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module _ (e : A ≃ B) where
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obverse : A → B
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obverse = proj₁ e
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reverse : B → A
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reverse = inverse e
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-- TODO Implement and push upstream.
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postulate
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verso-recto : reverse ∘ obverse ≡ Function.id
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recto-verso : obverse ∘ reverse ≡ Function.id
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module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
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private
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ℓ = ℓa ⊔ ℓb
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module ℂ = Category ℂ
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open ℂ using (Object ; Arrow)
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open NaturalTransformation ℂ ℂ
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open import Function using (_∘_ ; _$_)
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module M = Monoidal ℂ
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module K = Kleisli ℂ
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@ -162,7 +176,7 @@ module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
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∘ Monoidal→Kleisli
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∘ Kleisli→Monoidal
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∘ §2-3.§2.toMonad
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) m ≡⟨ u ⟩
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) m ≡⟨ cong (λ φ → φ m) t ⟩
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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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( §2-fromMonad
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@ -173,14 +187,12 @@ module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
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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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ve-re : Monoidal→Kleisli ∘ Kleisli→Monoidal ≡ Function.id
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ve-re = recto-verso Monoidal≃Kleisli
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t : (§2-fromMonad ∘ (Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ §2-3.§2.toMonad)
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≡ (§2-fromMonad ∘ §2-3.§2.toMonad)
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t = cong (λ φ → §2-fromMonad ∘ (λ{ {ω} → φ {{!????!}}}) ∘ §2-3.§2.toMonad) {!lem!}
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u : (§2-fromMonad ∘ (Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ §2-3.§2.toMonad) m
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≡ (§2-fromMonad ∘ §2-3.§2.toMonad) m
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u = cong (λ φ → φ m) t
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-- Why can I not give φ in the first hole like I do below in `backEq.t`?
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t = cong (λ φ → §2-fromMonad ∘ {!φ!} ∘ §2-3.§2.toMonad) ve-re
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backEq : ∀ m → (back ∘ forth) m ≡ m
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backEq m = begin
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@ -202,7 +214,12 @@ module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
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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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re-ve : Kleisli→Monoidal ∘ Monoidal→Kleisli ≡ Function.id
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re-ve = verso-recto Monoidal≃Kleisli
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t : §1-fromMonad ∘ Kleisli→Monoidal ∘ Monoidal→Kleisli ∘ §2-3.§1.toMonad
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≡ §1-fromMonad ∘ §2-3.§1.toMonad
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-- Why does `re-ve` not satisfy this goal?
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t = cong (λ φ → §1-fromMonad ∘ φ ∘ §2-3.§1.toMonad) ({!re-ve!})
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voe-isEquiv : isEquiv (§2-3.§1 omap pure) (§2-3.§2 omap pure) forth
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voe-isEquiv = gradLemma forth back forthEq backEq
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