Export TypeIsomorphism as an alias for Equivalence.Isomorphism
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@ -29,7 +29,11 @@
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module Cat.Category where
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module Cat.Category where
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open import Cat.Prelude
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open import Cat.Prelude
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open import Cat.Equivalence as Equivalence renaming (_≅_ to _≈_ ; Isomorphism to TypeIsomorphism) hiding (preorder≅)
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import Cat.Equivalence
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open Cat.Equivalence public using () renaming (Isomorphism to TypeIsomorphism)
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open Cat.Equivalence
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renaming (_≅_ to _≈_)
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hiding (preorder≅ ; Isomorphism)
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import Function
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import Function
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@ -485,7 +489,7 @@ module Opposite {ℓa ℓb : Level} where
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open IsPreCategory isPreCategory
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open IsPreCategory isPreCategory
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module _ {A B : ℂ.Object} where
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module _ {A B : ℂ.Object} where
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k : Equivalence.Isomorphism (ℂ.idToIso A B)
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k : TypeIsomorphism (ℂ.idToIso A B)
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k = toIso _ _ ℂ.univalent
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k = toIso _ _ ℂ.univalent
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open Σ k renaming (fst to η ; snd to inv-η)
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open Σ k renaming (fst to η ; snd to inv-η)
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open AreInverses inv-η
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open AreInverses inv-η
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@ -537,7 +541,7 @@ module Opposite {ℓa ℓb : Level} where
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x ∎)
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x ∎)
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}
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}
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h : Equivalence.Isomorphism (idToIso A B)
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h : TypeIsomorphism (idToIso A B)
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h = ζ , inv-ζ
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h = ζ , inv-ζ
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isCategory : IsCategory opRaw
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isCategory : IsCategory opRaw
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@ -209,3 +209,6 @@ 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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Monoidal≡Kleisli : M.Monad ≡ K.Monad
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Monoidal≡Kleisli = ua (forth , eqv)
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@ -172,9 +172,6 @@ module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
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where
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where
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t' : ((Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ §2-3.§2.toMonad {omap} {pure})
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t' : ((Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ §2-3.§2.toMonad {omap} {pure})
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≡ §2-3.§2.toMonad
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≡ §2-3.§2.toMonad
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cong-d : ∀ {ℓ} {A : Set ℓ} {ℓ'} {B : A → Set ℓ'} {x y : A}
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→ (f : (x : A) → B x) → (eq : x ≡ y) → PathP (\ i → B (eq i)) (f x) (f y)
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cong-d f p = λ i → f (p i)
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t' = cong (\ φ → φ ∘ §2-3.§2.toMonad) re-ve
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t' = cong (\ φ → φ ∘ §2-3.§2.toMonad) re-ve
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t : (§2-fromMonad ∘ (Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ §2-3.§2.toMonad {omap} {pure})
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t : (§2-fromMonad ∘ (Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ §2-3.§2.toMonad {omap} {pure})
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≡ (§2-fromMonad ∘ §2-3.§2.toMonad)
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≡ (§2-fromMonad ∘ §2-3.§2.toMonad)
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@ -203,12 +203,15 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
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: ((X , xa , xb) ≡ (Y , ya , yb))
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: ((X , xa , xb) ≡ (Y , ya , yb))
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≈ (Σ[ p ∈ (X ≡ Y) ] (PathP (λ i → ℂ.Arrow (p i) A) xa ya) × (PathP (λ i → ℂ.Arrow (p i) B) xb yb))
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≈ (Σ[ p ∈ (X ≡ Y) ] (PathP (λ i → ℂ.Arrow (p i) A) xa ya) × (PathP (λ i → ℂ.Arrow (p i) B) xb yb))
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step0
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step0
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= (λ p → (λ i → fst (p i)) , (λ i → fst (snd (p i))) , (λ i → snd (snd (p i))))
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= (λ p → cong fst p , cong-d (fst ⊙ snd) p , cong-d (snd ⊙ snd) p)
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, (λ x → λ i → fst x i , (fst (snd x) i) , (snd (snd x) i))
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-- , (λ x → λ i → fst x i , (fst (snd x) i) , (snd (snd x) i))
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, (λ{ (p , q , r) → Σ≡ p λ i → q i , r i})
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, record
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, record
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{ verso-recto = {!!}
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{ verso-recto = {!!}
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; recto-verso = {!!}
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; recto-verso = {!!}
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}
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}
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where
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open import Function renaming (_∘_ to _⊙_)
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step1
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step1
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: (Σ[ p ∈ (X ≡ Y) ] (PathP (λ i → ℂ.Arrow (p i) A) xa ya) × (PathP (λ i → ℂ.Arrow (p i) B) xb yb))
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: (Σ[ p ∈ (X ≡ Y) ] (PathP (λ i → ℂ.Arrow (p i) A) xa ya) × (PathP (λ i → ℂ.Arrow (p i) B) xb yb))
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≈ Σ (X ℂ.≅ Y) (λ iso
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≈ Σ (X ℂ.≅ Y) (λ iso
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