Add new type-synonym
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@ -49,6 +49,9 @@ record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
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IsIdentity id = {A B : Object} {f : Arrow A B}
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→ f ∘ id ≡ f × id ∘ f ≡ f
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ArrowsAreSets : Set (ℓa ⊔ ℓb)
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ArrowsAreSets = ∀ {A B : Object} → isSet (Arrow A B)
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IsInverseOf : ∀ {A B} → (Arrow A B) → (Arrow B A) → Set ℓb
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IsInverseOf = λ f g → g ∘ f ≡ 𝟙 × f ∘ g ≡ 𝟙
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@ -100,7 +103,7 @@ record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc
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field
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assoc : IsAssociative
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ident : IsIdentity 𝟙
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arrowIsSet : ∀ {A B : Object} → isSet (Arrow A B)
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arrowIsSet : ArrowsAreSets
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univalent : Univalent ident
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-- `IsCategory` is a mere proposition.
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@ -162,8 +165,13 @@ module _ {ℓa ℓb : Level} {C : RawCategory ℓa ℓb} where
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ident : (λ _ → IsIdentity 𝟙) [ X.ident ≡ Y.ident ]
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ident = propIsIdentity x X.ident Y.ident
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done : x ≡ y
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U : ∀ {a : IsIdentity 𝟙} → (λ _ → IsIdentity 𝟙) [ X.ident ≡ a ] → (b : Univalent a) → Set _
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U eqwal bbb = (λ i → Univalent (eqwal i)) [ X.univalent ≡ bbb ]
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U : ∀ {a : IsIdentity 𝟙}
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→ (λ _ → IsIdentity 𝟙) [ X.ident ≡ a ]
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→ (b : Univalent a)
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→ Set _
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U eqwal bbb =
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(λ i → Univalent (eqwal i))
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[ X.univalent ≡ bbb ]
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P : (y : IsIdentity 𝟙)
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→ (λ _ → IsIdentity 𝟙) [ X.ident ≡ y ] → Set _
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P y eq = ∀ (b' : Univalent y) → U eq b'
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