Move opposite- and span- category to own modules
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@ -10,19 +10,20 @@ Name & Link \\
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\hline
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Categories & \sourcelink{Cat.Category} \\
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Functors & \sourcelink{Cat.Category.Functor} \\
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Products & \sourcelink{Cat.Category.Products} \\
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Exponentials & \sourcelink{Cat.Category.Exponentials} \\
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Products & \sourcelink{Cat.Category.Product} \\
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Exponentials & \sourcelink{Cat.Category.Exponential} \\
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Cartesian closed categories & \sourcelink{Cat.Category.CartesianClosed} \\
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Natural transformations & \sourcelink{Cat.Category.NaturalTransformation} \\
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Yoneda embedding & \sourcelink{Cat.Category.Yoneda} \\
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Monads & \sourcelink{Cat.Category.Monads} \\
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Monads & \sourcelink{Cat.Category.Monad} \\
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Kleisli Monads & \sourcelink{Cat.Category.Monad.Kleisli} \\
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Monoidal Monads & \sourcelink{Cat.Category.Monad.Monoidal} \\
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Voevodsky's construction & \sourcelink{Cat.Category.Monad.Voevodsky} \\
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%% Categories & \null \\
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%%
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Opposite category &
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\href{\sourcebasepath Cat.Category.html#22744}{\texttt{Cat.Category.Opposite}} \\
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Opposite category & \sourcelink{Cat.Categories.Opposite} \\
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Category of sets & \sourcelink{Cat.Categories.Sets} \\
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Span category &
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\href{\sourcebasepath Cat.Category.Product.html#2919}{\texttt{Cat.Category.Product.Span}} \\
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Span category & \sourcelink{Cat.Categories.Span} \\
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%%
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\end{tabular}
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\end{center}
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@ -14,6 +14,7 @@ open import Cat.Category.Monad.Monoidal
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open import Cat.Category.Monad.Kleisli
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open import Cat.Category.Monad.Voevodsky
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open import Cat.Categories.Opposite
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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.Rel
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@ -5,6 +5,7 @@ open import Cat.Prelude
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open import Cat.Category
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open import Cat.Category.Functor
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open import Cat.Categories.Fam
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open import Cat.Categories.Opposite
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module _ {ℓa ℓb : Level} where
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record CwF : Set (lsuc (ℓa ⊔ ℓb)) where
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@ -3,11 +3,11 @@ module Cat.Categories.Fun where
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open import Cat.Prelude
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open import Cat.Equivalence
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open import Cat.Category
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open import Cat.Category.Functor
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import Cat.Category.NaturalTransformation
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as NaturalTransformation
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open import Cat.Categories.Opposite
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module Fun {ℓc ℓc' ℓd ℓd' : Level} (ℂ : Category ℓc ℓc') (𝔻 : Category ℓd ℓd') where
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open NaturalTransformation ℂ 𝔻 public hiding (module Properties)
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@ -0,0 +1,96 @@
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{-# OPTIONS --cubical #-}
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module Cat.Categories.Opposite where
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open import Cat.Prelude
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open import Cat.Equivalence
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open import Cat.Category
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-- | The opposite category
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--
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-- The opposite category is the category where the direction of the arrows are
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-- flipped.
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module _ {ℓa ℓb : Level} where
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module _ (ℂ : Category ℓa ℓb) where
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private
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module _ where
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module ℂ = Category ℂ
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opRaw : RawCategory ℓa ℓb
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RawCategory.Object opRaw = ℂ.Object
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RawCategory.Arrow opRaw = flip ℂ.Arrow
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RawCategory.identity opRaw = ℂ.identity
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RawCategory._<<<_ opRaw = ℂ._>>>_
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open RawCategory opRaw
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isPreCategory : IsPreCategory opRaw
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IsPreCategory.isAssociative isPreCategory = sym ℂ.isAssociative
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IsPreCategory.isIdentity isPreCategory = swap ℂ.isIdentity
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IsPreCategory.arrowsAreSets isPreCategory = ℂ.arrowsAreSets
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open IsPreCategory isPreCategory
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module _ {A B : ℂ.Object} where
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open Σ (toIso _ _ (ℂ.univalent {A} {B}))
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renaming (fst to idToIso* ; snd to inv*)
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open AreInverses {f = ℂ.idToIso A B} {idToIso*} inv*
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shuffle : A ≊ B → A ℂ.≊ B
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shuffle (f , g , inv) = g , f , inv
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shuffle~ : A ℂ.≊ B → A ≊ B
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shuffle~ (f , g , inv) = g , f , inv
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lem : (p : A ≡ B) → idToIso A B p ≡ shuffle~ (ℂ.idToIso A B p)
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lem p = isoEq refl
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isoToId* : A ≊ B → A ≡ B
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isoToId* = idToIso* ∘ shuffle
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inv : AreInverses (idToIso A B) isoToId*
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inv =
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( funExt (λ x → begin
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(isoToId* ∘ idToIso A B) x
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≡⟨⟩
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(idToIso* ∘ shuffle ∘ idToIso A B) x
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≡⟨ cong (λ φ → φ x) (cong (λ φ → idToIso* ∘ shuffle ∘ φ) (funExt lem)) ⟩
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(idToIso* ∘ shuffle ∘ shuffle~ ∘ ℂ.idToIso A B) x
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≡⟨⟩
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(idToIso* ∘ ℂ.idToIso A B) x
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≡⟨ (λ i → verso-recto i x) ⟩
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x ∎)
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, funExt (λ x → begin
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(idToIso A B ∘ idToIso* ∘ shuffle) x
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≡⟨ cong (λ φ → φ x) (cong (λ φ → φ ∘ idToIso* ∘ shuffle) (funExt lem)) ⟩
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(shuffle~ ∘ ℂ.idToIso A B ∘ idToIso* ∘ shuffle) x
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≡⟨ cong (λ φ → φ x) (cong (λ φ → shuffle~ ∘ φ ∘ shuffle) recto-verso) ⟩
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(shuffle~ ∘ shuffle) x
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≡⟨⟩
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x ∎)
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)
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isCategory : IsCategory opRaw
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IsCategory.isPreCategory isCategory = isPreCategory
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IsCategory.univalent isCategory
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= univalenceFromIsomorphism (isoToId* , inv)
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opposite : Category ℓa ℓb
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Category.raw opposite = opRaw
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Category.isCategory opposite = isCategory
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-- As demonstrated here a side-effect of having no-eta-equality on constructors
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-- means that we need to pick things apart to show that things are indeed
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-- definitionally equal. I.e; a thing that would normally be provable in one
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-- line now takes 13!! Admittedly it's a simple proof.
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module _ {ℂ : Category ℓa ℓb} where
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open Category ℂ
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private
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-- Since they really are definitionally equal we just need to pick apart
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-- the data-type.
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rawInv : Category.raw (opposite (opposite ℂ)) ≡ raw
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RawCategory.Object (rawInv _) = Object
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RawCategory.Arrow (rawInv _) = Arrow
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RawCategory.identity (rawInv _) = identity
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RawCategory._<<<_ (rawInv _) = _<<<_
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oppositeIsInvolution : opposite (opposite ℂ) ≡ ℂ
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oppositeIsInvolution = Category≡ rawInv
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@ -3,11 +3,11 @@
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module Cat.Categories.Sets where
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open import Cat.Prelude as P
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open import Cat.Equivalence
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open import Cat.Category
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open import Cat.Category.Functor
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open import Cat.Category.Product
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open import Cat.Equivalence
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open import Cat.Categories.Opposite
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_⊙_ : {ℓa ℓb ℓc : Level} {A : Set ℓa} {B : Set ℓb} {C : Set ℓc} → (A ≃ B) → (B ≃ C) → A ≃ C
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eqA ⊙ eqB = Equivalence.compose eqA eqB
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@ -0,0 +1,173 @@
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{-# OPTIONS --cubical --caching #-}
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module Cat.Categories.Span where
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open import Cat.Prelude as P hiding (_×_ ; fst ; snd)
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open import Cat.Equivalence
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open import Cat.Category
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module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb)
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(let module ℂ = Category ℂ) (𝒜 ℬ : ℂ.Object) where
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open P
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private
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module _ where
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raw : RawCategory _ _
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raw = record
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{ Object = Σ[ X ∈ ℂ.Object ] ℂ.Arrow X 𝒜 × ℂ.Arrow X ℬ
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; Arrow = λ{ (A , a0 , a1) (B , b0 , b1)
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→ Σ[ f ∈ ℂ.Arrow A B ]
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ℂ [ b0 ∘ f ] ≡ a0
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× ℂ [ b1 ∘ f ] ≡ a1
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}
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; identity = λ{ {X , f , g} → ℂ.identity {X} , ℂ.rightIdentity , ℂ.rightIdentity}
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; _<<<_ = λ { {_ , a0 , a1} {_ , b0 , b1} {_ , c0 , c1} (f , f0 , f1) (g , g0 , g1)
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→ (f ℂ.<<< g)
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, (begin
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ℂ [ c0 ∘ ℂ [ f ∘ g ] ] ≡⟨ ℂ.isAssociative ⟩
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ℂ [ ℂ [ c0 ∘ f ] ∘ g ] ≡⟨ cong (λ φ → ℂ [ φ ∘ g ]) f0 ⟩
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ℂ [ b0 ∘ g ] ≡⟨ g0 ⟩
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a0 ∎
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)
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, (begin
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ℂ [ c1 ∘ ℂ [ f ∘ g ] ] ≡⟨ ℂ.isAssociative ⟩
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ℂ [ ℂ [ c1 ∘ f ] ∘ g ] ≡⟨ cong (λ φ → ℂ [ φ ∘ g ]) f1 ⟩
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ℂ [ b1 ∘ g ] ≡⟨ g1 ⟩
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a1 ∎
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)
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}
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}
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module _ where
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open RawCategory raw
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propEqs : ∀ {X' : Object}{Y' : Object} (let X , xa , xb = X') (let Y , ya , yb = Y')
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→ (xy : ℂ.Arrow X Y) → isProp (ℂ [ ya ∘ xy ] ≡ xa × ℂ [ yb ∘ xy ] ≡ xb)
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propEqs xs = propSig (ℂ.arrowsAreSets _ _) (\ _ → ℂ.arrowsAreSets _ _)
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arrowEq : {X Y : Object} {f g : Arrow X Y} → fst f ≡ fst g → f ≡ g
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arrowEq {X} {Y} {f} {g} p = λ i → p i , lemPropF propEqs p {snd f} {snd g} i
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isAssociative : IsAssociative
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isAssociative {f = f , f0 , f1} {g , g0 , g1} {h , h0 , h1} = arrowEq ℂ.isAssociative
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isIdentity : IsIdentity identity
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isIdentity {AA@(A , a0 , a1)} {BB@(B , b0 , b1)} {f , f0 , f1} = arrowEq ℂ.leftIdentity , arrowEq ℂ.rightIdentity
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arrowsAreSets : ArrowsAreSets
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arrowsAreSets {X , x0 , x1} {Y , y0 , y1}
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= sigPresSet ℂ.arrowsAreSets λ a → propSet (propEqs _)
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isPreCat : IsPreCategory raw
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IsPreCategory.isAssociative isPreCat = isAssociative
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IsPreCategory.isIdentity isPreCat = isIdentity
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IsPreCategory.arrowsAreSets isPreCat = arrowsAreSets
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open IsPreCategory isPreCat
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module _ {𝕏 𝕐 : Object} where
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open Σ 𝕏 renaming (fst to X ; snd to x)
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open Σ x renaming (fst to xa ; snd to xb)
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open Σ 𝕐 renaming (fst to Y ; snd to y)
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open Σ y renaming (fst to ya ; snd to yb)
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open import Cat.Equivalence using (composeIso) renaming (_≅_ to _≅_)
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step0
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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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step0
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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{ (p , q , r) → Σ≡ p λ i → q i , r i})
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, funExt (λ{ p → refl})
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, funExt (λ{ (p , q , r) → refl})
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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 (X ℂ.≊ Y) (λ iso
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→ let p = ℂ.isoToId iso
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in
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( PathP (λ i → ℂ.Arrow (p i) 𝒜) xa ya)
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× PathP (λ i → ℂ.Arrow (p i) ℬ) xb yb
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)
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step1
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= symIso
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(isoSigFst
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{A = (X ℂ.≊ Y)}
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{B = (X ≡ Y)}
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(ℂ.groupoidObject _ _)
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{Q = \ p → (PathP (λ i → ℂ.Arrow (p i) 𝒜) xa ya) × (PathP (λ i → ℂ.Arrow (p i) ℬ) xb yb)}
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ℂ.isoToId
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(symIso (_ , ℂ.asTypeIso {X} {Y}) .snd)
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)
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step2
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: Σ (X ℂ.≊ Y) (λ iso
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→ let p = ℂ.isoToId iso
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in
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( PathP (λ i → ℂ.Arrow (p i) 𝒜) xa ya)
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× PathP (λ i → ℂ.Arrow (p i) ℬ) xb yb
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)
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≅ ((X , xa , xb) ≊ (Y , ya , yb))
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step2
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= ( λ{ (iso@(f , f~ , inv-f) , p , q)
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→ ( f , sym (ℂ.domain-twist-sym iso p) , sym (ℂ.domain-twist-sym iso q))
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, ( f~ , sym (ℂ.domain-twist iso p) , sym (ℂ.domain-twist iso q))
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, arrowEq (fst inv-f)
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, arrowEq (snd inv-f)
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}
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)
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, (λ{ (f , f~ , inv-f , inv-f~) →
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let
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iso : X ℂ.≊ Y
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iso = fst f , fst f~ , cong fst inv-f , cong fst inv-f~
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p : X ≡ Y
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p = ℂ.isoToId iso
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pA : ℂ.Arrow X 𝒜 ≡ ℂ.Arrow Y 𝒜
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pA = cong (λ x → ℂ.Arrow x 𝒜) p
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pB : ℂ.Arrow X ℬ ≡ ℂ.Arrow Y ℬ
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pB = cong (λ x → ℂ.Arrow x ℬ) p
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k0 = begin
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coe pB xb ≡⟨ ℂ.coe-dom iso ⟩
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xb ℂ.<<< fst f~ ≡⟨ snd (snd f~) ⟩
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yb ∎
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k1 = begin
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coe pA xa ≡⟨ ℂ.coe-dom iso ⟩
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xa ℂ.<<< fst f~ ≡⟨ fst (snd f~) ⟩
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ya ∎
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in iso , coe-lem-inv k1 , coe-lem-inv k0})
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, funExt (λ x → lemSig
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(λ x → propSig prop0 (λ _ → prop1))
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_ _
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(Σ≡ refl (ℂ.propIsomorphism _ _ _)))
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, funExt (λ{ (f , _) → lemSig propIsomorphism _ _ (Σ≡ refl (propEqs _ _ _))})
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where
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prop0 : ∀ {x} → isProp (PathP (λ i → ℂ.Arrow (ℂ.isoToId x i) 𝒜) xa ya)
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prop0 {x} = pathJ (λ y p → ∀ x → isProp (PathP (λ i → ℂ.Arrow (p i) 𝒜) xa x)) (λ x → ℂ.arrowsAreSets _ _) Y (ℂ.isoToId x) ya
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prop1 : ∀ {x} → isProp (PathP (λ i → ℂ.Arrow (ℂ.isoToId x i) ℬ) xb yb)
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prop1 {x} = pathJ (λ y p → ∀ x → isProp (PathP (λ i → ℂ.Arrow (p i) ℬ) xb x)) (λ x → ℂ.arrowsAreSets _ _) Y (ℂ.isoToId x) yb
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-- One thing to watch out for here is that the isomorphisms going forwards
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-- must compose to give idToIso
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iso
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: ((X , xa , xb) ≡ (Y , ya , yb))
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≅ ((X , xa , xb) ≊ (Y , ya , yb))
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iso = step0 ⊙ step1 ⊙ step2
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where
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infixl 5 _⊙_
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_⊙_ = composeIso
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equiv1
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: ((X , xa , xb) ≡ (Y , ya , yb))
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≃ ((X , xa , xb) ≊ (Y , ya , yb))
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equiv1 = _ , fromIso _ _ (snd iso)
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univalent : Univalent
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univalent = univalenceFrom≃ equiv1
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isCat : IsCategory raw
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IsCategory.isPreCategory isCat = isPreCat
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IsCategory.univalent isCat = univalent
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span : Category _ _
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span = record
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{ raw = raw
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; isCategory = isCat
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}
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@ -631,95 +631,3 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
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_[_∘_] : {A B C : Object} → (g : Arrow B C) → (f : Arrow A B) → Arrow A C
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_[_∘_] = _<<<_
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-- | The opposite category
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--
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-- The opposite category is the category where the direction of the arrows are
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-- flipped.
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module Opposite {ℓa ℓb : Level} where
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module _ (ℂ : Category ℓa ℓb) where
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private
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module _ where
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module ℂ = Category ℂ
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opRaw : RawCategory ℓa ℓb
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RawCategory.Object opRaw = ℂ.Object
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RawCategory.Arrow opRaw = flip ℂ.Arrow
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RawCategory.identity opRaw = ℂ.identity
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RawCategory._<<<_ opRaw = ℂ._>>>_
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open RawCategory opRaw
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isPreCategory : IsPreCategory opRaw
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IsPreCategory.isAssociative isPreCategory = sym ℂ.isAssociative
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IsPreCategory.isIdentity isPreCategory = swap ℂ.isIdentity
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IsPreCategory.arrowsAreSets isPreCategory = ℂ.arrowsAreSets
|
||||
|
||||
open IsPreCategory isPreCategory
|
||||
|
||||
module _ {A B : ℂ.Object} where
|
||||
open Σ (toIso _ _ (ℂ.univalent {A} {B}))
|
||||
renaming (fst to idToIso* ; snd to inv*)
|
||||
open AreInverses {f = ℂ.idToIso A B} {idToIso*} inv*
|
||||
|
||||
shuffle : A ≊ B → A ℂ.≊ B
|
||||
shuffle (f , g , inv) = g , f , inv
|
||||
|
||||
shuffle~ : A ℂ.≊ B → A ≊ B
|
||||
shuffle~ (f , g , inv) = g , f , inv
|
||||
|
||||
lem : (p : A ≡ B) → idToIso A B p ≡ shuffle~ (ℂ.idToIso A B p)
|
||||
lem p = isoEq refl
|
||||
|
||||
isoToId* : A ≊ B → A ≡ B
|
||||
isoToId* = idToIso* ∘ shuffle
|
||||
|
||||
inv : AreInverses (idToIso A B) isoToId*
|
||||
inv =
|
||||
( funExt (λ x → begin
|
||||
(isoToId* ∘ idToIso A B) x
|
||||
≡⟨⟩
|
||||
(idToIso* ∘ shuffle ∘ idToIso A B) x
|
||||
≡⟨ cong (λ φ → φ x) (cong (λ φ → idToIso* ∘ shuffle ∘ φ) (funExt lem)) ⟩
|
||||
(idToIso* ∘ shuffle ∘ shuffle~ ∘ ℂ.idToIso A B) x
|
||||
≡⟨⟩
|
||||
(idToIso* ∘ ℂ.idToIso A B) x
|
||||
≡⟨ (λ i → verso-recto i x) ⟩
|
||||
x ∎)
|
||||
, funExt (λ x → begin
|
||||
(idToIso A B ∘ idToIso* ∘ shuffle) x
|
||||
≡⟨ cong (λ φ → φ x) (cong (λ φ → φ ∘ idToIso* ∘ shuffle) (funExt lem)) ⟩
|
||||
(shuffle~ ∘ ℂ.idToIso A B ∘ idToIso* ∘ shuffle) x
|
||||
≡⟨ cong (λ φ → φ x) (cong (λ φ → shuffle~ ∘ φ ∘ shuffle) recto-verso) ⟩
|
||||
(shuffle~ ∘ shuffle) x
|
||||
≡⟨⟩
|
||||
x ∎)
|
||||
)
|
||||
|
||||
isCategory : IsCategory opRaw
|
||||
IsCategory.isPreCategory isCategory = isPreCategory
|
||||
IsCategory.univalent isCategory
|
||||
= univalenceFromIsomorphism (isoToId* , inv)
|
||||
|
||||
opposite : Category ℓa ℓb
|
||||
Category.raw opposite = opRaw
|
||||
Category.isCategory opposite = isCategory
|
||||
|
||||
-- As demonstrated here a side-effect of having no-eta-equality on constructors
|
||||
-- means that we need to pick things apart to show that things are indeed
|
||||
-- definitionally equal. I.e; a thing that would normally be provable in one
|
||||
-- line now takes 13!! Admittedly it's a simple proof.
|
||||
module _ {ℂ : Category ℓa ℓb} where
|
||||
open Category ℂ
|
||||
private
|
||||
-- Since they really are definitionally equal we just need to pick apart
|
||||
-- the data-type.
|
||||
rawInv : Category.raw (opposite (opposite ℂ)) ≡ raw
|
||||
RawCategory.Object (rawInv _) = Object
|
||||
RawCategory.Arrow (rawInv _) = Arrow
|
||||
RawCategory.identity (rawInv _) = identity
|
||||
RawCategory._<<<_ (rawInv _) = _<<<_
|
||||
|
||||
oppositeIsInvolution : opposite (opposite ℂ) ≡ ℂ
|
||||
oppositeIsInvolution = Category≡ rawInv
|
||||
|
||||
open Opposite public
|
||||
|
|
|
|||
|
|
@ -55,286 +55,122 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
|||
× ℂ [ g ∘ snd ]
|
||||
]
|
||||
|
||||
module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} {A B : Category.Object ℂ} where
|
||||
module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
|
||||
(let module ℂ = Category ℂ) {𝒜 ℬ : ℂ.Object} where
|
||||
private
|
||||
open Category ℂ
|
||||
module _ (raw : RawProduct ℂ A B) where
|
||||
module _ (x y : IsProduct ℂ A B raw) where
|
||||
private
|
||||
module x = IsProduct x
|
||||
module y = IsProduct y
|
||||
module _ (raw : RawProduct ℂ 𝒜 ℬ) where
|
||||
private
|
||||
open Category ℂ hiding (raw)
|
||||
module _ (x y : IsProduct ℂ 𝒜 ℬ raw) where
|
||||
private
|
||||
module x = IsProduct x
|
||||
module y = IsProduct y
|
||||
|
||||
module _ {X : Object} (f : ℂ [ X , A ]) (g : ℂ [ X , B ]) where
|
||||
module _ (f×g : Arrow X y.object) where
|
||||
help : isProp (∀{y} → (ℂ [ y.fst ∘ y ] ≡ f) P.× (ℂ [ y.snd ∘ y ] ≡ g) → f×g ≡ y)
|
||||
help = propPiImpl (λ _ → propPi (λ _ → arrowsAreSets _ _))
|
||||
module _ {X : Object} (f : ℂ [ X , 𝒜 ]) (g : ℂ [ X , ℬ ]) where
|
||||
module _ (f×g : Arrow X y.object) where
|
||||
help : isProp (∀{y} → (ℂ [ y.fst ∘ y ] ≡ f) P.× (ℂ [ y.snd ∘ y ] ≡ g) → f×g ≡ y)
|
||||
help = propPiImpl (λ _ → propPi (λ _ → arrowsAreSets _ _))
|
||||
|
||||
res = ∃-unique (x.ump f g) (y.ump f g)
|
||||
res = ∃-unique (x.ump f g) (y.ump f g)
|
||||
|
||||
prodAux : x.ump f g ≡ y.ump f g
|
||||
prodAux = lemSig ((λ f×g → propSig (propSig (arrowsAreSets _ _) λ _ → arrowsAreSets _ _) (λ _ → help f×g))) _ _ res
|
||||
prodAux : x.ump f g ≡ y.ump f g
|
||||
prodAux = lemSig ((λ f×g → propSig (propSig (arrowsAreSets _ _) λ _ → arrowsAreSets _ _) (λ _ → help f×g))) _ _ res
|
||||
|
||||
propIsProduct' : x ≡ y
|
||||
propIsProduct' i = record { ump = λ f g → prodAux f g i }
|
||||
propIsProduct' : x ≡ y
|
||||
propIsProduct' i = record { ump = λ f g → prodAux f g i }
|
||||
|
||||
propIsProduct : isProp (IsProduct ℂ A B raw)
|
||||
propIsProduct : isProp (IsProduct ℂ 𝒜 ℬ raw)
|
||||
propIsProduct = propIsProduct'
|
||||
|
||||
Product≡ : {x y : Product ℂ A B} → (Product.raw x ≡ Product.raw y) → x ≡ y
|
||||
Product≡ {x} {y} p i = record { raw = p i ; isProduct = q i }
|
||||
where
|
||||
q : (λ i → IsProduct ℂ A B (p i)) [ Product.isProduct x ≡ Product.isProduct y ]
|
||||
q = lemPropF propIsProduct p
|
||||
Product≡ : {x y : Product ℂ 𝒜 ℬ} → (Product.raw x ≡ Product.raw y) → x ≡ y
|
||||
Product≡ {x} {y} p i = record { raw = p i ; isProduct = q i }
|
||||
where
|
||||
q : (λ i → IsProduct ℂ 𝒜 ℬ (p i)) [ Product.isProduct x ≡ Product.isProduct y ]
|
||||
q = lemPropF propIsProduct p
|
||||
|
||||
module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
|
||||
(let module ℂ = Category ℂ) {𝒜 ℬ : ℂ.Object} where
|
||||
open P
|
||||
open import Cat.Categories.Span
|
||||
|
||||
open P
|
||||
open Category (span ℂ 𝒜 ℬ)
|
||||
|
||||
module _ where
|
||||
raw : RawCategory _ _
|
||||
raw = record
|
||||
{ Object = Σ[ X ∈ ℂ.Object ] ℂ.Arrow X 𝒜 × ℂ.Arrow X ℬ
|
||||
; Arrow = λ{ (A , a0 , a1) (B , b0 , b1)
|
||||
→ Σ[ f ∈ ℂ.Arrow A B ]
|
||||
ℂ [ b0 ∘ f ] ≡ a0
|
||||
× ℂ [ b1 ∘ f ] ≡ a1
|
||||
}
|
||||
; identity = λ{ {X , f , g} → ℂ.identity {X} , ℂ.rightIdentity , ℂ.rightIdentity}
|
||||
; _<<<_ = λ { {_ , a0 , a1} {_ , b0 , b1} {_ , c0 , c1} (f , f0 , f1) (g , g0 , g1)
|
||||
→ (f ℂ.<<< g)
|
||||
, (begin
|
||||
ℂ [ c0 ∘ ℂ [ f ∘ g ] ] ≡⟨ ℂ.isAssociative ⟩
|
||||
ℂ [ ℂ [ c0 ∘ f ] ∘ g ] ≡⟨ cong (λ φ → ℂ [ φ ∘ g ]) f0 ⟩
|
||||
ℂ [ b0 ∘ g ] ≡⟨ g0 ⟩
|
||||
a0 ∎
|
||||
)
|
||||
, (begin
|
||||
ℂ [ c1 ∘ ℂ [ f ∘ g ] ] ≡⟨ ℂ.isAssociative ⟩
|
||||
ℂ [ ℂ [ c1 ∘ f ] ∘ g ] ≡⟨ cong (λ φ → ℂ [ φ ∘ g ]) f1 ⟩
|
||||
ℂ [ b1 ∘ g ] ≡⟨ g1 ⟩
|
||||
a1 ∎
|
||||
)
|
||||
}
|
||||
}
|
||||
|
||||
module _ where
|
||||
open RawCategory raw
|
||||
|
||||
propEqs : ∀ {X' : Object}{Y' : Object} (let X , xa , xb = X') (let Y , ya , yb = Y')
|
||||
→ (xy : ℂ.Arrow X Y) → isProp (ℂ [ ya ∘ xy ] ≡ xa × ℂ [ yb ∘ xy ] ≡ xb)
|
||||
propEqs xs = propSig (ℂ.arrowsAreSets _ _) (\ _ → ℂ.arrowsAreSets _ _)
|
||||
|
||||
arrowEq : {X Y : Object} {f g : Arrow X Y} → fst f ≡ fst g → f ≡ g
|
||||
arrowEq {X} {Y} {f} {g} p = λ i → p i , lemPropF propEqs p {snd f} {snd g} i
|
||||
|
||||
private
|
||||
isAssociative : IsAssociative
|
||||
isAssociative {f = f , f0 , f1} {g , g0 , g1} {h , h0 , h1} = arrowEq ℂ.isAssociative
|
||||
|
||||
isIdentity : IsIdentity identity
|
||||
isIdentity {AA@(A , a0 , a1)} {BB@(B , b0 , b1)} {f , f0 , f1} = arrowEq ℂ.leftIdentity , arrowEq ℂ.rightIdentity
|
||||
|
||||
arrowsAreSets : ArrowsAreSets
|
||||
arrowsAreSets {X , x0 , x1} {Y , y0 , y1}
|
||||
= sigPresSet ℂ.arrowsAreSets λ a → propSet (propEqs _)
|
||||
|
||||
isPreCat : IsPreCategory raw
|
||||
IsPreCategory.isAssociative isPreCat = isAssociative
|
||||
IsPreCategory.isIdentity isPreCat = isIdentity
|
||||
IsPreCategory.arrowsAreSets isPreCat = arrowsAreSets
|
||||
|
||||
open IsPreCategory isPreCat
|
||||
|
||||
module _ {𝕏 𝕐 : Object} where
|
||||
open Σ 𝕏 renaming (fst to X ; snd to x)
|
||||
open Σ x renaming (fst to xa ; snd to xb)
|
||||
open Σ 𝕐 renaming (fst to Y ; snd to y)
|
||||
open Σ y renaming (fst to ya ; snd to yb)
|
||||
open import Cat.Equivalence using (composeIso) renaming (_≅_ to _≅_)
|
||||
step0
|
||||
: ((X , xa , xb) ≡ (Y , ya , yb))
|
||||
≅ (Σ[ p ∈ (X ≡ Y) ] (PathP (λ i → ℂ.Arrow (p i) 𝒜) xa ya) × (PathP (λ i → ℂ.Arrow (p i) ℬ) xb yb))
|
||||
step0
|
||||
= (λ p → cong fst p , cong-d (fst ∘ snd) p , cong-d (snd ∘ snd) p)
|
||||
-- , (λ x → λ i → fst x i , (fst (snd x) i) , (snd (snd x) i))
|
||||
, (λ{ (p , q , r) → Σ≡ p λ i → q i , r i})
|
||||
, funExt (λ{ p → refl})
|
||||
, funExt (λ{ (p , q , r) → refl})
|
||||
|
||||
step1
|
||||
: (Σ[ p ∈ (X ≡ Y) ] (PathP (λ i → ℂ.Arrow (p i) 𝒜) xa ya) × (PathP (λ i → ℂ.Arrow (p i) ℬ) xb yb))
|
||||
≅ Σ (X ℂ.≊ Y) (λ iso
|
||||
→ let p = ℂ.isoToId iso
|
||||
in
|
||||
( PathP (λ i → ℂ.Arrow (p i) 𝒜) xa ya)
|
||||
× PathP (λ i → ℂ.Arrow (p i) ℬ) xb yb
|
||||
)
|
||||
step1
|
||||
= symIso
|
||||
(isoSigFst
|
||||
{A = (X ℂ.≊ Y)}
|
||||
{B = (X ≡ Y)}
|
||||
(ℂ.groupoidObject _ _)
|
||||
{Q = \ p → (PathP (λ i → ℂ.Arrow (p i) 𝒜) xa ya) × (PathP (λ i → ℂ.Arrow (p i) ℬ) xb yb)}
|
||||
ℂ.isoToId
|
||||
(symIso (_ , ℂ.asTypeIso {X} {Y}) .snd)
|
||||
)
|
||||
|
||||
step2
|
||||
: Σ (X ℂ.≊ Y) (λ iso
|
||||
→ let p = ℂ.isoToId iso
|
||||
in
|
||||
( PathP (λ i → ℂ.Arrow (p i) 𝒜) xa ya)
|
||||
× PathP (λ i → ℂ.Arrow (p i) ℬ) xb yb
|
||||
)
|
||||
≅ ((X , xa , xb) ≊ (Y , ya , yb))
|
||||
step2
|
||||
= ( λ{ (iso@(f , f~ , inv-f) , p , q)
|
||||
→ ( f , sym (ℂ.domain-twist-sym iso p) , sym (ℂ.domain-twist-sym iso q))
|
||||
, ( f~ , sym (ℂ.domain-twist iso p) , sym (ℂ.domain-twist iso q))
|
||||
, arrowEq (fst inv-f)
|
||||
, arrowEq (snd inv-f)
|
||||
}
|
||||
)
|
||||
, (λ{ (f , f~ , inv-f , inv-f~) →
|
||||
let
|
||||
iso : X ℂ.≊ Y
|
||||
iso = fst f , fst f~ , cong fst inv-f , cong fst inv-f~
|
||||
p : X ≡ Y
|
||||
p = ℂ.isoToId iso
|
||||
pA : ℂ.Arrow X 𝒜 ≡ ℂ.Arrow Y 𝒜
|
||||
pA = cong (λ x → ℂ.Arrow x 𝒜) p
|
||||
pB : ℂ.Arrow X ℬ ≡ ℂ.Arrow Y ℬ
|
||||
pB = cong (λ x → ℂ.Arrow x ℬ) p
|
||||
k0 = begin
|
||||
coe pB xb ≡⟨ ℂ.coe-dom iso ⟩
|
||||
xb ℂ.<<< fst f~ ≡⟨ snd (snd f~) ⟩
|
||||
yb ∎
|
||||
k1 = begin
|
||||
coe pA xa ≡⟨ ℂ.coe-dom iso ⟩
|
||||
xa ℂ.<<< fst f~ ≡⟨ fst (snd f~) ⟩
|
||||
ya ∎
|
||||
in iso , coe-lem-inv k1 , coe-lem-inv k0})
|
||||
, funExt (λ x → lemSig
|
||||
(λ x → propSig prop0 (λ _ → prop1))
|
||||
_ _
|
||||
(Σ≡ refl (ℂ.propIsomorphism _ _ _)))
|
||||
, funExt (λ{ (f , _) → lemSig propIsomorphism _ _ (Σ≡ refl (propEqs _ _ _))})
|
||||
where
|
||||
prop0 : ∀ {x} → isProp (PathP (λ i → ℂ.Arrow (ℂ.isoToId x i) 𝒜) xa ya)
|
||||
prop0 {x} = pathJ (λ y p → ∀ x → isProp (PathP (λ i → ℂ.Arrow (p i) 𝒜) xa x)) (λ x → ℂ.arrowsAreSets _ _) Y (ℂ.isoToId x) ya
|
||||
prop1 : ∀ {x} → isProp (PathP (λ i → ℂ.Arrow (ℂ.isoToId x i) ℬ) xb yb)
|
||||
prop1 {x} = pathJ (λ y p → ∀ x → isProp (PathP (λ i → ℂ.Arrow (p i) ℬ) xb x)) (λ x → ℂ.arrowsAreSets _ _) Y (ℂ.isoToId x) yb
|
||||
-- One thing to watch out for here is that the isomorphisms going forwards
|
||||
-- must compose to give idToIso
|
||||
iso
|
||||
: ((X , xa , xb) ≡ (Y , ya , yb))
|
||||
≅ ((X , xa , xb) ≊ (Y , ya , yb))
|
||||
iso = step0 ⊙ step1 ⊙ step2
|
||||
lemma : Terminal ≃ Product ℂ 𝒜 ℬ
|
||||
lemma = fromIsomorphism Terminal (Product ℂ 𝒜 ℬ) (f , g , inv)
|
||||
where
|
||||
f : Terminal → Product ℂ 𝒜 ℬ
|
||||
f ((X , x0 , x1) , uniq) = p
|
||||
where
|
||||
infixl 5 _⊙_
|
||||
_⊙_ = composeIso
|
||||
equiv1
|
||||
: ((X , xa , xb) ≡ (Y , ya , yb))
|
||||
≃ ((X , xa , xb) ≊ (Y , ya , yb))
|
||||
equiv1 = _ , fromIso _ _ (snd iso)
|
||||
|
||||
univalent : Univalent
|
||||
univalent = univalenceFrom≃ equiv1
|
||||
|
||||
isCat : IsCategory raw
|
||||
IsCategory.isPreCategory isCat = isPreCat
|
||||
IsCategory.univalent isCat = univalent
|
||||
|
||||
cat : Category _ _
|
||||
cat = record
|
||||
{ raw = raw
|
||||
; isCategory = isCat
|
||||
}
|
||||
|
||||
open Category cat
|
||||
|
||||
lemma : Terminal ≃ Product ℂ 𝒜 ℬ
|
||||
lemma = fromIsomorphism Terminal (Product ℂ 𝒜 ℬ) (f , g , inv)
|
||||
-- C-x 8 RET MATHEMATICAL BOLD SCRIPT CAPITAL A
|
||||
-- 𝒜
|
||||
where
|
||||
f : Terminal → Product ℂ 𝒜 ℬ
|
||||
f ((X , x0 , x1) , uniq) = p
|
||||
where
|
||||
rawP : RawProduct ℂ 𝒜 ℬ
|
||||
rawP = record
|
||||
{ object = X
|
||||
; fst = x0
|
||||
; snd = x1
|
||||
}
|
||||
-- open RawProduct rawP renaming (fst to x0 ; snd to x1)
|
||||
module _ {Y : ℂ.Object} (p0 : ℂ [ Y , 𝒜 ]) (p1 : ℂ [ Y , ℬ ]) where
|
||||
uy : isContr (Arrow (Y , p0 , p1) (X , x0 , x1))
|
||||
uy = uniq {Y , p0 , p1}
|
||||
open Σ uy renaming (fst to Y→X ; snd to contractible)
|
||||
open Σ Y→X renaming (fst to p0×p1 ; snd to cond)
|
||||
ump : ∃![ f×g ] (ℂ [ x0 ∘ f×g ] ≡ p0 P.× ℂ [ x1 ∘ f×g ] ≡ p1)
|
||||
ump = p0×p1 , cond , λ {f} cond-f → cong fst (contractible (f , cond-f))
|
||||
isP : IsProduct ℂ 𝒜 ℬ rawP
|
||||
isP = record { ump = ump }
|
||||
p : Product ℂ 𝒜 ℬ
|
||||
p = record
|
||||
{ raw = rawP
|
||||
; isProduct = isP
|
||||
}
|
||||
g : Product ℂ 𝒜 ℬ → Terminal
|
||||
g p = 𝒳 , t
|
||||
where
|
||||
open Product p renaming (object to X ; fst to x₀ ; snd to x₁) using ()
|
||||
module p = Product p
|
||||
module isp = IsProduct p.isProduct
|
||||
𝒳 : Object
|
||||
𝒳 = X , x₀ , x₁
|
||||
module _ {𝒴 : Object} where
|
||||
open Σ 𝒴 renaming (fst to Y)
|
||||
open Σ (snd 𝒴) renaming (fst to y₀ ; snd to y₁)
|
||||
ump = p.ump y₀ y₁
|
||||
open Σ ump renaming (fst to f')
|
||||
open Σ (snd ump) renaming (fst to f'-cond)
|
||||
𝒻 : Arrow 𝒴 𝒳
|
||||
𝒻 = f' , f'-cond
|
||||
contractible : (f : Arrow 𝒴 𝒳) → 𝒻 ≡ f
|
||||
contractible ff@(f , f-cond) = res
|
||||
where
|
||||
k : f' ≡ f
|
||||
k = snd (snd ump) f-cond
|
||||
prp : (a : ℂ.Arrow Y X) → isProp
|
||||
( (ℂ [ x₀ ∘ a ] ≡ y₀)
|
||||
× (ℂ [ x₁ ∘ a ] ≡ y₁)
|
||||
)
|
||||
prp f f0 f1 = Σ≡
|
||||
(ℂ.arrowsAreSets _ _ (fst f0) (fst f1))
|
||||
(ℂ.arrowsAreSets _ _ (snd f0) (snd f1))
|
||||
h :
|
||||
( λ i
|
||||
→ ℂ [ x₀ ∘ k i ] ≡ y₀
|
||||
× ℂ [ x₁ ∘ k i ] ≡ y₁
|
||||
) [ f'-cond ≡ f-cond ]
|
||||
h = lemPropF prp k
|
||||
res : (f' , f'-cond) ≡ (f , f-cond)
|
||||
res = Σ≡ k h
|
||||
t : IsTerminal 𝒳
|
||||
t {𝒴} = 𝒻 , contractible
|
||||
ve-re : ∀ x → g (f x) ≡ x
|
||||
ve-re x = Propositionality.propTerminal _ _
|
||||
re-ve : ∀ p → f (g p) ≡ p
|
||||
re-ve p = Product≡ e
|
||||
where
|
||||
module p = Product p
|
||||
-- RawProduct does not have eta-equality.
|
||||
e : Product.raw (f (g p)) ≡ Product.raw p
|
||||
RawProduct.object (e i) = p.object
|
||||
RawProduct.fst (e i) = p.fst
|
||||
RawProduct.snd (e i) = p.snd
|
||||
inv : AreInverses f g
|
||||
inv = funExt ve-re , funExt re-ve
|
||||
rawP : RawProduct ℂ 𝒜 ℬ
|
||||
rawP = record
|
||||
{ object = X
|
||||
; fst = x0
|
||||
; snd = x1
|
||||
}
|
||||
-- open RawProduct rawP renaming (fst to x0 ; snd to x1)
|
||||
module _ {Y : ℂ.Object} (p0 : ℂ [ Y , 𝒜 ]) (p1 : ℂ [ Y , ℬ ]) where
|
||||
uy : isContr (Arrow (Y , p0 , p1) (X , x0 , x1))
|
||||
uy = uniq {Y , p0 , p1}
|
||||
open Σ uy renaming (fst to Y→X ; snd to contractible)
|
||||
open Σ Y→X renaming (fst to p0×p1 ; snd to cond)
|
||||
ump : ∃![ f×g ] (ℂ [ x0 ∘ f×g ] ≡ p0 P.× ℂ [ x1 ∘ f×g ] ≡ p1)
|
||||
ump = p0×p1 , cond , λ {f} cond-f → cong fst (contractible (f , cond-f))
|
||||
isP : IsProduct ℂ 𝒜 ℬ rawP
|
||||
isP = record { ump = ump }
|
||||
p : Product ℂ 𝒜 ℬ
|
||||
p = record
|
||||
{ raw = rawP
|
||||
; isProduct = isP
|
||||
}
|
||||
g : Product ℂ 𝒜 ℬ → Terminal
|
||||
g p = 𝒳 , t
|
||||
where
|
||||
open Product p renaming (object to X ; fst to x₀ ; snd to x₁) using ()
|
||||
module p = Product p
|
||||
module isp = IsProduct p.isProduct
|
||||
𝒳 : Object
|
||||
𝒳 = X , x₀ , x₁
|
||||
module _ {𝒴 : Object} where
|
||||
open Σ 𝒴 renaming (fst to Y)
|
||||
open Σ (snd 𝒴) renaming (fst to y₀ ; snd to y₁)
|
||||
ump = p.ump y₀ y₁
|
||||
open Σ ump renaming (fst to f')
|
||||
open Σ (snd ump) renaming (fst to f'-cond)
|
||||
𝒻 : Arrow 𝒴 𝒳
|
||||
𝒻 = f' , f'-cond
|
||||
contractible : (f : Arrow 𝒴 𝒳) → 𝒻 ≡ f
|
||||
contractible ff@(f , f-cond) = res
|
||||
where
|
||||
k : f' ≡ f
|
||||
k = snd (snd ump) f-cond
|
||||
prp : (a : ℂ.Arrow Y X) → isProp
|
||||
( (ℂ [ x₀ ∘ a ] ≡ y₀)
|
||||
× (ℂ [ x₁ ∘ a ] ≡ y₁)
|
||||
)
|
||||
prp f f0 f1 = Σ≡
|
||||
(ℂ.arrowsAreSets _ _ (fst f0) (fst f1))
|
||||
(ℂ.arrowsAreSets _ _ (snd f0) (snd f1))
|
||||
h :
|
||||
( λ i
|
||||
→ ℂ [ x₀ ∘ k i ] ≡ y₀
|
||||
× ℂ [ x₁ ∘ k i ] ≡ y₁
|
||||
) [ f'-cond ≡ f-cond ]
|
||||
h = lemPropF prp k
|
||||
res : (f' , f'-cond) ≡ (f , f-cond)
|
||||
res = Σ≡ k h
|
||||
t : IsTerminal 𝒳
|
||||
t {𝒴} = 𝒻 , contractible
|
||||
ve-re : ∀ x → g (f x) ≡ x
|
||||
ve-re x = Propositionality.propTerminal _ _
|
||||
re-ve : ∀ p → f (g p) ≡ p
|
||||
re-ve p = Product≡ e
|
||||
where
|
||||
module p = Product p
|
||||
-- RawProduct does not have eta-equality.
|
||||
e : Product.raw (f (g p)) ≡ Product.raw p
|
||||
RawProduct.object (e i) = p.object
|
||||
RawProduct.fst (e i) = p.fst
|
||||
RawProduct.snd (e i) = p.snd
|
||||
inv : AreInverses f g
|
||||
inv = funExt ve-re , funExt re-ve
|
||||
|
||||
propProduct : isProp (Product ℂ 𝒜 ℬ)
|
||||
propProduct = equivPreservesNType {n = ⟨-1⟩} lemma Propositionality.propTerminal
|
||||
|
|
@ -348,10 +184,7 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} {A B : Category.Object
|
|||
module y = HasProducts y
|
||||
|
||||
productEq : x.product ≡ y.product
|
||||
productEq = funExt λ A → funExt λ B → Try0.propProduct _ _
|
||||
productEq = funExt λ A → funExt λ B → propProduct _ _
|
||||
|
||||
propHasProducts : isProp (HasProducts ℂ)
|
||||
propHasProducts x y i = record { product = productEq x y i }
|
||||
|
||||
fmap≡ : {A : Set} {a0 a1 : A} {B : Set} → (f : A → B) → Path a0 a1 → Path (f a0) (f a1)
|
||||
fmap≡ = cong
|
||||
|
|
|
|||
|
|
@ -9,9 +9,9 @@ open import Cat.Category.Functor
|
|||
open import Cat.Category.NaturalTransformation
|
||||
renaming (module Properties to F)
|
||||
using ()
|
||||
|
||||
open import Cat.Categories.Fun using (module Fun)
|
||||
open import Cat.Categories.Opposite
|
||||
open import Cat.Categories.Sets hiding (presheaf)
|
||||
open import Cat.Categories.Fun using (module Fun)
|
||||
|
||||
-- There is no (small) category of categories. So we won't use _⇑_ from
|
||||
-- `HasExponential`
|
||||
|
|
|
|||
Loading…
Reference in New Issue