Stuff about univalence in the category of sets
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Subproject commit a487c76a5f3ecf2752dabc9e5c3a8866fda28a19
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Subproject commit 3a125a0cb010903e6aa80569a0ca5339424eaf86
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@ -4,10 +4,15 @@ module Cat.Categories.Sets where
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open import Agda.Primitive
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open import Data.Product
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import Function
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open import Function using (_∘_)
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open import Cubical hiding (inverse ; _≃_ {- ; obverse ; recto-verso ; verso-recto -} )
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open import Cubical.Univalence using (_≃_ ; ua)
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open import Cubical hiding (_≃_ ; inverse)
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open import Cubical.Equivalence
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renaming
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( _≅_ to _A≅_ )
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using
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(_≃_ ; con ; AreInverses)
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open import Cubical.Univalence
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open import Cubical.GradLemma
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open import Cat.Category
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@ -27,7 +32,7 @@ module _ (ℓ : Level) where
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RawCategory.𝟙 SetsRaw = Function.id
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RawCategory._∘_ SetsRaw = Function._∘′_
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open RawCategory SetsRaw
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open RawCategory SetsRaw hiding (_∘_)
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open Univalence SetsRaw
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isIdentity : IsIdentity Function.id
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@ -62,48 +67,100 @@ module _ (ℓ : Level) where
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-- ordering should be swapped.
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areInverses : IsInverseOf {A = hA} {hB} obverse inverse
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areInverses = proj₂ (proj₂ iso)
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verso-recto : ∀ a → (inverse Function.∘ obverse) a ≡ a
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verso-recto : ∀ a → (inverse ∘ obverse) a ≡ a
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verso-recto a i = proj₁ areInverses i a
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recto-verso : ∀ b → (obverse Function.∘ inverse) b ≡ b
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recto-verso b i = proj₂ areInverses i b
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univalent : isEquiv (hA ≡ hB) (hA ≅ hB) (id-to-iso (λ {A} {B} → isIdentity {A} {B}) hA hB)
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univalent = gradLemma obverse inverse verso-recto recto-verso
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where
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private
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univIso : (A ≡ B) A≅ (A ≃ B)
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univIso = _≃_.toIsomorphism univalence
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obverse' : A ≡ B → A ≃ B
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obverse' = proj₁ univIso
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inverse' : A ≃ B → A ≡ B
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inverse' = proj₁ (proj₂ univIso)
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-- Drop proof of being a set from both sides of an equality.
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dropP : hA ≡ hB → A ≡ B
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dropP eq i = proj₁ (eq i)
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-- Add proof of being a set to both sides of a set-theoretic equivalence
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-- returning a category-theoretic equivalence.
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addE : A A≅ B → hA ≅ hB
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addE eqv = proj₁ eqv , (proj₁ (proj₂ eqv)) , asPair
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where
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areeqv = proj₂ (proj₂ eqv)
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asPair =
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let module Inv = AreInverses areeqv
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in Inv.verso-recto , Inv.recto-verso
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obverse : hA ≡ hB → hA ≅ hB
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obverse eq = {!res!}
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where
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-- Problem: How do I extract this equality from `eq`?
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eqq : A ≡ B
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eqq = {!!}
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eq' : A ≃ B
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eq' = fromEquality eqq
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-- Problem: Why does this not satisfy the goal?
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res : hA ≅ hB
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res = toIsomorphism eq'
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obverse = addE ∘ _≃_.toIsomorphism ∘ obverse' ∘ dropP
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-- Drop proof of being a set form both sides of a category-theoretic
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-- equivalence returning a set-theoretic equivalence.
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dropE : hA ≅ hB → A A≅ B
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dropE eqv = obv , inv , asAreInverses
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where
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obv = proj₁ eqv
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inv = proj₁ (proj₂ eqv)
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areEq = proj₂ (proj₂ eqv)
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asAreInverses : AreInverses A B obv inv
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asAreInverses = record { verso-recto = proj₁ areEq ; recto-verso = proj₂ areEq }
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-- Dunno if this is a thing.
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isoToEquiv : A A≅ B → A ≃ B
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isoToEquiv = {!!}
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-- Add proof of being a set to both sides of an equality.
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addP : A ≡ B → hA ≡ hB
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addP p = lemSig (λ X → propPi λ x → propPi (λ y → propIsProp)) hA hB p
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inverse : hA ≅ hB → hA ≡ hB
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inverse iso = res
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where
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eq : A ≡ B
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eq = ua (fromIsomorphism iso)
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inverse = addP ∘ inverse' ∘ isoToEquiv ∘ dropE
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-- Use the fact that being an h-level level is a mere proposition.
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-- This is almost provable using `Wishlist.isSetIsProp` - although
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-- this creates homogenous paths.
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isSetEq : (λ i → isSet (eq i)) [ isSetA ≡ isSetB ]
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isSetEq = {!!}
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-- open AreInverses (proj₂ (proj₂ univIso)) renaming
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-- ( verso-recto to verso-recto'
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-- ; recto-verso to recto-verso'
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-- )
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-- I can just open them but I wanna be able to see the type annotations.
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verso-recto' : inverse' ∘ obverse' ≡ Function.id
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verso-recto' = AreInverses.verso-recto (proj₂ (proj₂ univIso))
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recto-verso' : obverse' ∘ inverse' ≡ Function.id
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recto-verso' = AreInverses.recto-verso (proj₂ (proj₂ univIso))
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verso-recto : (iso : hA ≅ hB) → obverse (inverse iso) ≡ iso
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verso-recto iso = begin
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obverse (inverse iso) ≡⟨⟩
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( addE ∘ _≃_.toIsomorphism
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∘ obverse' ∘ dropP ∘ addP
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∘ inverse' ∘ isoToEquiv
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∘ dropE) iso
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≡⟨⟩
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( addE ∘ _≃_.toIsomorphism
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∘ obverse'
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∘ inverse' ∘ isoToEquiv
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∘ dropE) iso
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≡⟨ {!!} ⟩ -- obverse' inverse' are inverses
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( addE ∘ _≃_.toIsomorphism ∘ isoToEquiv ∘ dropE) iso
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≡⟨ {!!} ⟩ -- should be easy to prove
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-- _≃_.toIsomorphism ∘ isoToEquiv ≡ id
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(addE ∘ dropE) iso
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≡⟨⟩
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iso ∎
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res : hA ≡ hB
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proj₁ (res i) = eq i
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proj₂ (res i) = isSetEq i
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-- Similar to above.
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recto-verso : (eq : hA ≡ hB) → inverse (obverse eq) ≡ eq
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recto-verso eq = begin
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inverse (obverse eq) ≡⟨ {!!} ⟩
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eq ∎
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-- FIXME Either the name of inverse/obverse is flipped or
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-- recto-verso/verso-recto is flipped.
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recto-verso : ∀ y → (inverse Function.∘ obverse) y ≡ y
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recto-verso x = {!!}
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verso-recto : ∀ x → (obverse Function.∘ inverse) x ≡ x
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verso-recto x = {!!}
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-- Use the fact that being an h-level is a mere proposition.
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-- This is almost provable using `Wishlist.isSetIsProp` - although
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-- this creates homogenous paths.
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isSetEq : (p : A ≡ B) → (λ i → isSet (p i)) [ isSetA ≡ isSetB ]
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isSetEq = {!!}
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res : hA ≡ hB
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proj₁ (res i) = {!!}
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proj₂ (res i) = isSetEq {!!} i
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univalent : isEquiv (hA ≡ hB) (hA ≅ hB) (id-to-iso (λ {A} {B} → isIdentity {A} {B}) hA hB)
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univalent = {!gradLemma obverse inverse verso-recto recto-verso!}
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SetsIsCategory : IsCategory SetsRaw
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IsCategory.isAssociative SetsIsCategory = refl
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@ -129,7 +129,9 @@ record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
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Terminal : Set (ℓa ⊔ ℓb)
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Terminal = Σ Object IsTerminal
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-- Univalence is indexed by a raw category as well as an identity proof.
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-- | Univalence is indexed by a raw category as well as an identity proof.
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--
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-- FIXME Put this in `RawCategory` and index it on the witness to `isIdentity`.
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module Univalence {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
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open RawCategory ℂ
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module _ (isIdentity : IsIdentity 𝟙) where
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@ -150,6 +152,8 @@ module Univalence {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
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-- iso-is-epi : Isomorphism f → Epimorphism {X = X} f
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-- iso-is-mono : Isomorphism f → Monomorphism {X = X} f
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--
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-- Sans `univalent` this would be what is referred to as a pre-category in
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-- [HoTT].
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record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc (ℓa ⊔ ℓb)) where
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open RawCategory ℂ public
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open Univalence ℂ public
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@ -248,51 +252,66 @@ module Propositionality {ℓa ℓb : Level} {C : RawCategory ℓa ℓb} where
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-- adverse effects this may have.
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isIdentity : (λ _ → IsIdentity 𝟙) [ X.isIdentity ≡ Y.isIdentity ]
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isIdentity = propIsIdentity x X.isIdentity Y.isIdentity
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done : x ≡ y
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U : ∀ {a : IsIdentity 𝟙}
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→ (λ _ → IsIdentity 𝟙) [ X.isIdentity ≡ a ]
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→ (b : Univalent a)
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→ Set _
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U eqwal bbb =
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U eqwal univ =
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(λ i → Univalent (eqwal i))
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[ X.univalent ≡ bbb ]
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[ X.univalent ≡ univ ]
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P : (y : IsIdentity 𝟙)
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→ (λ _ → IsIdentity 𝟙) [ X.isIdentity ≡ y ] → Set _
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P y eq = ∀ (b' : Univalent y) → U eq b'
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helper : ∀ (b' : Univalent X.isIdentity)
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P y eq = ∀ (univ : Univalent y) → U eq univ
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p : ∀ (b' : Univalent X.isIdentity)
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→ (λ _ → Univalent X.isIdentity) [ X.univalent ≡ b' ]
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helper univ = propUnivalent x X.univalent univ
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foo = pathJ P helper Y.isIdentity isIdentity
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p univ = propUnivalent x X.univalent univ
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helper : P Y.isIdentity isIdentity
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helper = pathJ P p Y.isIdentity isIdentity
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eqUni : U isIdentity Y.univalent
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eqUni = foo Y.univalent
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IC.isAssociative (done i) = propIsAssociative x X.isAssociative Y.isAssociative i
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IC.isIdentity (done i) = isIdentity i
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eqUni = helper Y.univalent
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done : x ≡ y
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IC.isAssociative (done i) = propIsAssociative x X.isAssociative Y.isAssociative i
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IC.isIdentity (done i) = isIdentity i
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IC.arrowsAreSets (done i) = propArrowIsSet x X.arrowsAreSets Y.arrowsAreSets i
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IC.univalent (done i) = eqUni i
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IC.univalent (done i) = eqUni i
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propIsCategory : isProp (IsCategory C)
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propIsCategory = done
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-- | Univalent categories
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--
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-- Just bundles up the data with witnesses inhabting the propositions.
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-- Just bundles up the data with witnesses inhabiting the propositions.
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record Category (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
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field
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raw : RawCategory ℓa ℓb
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raw : RawCategory ℓa ℓb
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{{isCategory}} : IsCategory raw
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open IsCategory isCategory public
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Category≡ : {ℓa ℓb : Level} {ℂ 𝔻 : Category ℓa ℓb} → Category.raw ℂ ≡ Category.raw 𝔻 → ℂ ≡ 𝔻
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Category≡ {ℂ = ℂ} {𝔻} eq i = record
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{ raw = eq i
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; isCategory = isCategoryEq i
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}
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where
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open Category
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module ℂ = Category ℂ
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isCategoryEq : (λ i → IsCategory (eq i)) [ isCategory ℂ ≡ isCategory 𝔻 ]
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isCategoryEq = {!!}
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-- The fact that being a category is a mere proposition gives rise to this
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-- equality principle for categories.
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module _ {ℓa ℓb : Level} {ℂ 𝔻 : Category ℓa ℓb} where
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private
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module ℂ = Category ℂ
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module 𝔻 = Category 𝔻
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module _ (rawEq : ℂ.raw ≡ 𝔻.raw) where
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private
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P : (target : RawCategory ℓa ℓb) → ({!!} ≡ target) → Set _
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P _ eq = ∀ isCategory' → (λ i → IsCategory (eq i)) [ ℂ.isCategory ≡ isCategory' ]
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p : P ℂ.raw refl
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p isCategory' = Propositionality.propIsCategory {!!} {!!}
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-- TODO Make and use heterogeneous version of Category≡
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isCategoryEq : (λ i → IsCategory (rawEq i)) [ ℂ.isCategory ≡ 𝔻.isCategory ]
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isCategoryEq = {!!}
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Category≡ : ℂ ≡ 𝔻
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Category≡ i = record
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{ raw = rawEq i
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; isCategory = isCategoryEq i
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}
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-- | Syntax for arrows- and composition in a given category.
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module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
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@ -318,7 +337,7 @@ module Opposite {ℓa ℓb : Level} where
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RawCategory._∘_ opRaw = Function.flip ℂ._∘_
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open RawCategory opRaw
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open Univalence opRaw
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open Univalence opRaw
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isIdentity : IsIdentity 𝟙
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isIdentity = swap ℂ.isIdentity
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@ -735,7 +735,7 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
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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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lem = {!!} -- verso-recto Monoidal≃Kleisli
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t : {ℓ : Level} {A B : Set ℓ} {a : _ → A} {b : B → _}
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→ a ∘ (Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ b ≡ a ∘ b
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t {a = a} {b} = cong (λ φ → a ∘ φ ∘ b) lem
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@ -763,7 +763,7 @@ module _ {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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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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voe-isEquiv : isEquiv (voe-2-3-1 omap pure) (voe-2-3-2 omap pure) forth
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voe-isEquiv = gradLemma forth back forthEq backEq
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@ -29,7 +29,7 @@ module _ {ℓ : Level} {ℂ : Category ℓ ℓ} where
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--
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-- In stead we'll use an ad-hoc definition -- which is definitionally
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-- equivalent to that other one.
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_⇑_ = CatExponential.prodObj
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_⇑_ = CatExponential.object
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module _ {A B : ℂ.Object} (f : ℂ [ A , B ]) where
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fmap : Transformation (prshf A) (prshf B)
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@ -4,12 +4,13 @@ open import Level
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open import Cubical.NType
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open import Data.Nat using (_≤_ ; z≤n ; s≤s)
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open import Cubical.NType.Properties public using (propHasLevel)
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postulate ntypeCommulative : ∀ {ℓ n m} {A : Set ℓ} → n ≤ m → HasLevel ⟨ n ⟩₋₂ A → HasLevel ⟨ m ⟩₋₂ A
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module _ {ℓ : Level} {A : Set ℓ} where
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-- This is §7.1.10 in [HoTT]. Andrea says the proof is in `cubical` but I
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-- can't find it.
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postulate propHasLevel : ∀ n → isProp (HasLevel n A)
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isSetIsProp : isProp (isSet A)
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isSetIsProp = propHasLevel (S (S ⟨-2⟩))
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propIsProp : isProp (isProp A)
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propIsProp = propHasLevel (S ⟨-2⟩)
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