isProp (Product C A B) setup
This commit is contained in:
Andrea Vezzosi
parent
facd1167e0
commit
8ac6b97213
|
|
@ -253,6 +253,10 @@ record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc
|
|||
res : (fx , cx) ≡ (fy , cy)
|
||||
res i = fp i , cp i
|
||||
|
||||
-- this needs the univalence of the category
|
||||
propTerminal : isProp Terminal
|
||||
propTerminal = {!!}
|
||||
|
||||
-- Merely the dual of the above statement.
|
||||
propIsInitial : ∀ I → isProp (IsInitial I)
|
||||
propIsInitial I x y i {X} = res X i
|
||||
|
|
@ -269,6 +273,9 @@ record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc
|
|||
res : (fx , cx) ≡ (fy , cy)
|
||||
res i = fp i , cp i
|
||||
|
||||
propInitial : isProp Initial
|
||||
propInitial = {!!}
|
||||
|
||||
-- | Propositionality of being a category
|
||||
module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
|
||||
open RawCategory ℂ
|
||||
|
|
|
|||
|
|
@ -1,6 +1,7 @@
|
|||
{-# OPTIONS --allow-unsolved-metas --cubical #-}
|
||||
module Cat.Category.Product where
|
||||
|
||||
open import Cubical.NType.Properties
|
||||
open import Cat.Prelude hiding (_×_ ; proj₁ ; proj₂)
|
||||
import Data.Product as P
|
||||
|
||||
|
|
@ -128,108 +129,120 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
|
|||
raw : RawCategory _ _
|
||||
raw = record
|
||||
{ Object = Σ[ X ∈ ℂ.Object ] ℂ.Arrow X A × ℂ.Arrow X B
|
||||
; Arrow = λ{ (A , _) (B , _) → ℂ.Arrow A B}
|
||||
; 𝟙 = λ{ {A , _} → ℂ.𝟙 {A}}
|
||||
; _∘_ = ℂ._∘_
|
||||
}
|
||||
|
||||
open RawCategory raw
|
||||
open Univalence ℂ.isIdentity
|
||||
open import Cat.Equivalence hiding (_≅_)
|
||||
|
||||
k : {A B : ℂ.Object} → isEquiv (A ≡ B) (A ℂ.≅ B) (ℂ.id-to-iso A B)
|
||||
k = ℂ.univalent
|
||||
|
||||
module _ {X' Y' : Σ[ X ∈ ℂ.Object ] (ℂ [ X , A ] × ℂ [ X , B ])} where
|
||||
open Σ X' renaming (proj₁ to X) using ()
|
||||
open Σ (proj₂ X') renaming (proj₁ to Xxa ; proj₂ to Xxb)
|
||||
open Σ Y' renaming (proj₁ to Y) using ()
|
||||
open Σ (proj₂ Y') renaming (proj₁ to Yxa ; proj₂ to Yxb)
|
||||
module _ (p : X ≡ Y) where
|
||||
D : ∀ y → X ≡ y → Set _
|
||||
D y q = ∀ b → (λ i → ℂ [ q i , A ]) [ Xxa ≡ b ]
|
||||
-- Not sure this is actually provable - but if it were it might involve
|
||||
-- something like the ump of the product -- in which case perhaps the
|
||||
-- objects of the category I'm constructing should not merely be the
|
||||
-- data-part of the product but also the laws.
|
||||
|
||||
-- d : D X refl
|
||||
d : ∀ b → (λ i → ℂ [ X , A ]) [ Xxa ≡ b ]
|
||||
d b = {!!}
|
||||
kk : D Y p
|
||||
kk = pathJ D d Y p
|
||||
a : (λ i → ℂ [ p i , A ]) [ Xxa ≡ Yxa ]
|
||||
a = kk Yxa
|
||||
b : (λ i → ℂ [ p i , B ]) [ Xxb ≡ Yxb ]
|
||||
b = {!!}
|
||||
f : X' ≡ Y'
|
||||
f i = p i , a i , b i
|
||||
|
||||
module _ (p : X' ≡ Y') where
|
||||
g : X ≡ Y
|
||||
g i = proj₁ (p i)
|
||||
|
||||
step0 : (X' ≡ Y') ≃ (X ≡ Y)
|
||||
step0 = Equiv≃.fromIsomorphism _ _ (g , f , record { verso-recto = {!refl!} ; recto-verso = refl})
|
||||
|
||||
step1 : (X ≡ Y) ≃ X ℂ.≅ Y
|
||||
step1 = ℂ.univalent≃
|
||||
|
||||
-- Just a reminder
|
||||
step1-5 : (X' ≅ Y') ≡ (X ℂ.≅ Y)
|
||||
step1-5 = refl
|
||||
|
||||
step2 : (X' ≡ Y') ≃ (X ℂ.≅ Y)
|
||||
step2 = Equivalence.compose step0 step1
|
||||
|
||||
univalent : isEquiv (X' ≡ Y') (X ℂ.≅ Y) (id-to-iso X' Y')
|
||||
univalent = proj₂ step2
|
||||
|
||||
isCategory : IsCategory raw
|
||||
isCategory = record
|
||||
{ isAssociative = ℂ.isAssociative
|
||||
; isIdentity = ℂ.isIdentity
|
||||
; arrowsAreSets = ℂ.arrowsAreSets
|
||||
; univalent = univalent
|
||||
}
|
||||
|
||||
category : Category _ _
|
||||
category = record
|
||||
{ raw = raw
|
||||
; isCategory = isCategory
|
||||
}
|
||||
|
||||
open Category category hiding (IsTerminal ; Object)
|
||||
|
||||
-- Essential turns `p : Product ℂ A B` into a triple
|
||||
productObject : Object
|
||||
productObject = Product.object p , Product.proj₁ p , Product.proj₂ p
|
||||
|
||||
productObjectIsTerminal : IsTerminal productObject
|
||||
productObjectIsTerminal = {!!}
|
||||
|
||||
proppp : isProp (IsTerminal productObject)
|
||||
proppp = Propositionality.propIsTerminal productObject
|
||||
|
||||
module Try1 {ℓa ℓb : Level} (A B : Set) where
|
||||
open import Data.Product
|
||||
raw : RawCategory _ _
|
||||
raw = record
|
||||
{ Object = Σ[ X ∈ Set ] (X → A) × (X → B)
|
||||
; Arrow = λ{ (X0 , f0 , g0) (X1 , f1 , g1) → X0 → X1}
|
||||
; 𝟙 = λ x → x
|
||||
; _∘_ = λ x x₁ x₂ → x (x₁ x₂)
|
||||
; Arrow = λ{ (X , xa , xb) (Y , ya , yb) → Σ[ xy ∈ ℂ.Arrow X Y ] (ℂ [ ya ∘ xy ] ≡ xa) × (ℂ [ yb ∘ xy ] ≡ xb) }
|
||||
; 𝟙 = λ{ {A , _} → ℂ.𝟙 {A} , {!!}}
|
||||
; _∘_ = \ { (f , p) (g , q) → ℂ._∘_ f g , {!!} }
|
||||
}
|
||||
|
||||
open RawCategory raw
|
||||
|
||||
isCategory : IsCategory raw
|
||||
isCategory = record
|
||||
{ isAssociative = refl
|
||||
; isIdentity = refl , refl
|
||||
; arrowsAreSets = {!!}
|
||||
; univalent = {!!}
|
||||
}
|
||||
cat : IsCategory raw
|
||||
cat = {!!}
|
||||
|
||||
t : IsTerminal ((A × B) , proj₁ , proj₂)
|
||||
t = {!!}
|
||||
module cat = IsCategory cat
|
||||
|
||||
lemma : Terminal ≃ Product ℂ A B
|
||||
lemma = {!!}
|
||||
|
||||
thm : isProp (Product ℂ A B)
|
||||
thm = equivPreservesNType {n = ⟨-1⟩} lemma cat.Propositionality.propTerminal
|
||||
|
||||
-- open Univalence ℂ.isIdentity
|
||||
-- open import Cat.Equivalence hiding (_≅_)
|
||||
|
||||
-- k : {A B : ℂ.Object} → isEquiv (A ≡ B) (A ℂ.≅ B) (ℂ.id-to-iso A B)
|
||||
-- k = ℂ.univalent
|
||||
|
||||
-- module _ {X' Y' : Σ[ X ∈ ℂ.Object ] (ℂ [ X , A ] × ℂ [ X , B ])} where
|
||||
-- open Σ X' renaming (proj₁ to X) using ()
|
||||
-- open Σ (proj₂ X') renaming (proj₁ to Xxa ; proj₂ to Xxb)
|
||||
-- open Σ Y' renaming (proj₁ to Y) using ()
|
||||
-- open Σ (proj₂ Y') renaming (proj₁ to Yxa ; proj₂ to Yxb)
|
||||
-- module _ (p : X ≡ Y) where
|
||||
-- D : ∀ y → X ≡ y → Set _
|
||||
-- D y q = ∀ b → (λ i → ℂ [ q i , A ]) [ Xxa ≡ b ]
|
||||
-- -- Not sure this is actually provable - but if it were it might involve
|
||||
-- -- something like the ump of the product -- in which case perhaps the
|
||||
-- -- objects of the category I'm constructing should not merely be the
|
||||
-- -- data-part of the product but also the laws.
|
||||
|
||||
-- -- d : D X refl
|
||||
-- d : ∀ b → (λ i → ℂ [ X , A ]) [ Xxa ≡ b ]
|
||||
-- d b = {!!}
|
||||
-- kk : D Y p
|
||||
-- kk = pathJ D d Y p
|
||||
-- a : (λ i → ℂ [ p i , A ]) [ Xxa ≡ Yxa ]
|
||||
-- a = kk Yxa
|
||||
-- b : (λ i → ℂ [ p i , B ]) [ Xxb ≡ Yxb ]
|
||||
-- b = {!!}
|
||||
-- f : X' ≡ Y'
|
||||
-- f i = p i , a i , b i
|
||||
|
||||
-- module _ (p : X' ≡ Y') where
|
||||
-- g : X ≡ Y
|
||||
-- g i = proj₁ (p i)
|
||||
|
||||
-- step0 : (X' ≡ Y') ≃ (X ≡ Y)
|
||||
-- step0 = Equiv≃.fromIsomorphism _ _ (g , f , record { verso-recto = {!refl!} ; recto-verso = refl})
|
||||
|
||||
-- step1 : (X ≡ Y) ≃ X ℂ.≅ Y
|
||||
-- step1 = ℂ.univalent≃
|
||||
|
||||
-- -- Just a reminder
|
||||
-- step1-5 : (X' ≅ Y') ≡ (X ℂ.≅ Y)
|
||||
-- step1-5 = refl
|
||||
|
||||
-- step2 : (X' ≡ Y') ≃ (X ℂ.≅ Y)
|
||||
-- step2 = Equivalence.compose step0 step1
|
||||
|
||||
-- univalent : isEquiv (X' ≡ Y') (X ℂ.≅ Y) (id-to-iso X' Y')
|
||||
-- univalent = proj₂ step2
|
||||
|
||||
-- isCategory : IsCategory raw
|
||||
-- isCategory = record
|
||||
-- { isAssociative = ℂ.isAssociative
|
||||
-- ; isIdentity = ℂ.isIdentity
|
||||
-- ; arrowsAreSets = ℂ.arrowsAreSets
|
||||
-- ; univalent = univalent
|
||||
-- }
|
||||
|
||||
-- category : Category _ _
|
||||
-- category = record
|
||||
-- { raw = raw
|
||||
-- ; isCategory = isCategory
|
||||
-- }
|
||||
|
||||
-- open Category category hiding (IsTerminal ; Object)
|
||||
|
||||
-- -- Essential turns `p : Product ℂ A B` into a triple
|
||||
-- productObject : Object
|
||||
-- productObject = Product.object p , Product.proj₁ p , Product.proj₂ p
|
||||
|
||||
-- productObjectIsTerminal : IsTerminal productObject
|
||||
-- productObjectIsTerminal = {!!}
|
||||
|
||||
-- proppp : isProp (IsTerminal productObject)
|
||||
-- proppp = Propositionality.propIsTerminal productObject
|
||||
|
||||
-- module Try1 {ℓa ℓb : Level} (A B : Set) where
|
||||
-- open import Data.Product
|
||||
-- raw : RawCategory _ _
|
||||
-- raw = record
|
||||
-- { Object = Σ[ X ∈ Set ] (X → A) × (X → B)
|
||||
-- ; Arrow = λ{ (X0 , f0 , g0) (X1 , f1 , g1) → X0 → X1}
|
||||
-- ; 𝟙 = λ x → x
|
||||
-- ; _∘_ = λ x x₁ x₂ → x (x₁ x₂)
|
||||
-- }
|
||||
|
||||
-- open RawCategory raw
|
||||
|
||||
-- isCategory : IsCategory raw
|
||||
-- isCategory = record
|
||||
-- { isAssociative = refl
|
||||
-- ; isIdentity = refl , refl
|
||||
-- ; arrowsAreSets = {!!}
|
||||
-- ; univalent = {!!}
|
||||
-- }
|
||||
|
||||
-- t : IsTerminal ((A × B) , proj₁ , proj₂)
|
||||
-- t = {!!}
|
||||
|
|
|
|||
Loading…
Reference in New Issue