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Rename variable

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Frederik Hanghøj Iversen 2018-04-25 08:19:36 +02:00
parent 129eef1150
commit 6b6e6672e0
1 changed files with 30 additions and 30 deletions
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src/Cat/Category/Product.agda
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@ -87,18 +87,18 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} {A B : Category.Object
q = lemPropF propIsProduct p q = lemPropF propIsProduct p
module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
(let module ℂ = Category ℂ) {A B : ℂ.Object} where (let module ℂ = Category ℂ) {𝓐 𝓑 : ℂ.Object} where
open P open P
module _ where module _ where
raw : RawCategory _ _ raw : RawCategory _ _
raw = record raw = record
{ Object = Σ[ X ∈ ℂ.Object ] ℂ.Arrow X A × ℂ.Arrow X B { Object = Σ[ X ∈ ℂ.Object ] ℂ.Arrow X 𝓐 × ℂ.Arrow X 𝓑
; Arrow = λ{ (X , x0 , x1) (Y , y0 , y1) ; Arrow = λ{ (A , a0 , a1) (B , b0 , b1)
→ Σ[ f ∈ ℂ.Arrow X Y ] → Σ[ f ∈ ℂ.Arrow A B ]
ℂ [ y0 ∘ f ] ≡ x0 ℂ [ b0 ∘ f ] ≡ a0
× ℂ [ y1 ∘ f ] ≡ x1 × ℂ [ b1 ∘ f ] ≡ a1
} }
; identity = λ{ {X , f , g} → ℂ.identity {X} , ℂ.rightIdentity , ℂ.rightIdentity} ; 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) ; _<<<_ = λ { {_ , a0 , a1} {_ , b0 , b1} {_ , c0 , c1} (f , f0 , f1) (g , g0 , g1)
@ -176,7 +176,7 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
open import Cat.Equivalence using (composeIso) renaming (_≅_ to _≅_) open import Cat.Equivalence using (composeIso) renaming (_≅_ to _≅_)
step0 step0
: ((X , xa , xb) ≡ (Y , ya , yb)) : ((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)) ≅ (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 step0
= (λ p → cong fst p , cong-d (fst ∘ snd) p , cong-d (snd ∘ snd) p) = (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 x → λ i → fst x i , (fst (snd x) i) , (snd (snd x) i))
@ -203,12 +203,12 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
p p
step1 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[ 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 ≅ Σ (X ℂ.≊ Y) (λ iso
→ let p = ℂ.isoToId iso → let p = ℂ.isoToId iso
in in
( PathP (λ i → ℂ.Arrow (p i) A) xa ya) ( PathP (λ i → ℂ.Arrow (p i) 𝓐) xa ya)
× PathP (λ i → ℂ.Arrow (p i) B) xb yb × PathP (λ i → ℂ.Arrow (p i) 𝓑) xb yb
) )
step1 step1
= symIso = symIso
@ -216,7 +216,7 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
{A = (X ℂ.≊ Y)} {A = (X ℂ.≊ Y)}
{B = (X ≡ Y)} {B = (X ≡ Y)}
(ℂ.groupoidObject _ _) (ℂ.groupoidObject _ _)
{Q = \ p → (PathP (λ i → ℂ.Arrow (p i) A) xa ya) × (PathP (λ i → ℂ.Arrow (p i) B) xb yb)} {Q = \ p → (PathP (λ i → ℂ.Arrow (p i) 𝓐) xa ya) × (PathP (λ i → ℂ.Arrow (p i) 𝓑) xb yb)}
ℂ.isoToId ℂ.isoToId
(symIso (_ , ℂ.asTypeIso {X} {Y}) .snd) (symIso (_ , ℂ.asTypeIso {X} {Y}) .snd)
) )
@ -225,8 +225,8 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
: Σ (X ℂ.≊ Y) (λ iso : Σ (X ℂ.≊ Y) (λ iso
→ let p = ℂ.isoToId iso → let p = ℂ.isoToId iso
in in
( PathP (λ i → ℂ.Arrow (p i) A) xa ya) ( PathP (λ i → ℂ.Arrow (p i) 𝓐) xa ya)
× PathP (λ i → ℂ.Arrow (p i) B) xb yb × PathP (λ i → ℂ.Arrow (p i) 𝓑) xb yb
) )
≅ ((X , xa , xb) ≊ (Y , ya , yb)) ≅ ((X , xa , xb) ≊ (Y , ya , yb))
step2 step2
@ -243,10 +243,10 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
iso = fst f , fst f~ , cong fst inv-f , cong fst inv-f~ iso = fst f , fst f~ , cong fst inv-f , cong fst inv-f~
p : X ≡ Y p : X ≡ Y
p = ℂ.isoToId iso p = ℂ.isoToId iso
pA : ℂ.Arrow X A ≡ ℂ.Arrow Y A pA : ℂ.Arrow X 𝓐 ≡ ℂ.Arrow Y 𝓐
pA = cong (λ x → ℂ.Arrow x A) p pA = cong (λ x → ℂ.Arrow x 𝓐) p
pB : ℂ.Arrow X B ≡ ℂ.Arrow Y B pB : ℂ.Arrow X 𝓑 ≡ ℂ.Arrow Y 𝓑
pB = cong (λ x → ℂ.Arrow x B) p pB = cong (λ x → ℂ.Arrow x 𝓑) p
k0 = begin k0 = begin
coe pB xb ≡⟨ ℂ.coe-dom iso ⟩ coe pB xb ≡⟨ ℂ.coe-dom iso ⟩
xb ℂ.<<< fst f~ ≡⟨ snd (snd f~) ⟩ xb ℂ.<<< fst f~ ≡⟨ snd (snd f~) ⟩
@ -264,10 +264,10 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
(Σ≡ refl (ℂ.propIsomorphism _ _ _))) (Σ≡ refl (ℂ.propIsomorphism _ _ _)))
, funExt (λ{ (f , _) → lemSig propIsomorphism _ _ (Σ≡ refl (propEqs _ _ _))}) , funExt (λ{ (f , _) → lemSig propIsomorphism _ _ (Σ≡ refl (propEqs _ _ _))})
where where
prop0 : ∀ {x} → isProp (PathP (λ i → ℂ.Arrow (ℂ.isoToId x i) A) xa ya) 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 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} → 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 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 -- One thing to watch out for here is that the isomorphisms going forwards
-- must compose to give idToIso -- must compose to give idToIso
iso iso
@ -297,34 +297,34 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
open Category cat open Category cat
lemma : Terminal ≃ Product ℂ A B lemma : Terminal ≃ Product ℂ 𝓐 𝓑
lemma = fromIsomorphism Terminal (Product ℂ A B) (f , g , inv) lemma = fromIsomorphism Terminal (Product ℂ 𝓐 𝓑) (f , g , inv)
where where
f : Terminal → Product ℂ A B f : Terminal → Product ℂ 𝓐 𝓑
f ((X , x0 , x1) , uniq) = p f ((X , x0 , x1) , uniq) = p
where where
rawP : RawProduct ℂ A B rawP : RawProduct ℂ 𝓐 𝓑
rawP = record rawP = record
{ object = X { object = X
; fst = x0 ; fst = x0
; snd = x1 ; snd = x1
} }
-- open RawProduct rawP renaming (fst to x0 ; snd to x1) -- open RawProduct rawP renaming (fst to x0 ; snd to x1)
module _ {Y : ℂ.Object} (p0 : ℂ [ Y , A ]) (p1 : ℂ [ Y , B ]) where module _ {Y : ℂ.Object} (p0 : ℂ [ Y , 𝓐 ]) (p1 : ℂ [ Y , 𝓑 ]) where
uy : isContr (Arrow (Y , p0 , p1) (X , x0 , x1)) uy : isContr (Arrow (Y , p0 , p1) (X , x0 , x1))
uy = uniq {Y , p0 , p1} uy = uniq {Y , p0 , p1}
open Σ uy renaming (fst to Y→X ; snd to contractible) open Σ uy renaming (fst to Y→X ; snd to contractible)
open Σ Y→X renaming (fst to p0×p1 ; snd to cond) 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 : ∃![ f×g ] (ℂ [ x0 ∘ f×g ] ≡ p0 P.× ℂ [ x1 ∘ f×g ] ≡ p1)
ump = p0×p1 , cond , λ {y} x → let k = contractible (y , x) in λ i → fst (k i) ump = p0×p1 , cond , λ {y} x → let k = contractible (y , x) in λ i → fst (k i)
isP : IsProduct ℂ A B rawP isP : IsProduct ℂ 𝓐 𝓑 rawP
isP = record { ump = ump } isP = record { ump = ump }
p : Product ℂ A B p : Product ℂ 𝓐 𝓑
p = record p = record
{ raw = rawP { raw = rawP
; isProduct = isP ; isProduct = isP
} }
g : Product ℂ A B → Terminal g : Product ℂ 𝓐 𝓑 → Terminal
g p = o , t g p = o , t
where where
module p = Product p module p = Product p
@ -375,7 +375,7 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
inv : AreInverses f g inv : AreInverses f g
inv = funExt ve-re , funExt re-ve inv = funExt ve-re , funExt re-ve
propProduct : isProp (Product ℂ A B) propProduct : isProp (Product ℂ 𝓐 𝓑)
propProduct = equivPreservesNType {n = ⟨-1⟩} lemma Propositionality.propTerminal propProduct = equivPreservesNType {n = ⟨-1⟩} lemma Propositionality.propTerminal
module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} {A B : Category.Object ℂ} where module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} {A B : Category.Object ℂ} where