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Use correct order for left- and right identity 

Define and use helpers left- and right identity
This commit is contained in:
Frederik Hanghøj Iversen 2018-03-21 11:46:36 +01:00
parent 71d9acff9a
commit 4beb48e066
11 changed files with 65 additions and 61 deletions
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8
src/Cat/Categories/Cat.agda
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@ -47,7 +47,7 @@ module _ (ℓ ℓ' : Level) where
isAssociative : IsAssociative
isAssociative {f = F} {G} {H} = assc {F = F} {G = G} {H = H}
ident : IsIdentity identity
ident = ident-r , ident-l
ident = ident-l , ident-r
-- NB! `ArrowsAreSets RawCat` is *not* provable. The type of functors,
-- however, form a groupoid! Therefore there is no (1-)category of
@ -241,10 +241,10 @@ module CatExponential {ℓ : Level} (ℂ 𝔻 : Category ℓ ℓ) where
ident : fmap {c} {c} (NT.identity F , 𝟙 ℂ {A = proj₂ c}) ≡ 𝟙 𝔻
ident = begin
fmap {c} {c} (𝟙 (object ⊗ ℂ) {c}) ≡⟨⟩
fmap {c} {c} (idN F , 𝟙 ℂ) ≡⟨⟩
fmap {c} {c} (𝟙 (object ⊗ ℂ) {c}) ≡⟨⟩
fmap {c} {c} (idN F , 𝟙 ℂ) ≡⟨⟩
𝔻 [ identityTrans F C ∘ F.fmap (𝟙 ℂ)] ≡⟨⟩
𝔻 [ 𝟙 𝔻 ∘ F.fmap (𝟙 ℂ)] ≡⟨ proj₂ 𝔻.isIdentity ⟩
𝔻 [ 𝟙 𝔻 ∘ F.fmap (𝟙 ℂ)] ≡⟨ 𝔻.leftIdentity ⟩
F.fmap (𝟙 ℂ) ≡⟨ F.isIdentity ⟩
𝟙 𝔻 ∎
where

2
src/Cat/Categories/Free.agda
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@ -55,7 +55,7 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
ident-l = refl
isIdentity : IsIdentity 𝟙
isIdentity = ident-r , ident-l
isIdentity = ident-l , ident-r
open Univalence isIdentity

10
src/Cat/Categories/Fun.agda
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@ -45,18 +45,18 @@ module Fun {ℓc ℓc' ℓd ℓd' : Level} (ℂ : Category ℓc ℓc') (𝔻 : C
eq-r : ∀ C → (𝔻 [ f' C ∘ identityTrans A C ]) ≡ f' C
eq-r C = begin
𝔻 [ f' C ∘ identityTrans A C ] ≡⟨⟩
𝔻 [ f' C ∘ 𝔻.𝟙 ] ≡⟨ proj₁ 𝔻.isIdentity ⟩
𝔻 [ f' C ∘ 𝔻.𝟙 ] ≡⟨ 𝔻.rightIdentity ⟩
f' C ∎
eq-l : ∀ C → (𝔻 [ identityTrans B C ∘ f' C ]) ≡ f' C
eq-l C = proj₂ 𝔻.isIdentity
eq-l C = 𝔻.leftIdentity
ident-r : (NT[_∘_] {A} {A} {B} f (NT.identity A)) ≡ f
ident-r = lemSig allNatural _ _ (funExt eq-r)
ident-l : (NT[_∘_] {A} {B} {B} (NT.identity B) f) ≡ f
ident-l = lemSig allNatural _ _ (funExt eq-l)
isIdentity
: (NT[_∘_] {A} {A} {B} f (NT.identity A)) ≡ f
× (NT[_∘_] {A} {B} {B} (NT.identity B) f) ≡ f
isIdentity = ident-r , ident-l
: (NT[_∘_] {A} {B} {B} (NT.identity B) f) ≡ f
× (NT[_∘_] {A} {A} {B} f (NT.identity A)) ≡ f
isIdentity = ident-l , ident-r
-- Functor categories. Objects are functors, arrows are natural transformations.
RawFun : RawCategory (ℓc ⊔ ℓc' ⊔ ℓd ⊔ ℓd') (ℓc ⊔ ℓc' ⊔ ℓd')
RawFun = record

8
src/Cat/Categories/Rel.agda
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@ -76,9 +76,9 @@ module _ {A B : Set} {S : Subset (A × B)} (ab : A × B) where
≃ (a , b) ∈ S
equi = backwards Cubical.FromStdLib., isequiv
ident-l : (Σ[ a' ∈ A ] (a , a') ∈ Diag A × (a' , b) ∈ S)
ident-r : (Σ[ a' ∈ A ] (a , a') ∈ Diag A × (a' , b) ∈ S)
≡ (a , b) ∈ S
ident-l = equivToPath equi
ident-r = equivToPath equi
module _ where
private
@ -110,9 +110,9 @@ module _ {A B : Set} {S : Subset (A × B)} (ab : A × B) where
≃ ab ∈ S
equi = backwards Cubical.FromStdLib., isequiv
ident-r : (Σ[ b' ∈ B ] (a , b') ∈ S × (b' , b) ∈ Diag B)
ident-l : (Σ[ b' ∈ B ] (a , b') ∈ S × (b' , b) ∈ Diag B)
≡ ab ∈ S
ident-r = equivToPath equi
ident-l = equivToPath equi
module _ {A B C D : Set} {S : Subset (A × B)} {R : Subset (B × C)} {Q : Subset (C × D)} (ad : A × D) where
private

44
src/Cat/Categories/Sets.agda
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@ -287,39 +287,35 @@ module _ {ℓ : Level} where
open Category 𝓢
open import Cubical.Sigma
module _ (0A 0B : Object) where
module _ (hA hB : Object) where
open Σ hA renaming (proj₁ to A ; proj₂ to sA)
open Σ hB renaming (proj₁ to B ; proj₂ to sB)
private
A : Set ℓ
A = proj₁ 0A
sA : isSet A
sA = proj₂ 0A
B : Set ℓ
B = proj₁ 0B
sB : isSet B
sB = proj₂ 0B
0A×0B : Object
0A×0B = (A × B) , sigPresSet sA λ _ → sB
productObject : Object
productObject = (A × B) , sigPresSet sA λ _ → sB
module _ {X A B : Set ℓ} (f : X → A) (g : X → B) where
_&&&_ : (X → A × B)
_&&&_ x = f x , g x
module _ {0X : Object} where
X = proj₁ 0X
module _ (f : X → A ) (g : X → B) where
lem : proj₁ Function.∘′ (f &&& g) ≡ f × proj₂ Function.∘′ (f &&& g) ≡ g
proj₁ lem = refl
proj₂ lem = refl
rawProduct : RawProduct 𝓢 0A 0B
RawProduct.object rawProduct = 0A×0B
module _ (hX : Object) where
open Σ hX renaming (proj₁ to X)
module _ (f : X → A ) (g : X → B) where
ump : proj₁ Function.∘′ (f &&& g) ≡ f × proj₂ Function.∘′ (f &&& g) ≡ g
proj₁ ump = refl
proj₂ ump = refl
rawProduct : RawProduct 𝓢 hA hB
RawProduct.object rawProduct = productObject
RawProduct.proj₁ rawProduct = Data.Product.proj₁
RawProduct.proj₂ rawProduct = Data.Product.proj₂
isProduct : IsProduct 𝓢 _ _ rawProduct
IsProduct.ump isProduct {X = X} f g
= (f &&& g) , lem {0X = X} f g
IsProduct.ump isProduct {X = hX} f g
= (f &&& g) , ump hX f g
product : Product 𝓢 0A 0B
product : Product 𝓢 hA hB
Product.raw product = rawProduct
Product.isProduct product = isProduct
@ -346,7 +342,7 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
; fmap = ℂ [_∘_]
}
; isFunctor = record
{ isIdentity = funExt λ _ → proj₂ isIdentity
{ isIdentity = funExt λ _ → leftIdentity
; isDistributive = funExt λ x → sym isAssociative
}
}
@ -359,7 +355,7 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
; fmap = λ f g → ℂ [ g ∘ f ]
}
; isFunctor = record
{ isIdentity = funExt λ x → proj₁ isIdentity
{ isIdentity = funExt λ x → rightIdentity
; isDistributive = funExt λ x → isAssociative
}
}

24
src/Cat/Category.agda
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@ -96,7 +96,7 @@ record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
IsIdentity : ({A : Object} → Arrow A A) → Set (ℓa ⊔ ℓb)
IsIdentity id = {A B : Object} {f : Arrow A B}
→ f ∘ id ≡ f × id ∘ f ≡ f
→ id ∘ f ≡ f × f ∘ id ≡ f
ArrowsAreSets : Set (ℓa ⊔ ℓb)
ArrowsAreSets = ∀ {A B : Object} → isSet (Arrow A B)
@ -166,29 +166,37 @@ record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc
field
univalent : Univalent
leftIdentity : {A B : Object} {f : Arrow A B} → 𝟙 ∘ f ≡ f
leftIdentity {A} {B} {f} = fst (isIdentity {A = A} {B} {f})
-- leftIdentity {A} {B} {f} = snd (isIdentity {A = A} {B} {f})
rightIdentity : {A B : Object} {f : Arrow A B} → f ∘ 𝟙 ≡ f
rightIdentity {A} {B} {f} = snd (isIdentity {A = A} {B} {f})
-- rightIdentity {A} {B} {f} = fst (isIdentity {A = A} {B} {f})
-- Some common lemmas about categories.
module _ {A B : Object} {X : Object} (f : Arrow A B) where
iso-is-epi : Isomorphism f → Epimorphism {X = X} f
iso-is-epi (f- , left-inv , right-inv) g₀ g₁ eq = begin
g₀ ≡⟨ sym (fst isIdentity) ⟩
g₀ ≡⟨ sym rightIdentity ⟩
g₀ ∘ 𝟙 ≡⟨ cong (_∘_ g₀) (sym right-inv) ⟩
g₀ ∘ (f ∘ f-) ≡⟨ isAssociative ⟩
(g₀ ∘ f) ∘ f- ≡⟨ cong (λ φ → φ ∘ f-) eq ⟩
(g₁ ∘ f) ∘ f- ≡⟨ sym isAssociative ⟩
g₁ ∘ (f ∘ f-) ≡⟨ cong (_∘_ g₁) right-inv ⟩
g₁ ∘ 𝟙 ≡⟨ fst isIdentity ⟩
g₁ ∘ 𝟙 ≡⟨ rightIdentity ⟩
g₁ ∎
iso-is-mono : Isomorphism f → Monomorphism {X = X} f
iso-is-mono (f- , (left-inv , right-inv)) g₀ g₁ eq =
begin
g₀ ≡⟨ sym (snd isIdentity) ⟩
g₀ ≡⟨ sym leftIdentity ⟩
𝟙 ∘ g₀ ≡⟨ cong (λ φ → φ ∘ g₀) (sym left-inv) ⟩
(f- ∘ f) ∘ g₀ ≡⟨ sym isAssociative ⟩
f- ∘ (f ∘ g₀) ≡⟨ cong (_∘_ f-) eq ⟩
f- ∘ (f ∘ g₁) ≡⟨ isAssociative ⟩
(f- ∘ f) ∘ g₁ ≡⟨ cong (λ φ → φ ∘ g₁) left-inv ⟩
𝟙 ∘ g₁ ≡⟨ snd isIdentity ⟩
𝟙 ∘ g₁ ≡⟨ leftIdentity ⟩
g₁ ∎
iso-is-epi-mono : Isomorphism f → Epimorphism {X = X} f × Monomorphism {X = X} f
@ -201,7 +209,7 @@ record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc
module Propositionality {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
open RawCategory ℂ
module _ (ℂ : IsCategory ℂ) where
open IsCategory ℂ using (isAssociative ; arrowsAreSets ; isIdentity ; Univalent)
open IsCategory ℂ using (isAssociative ; arrowsAreSets ; Univalent ; leftIdentity ; rightIdentity)
open import Cubical.NType
open import Cubical.NType.Properties
@ -233,11 +241,11 @@ module Propositionality {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
open Cubical.NType.Properties
geq : g ≡ g'
geq = begin
g ≡⟨ sym (fst isIdentity) ⟩
g ≡⟨ sym rightIdentity ⟩
g ∘ 𝟙 ≡⟨ cong (λ φ → g ∘ φ) (sym ε') ⟩
g ∘ (f ∘ g') ≡⟨ isAssociative ⟩
(g ∘ f) ∘ g' ≡⟨ cong (λ φ → φ ∘ g') η ⟩
𝟙 ∘ g' ≡⟨ snd isIdentity ⟩
𝟙 ∘ g' ≡⟨ leftIdentity ⟩
g' ∎
propUnivalent : isProp Univalent

8
src/Cat/Category/Monad.agda
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@ -124,7 +124,7 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
bind (f >>> (pure >>> bind 𝟙))
≡⟨ cong (λ φ → bind (f >>> φ)) (isNatural _) ⟩
bind (f >>> 𝟙)
≡⟨ cong bind (proj₂ ℂ.isIdentity) ⟩
≡⟨ cong bind ℂ.leftIdentity ⟩
bind f ∎
forthRawEq : forthRaw (backRaw m) ≡ K.Monad.raw m
@ -155,7 +155,7 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
KM.bind 𝟙 ≡⟨⟩
bind 𝟙 ≡⟨⟩
joinT X ∘ Rfmap 𝟙 ≡⟨ cong (λ φ → _ ∘ φ) R.isIdentity ⟩
joinT X ∘ 𝟙 ≡⟨ proj₁ ℂ.isIdentity ⟩
joinT X ∘ 𝟙 ≡⟨ ℂ.rightIdentity ⟩
joinT X ∎
fmapEq : ∀ {A B} → KM.fmap {A} {B} ≡ Rfmap
@ -167,7 +167,7 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
Rfmap (f >>> pureT B) >>> joinT B ≡⟨ cong (λ φ → φ >>> joinT B) R.isDistributive ⟩
Rfmap f >>> Rfmap (pureT B) >>> joinT B ≡⟨ ℂ.isAssociative ⟩
joinT B ∘ Rfmap (pureT B) ∘ Rfmap f ≡⟨ cong (λ φ → φ ∘ Rfmap f) (proj₂ isInverse) ⟩
𝟙 ∘ Rfmap f ≡⟨ proj₂ ℂ.isIdentity ⟩
𝟙 ∘ Rfmap f ≡⟨ ℂ.leftIdentity ⟩
Rfmap f ∎
)
@ -192,7 +192,7 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
M.RawMonad.joinT (backRaw (forth m)) X ≡⟨⟩
KM.join ≡⟨⟩
joinT X ∘ Rfmap 𝟙 ≡⟨ cong (λ φ → joinT X ∘ φ) R.isIdentity ⟩
joinT X ∘ 𝟙 ≡⟨ proj₁ ℂ.isIdentity ⟩
joinT X ∘ 𝟙 ≡⟨ ℂ.rightIdentity ⟩
joinT X ∎)
joinNTEq : (λ i → NaturalTransformation F[ Req i ∘ Req i ] (Req i))

12
src/Cat/Category/Monad/Kleisli.agda
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@ -104,7 +104,7 @@ record IsMonad (raw : RawMonad) : Set ℓ where
isFunctorR : IsFunctor ℂ ℂ rawR
IsFunctor.isIdentity isFunctorR = begin
bind (pure ∘ 𝟙) ≡⟨ cong bind (proj₁ ℂ.isIdentity) ⟩
bind (pure ∘ 𝟙) ≡⟨ cong bind (ℂ.rightIdentity) ⟩
bind pure ≡⟨ isIdentity ⟩
𝟙 ∎
@ -156,9 +156,9 @@ record IsMonad (raw : RawMonad) : Set ℓ where
bind (bind (f >>> pure) >>> (pure >>> bind 𝟙))
≡⟨ cong (λ φ → bind (bind (f >>> pure) >>> φ)) (isNatural _) ⟩
bind (bind (f >>> pure) >>> 𝟙)
≡⟨ cong bind (proj₂ ℂ.isIdentity) ⟩
≡⟨ cong bind ℂ.leftIdentity ⟩
bind (bind (f >>> pure))
≡⟨ cong bind (sym (proj₁ ℂ.isIdentity)) ⟩
≡⟨ cong bind (sym ℂ.rightIdentity) ⟩
bind (𝟙 >>> bind (f >>> pure)) ≡⟨⟩
bind (𝟙 >=> (f >>> pure))
≡⟨ sym (isDistributive _ _) ⟩
@ -186,10 +186,10 @@ record IsMonad (raw : RawMonad) : Set ℓ where
bind (join >>> (pure >>> bind 𝟙))
≡⟨ cong (λ φ → bind (join >>> φ)) (isNatural _) ⟩
bind (join >>> 𝟙)
≡⟨ cong bind (proj₂ ℂ.isIdentity) ⟩
≡⟨ cong bind ℂ.leftIdentity ⟩
bind join ≡⟨⟩
bind (bind 𝟙)
≡⟨ cong bind (sym (proj₁ ℂ.isIdentity)) ⟩
≡⟨ cong bind (sym ℂ.rightIdentity) ⟩
bind (𝟙 >>> bind 𝟙) ≡⟨⟩
bind (𝟙 >=> 𝟙) ≡⟨ sym (isDistributive _ _) ⟩
bind 𝟙 >>> bind 𝟙 ≡⟨⟩
@ -212,7 +212,7 @@ record IsMonad (raw : RawMonad) : Set ℓ where
bind (pure >>> (pure >>> bind 𝟙))
≡⟨ cong (λ φ → bind (pure >>> φ)) (isNatural _) ⟩
bind (pure >>> 𝟙)
≡⟨ cong bind (proj₂ ℂ.isIdentity) ⟩
≡⟨ cong bind ℂ.leftIdentity ⟩
bind pure ≡⟨ isIdentity ⟩
𝟙 ∎

4
src/Cat/Category/Monad/Monoidal.agda
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@ -75,8 +75,8 @@ record IsMonad (raw : RawMonad) : Set ℓ where
joinT Y ∘ (R.fmap f ∘ pureT X) ≡⟨ cong (λ φ → joinT Y ∘ φ) (sym (pureN f)) ⟩
joinT Y ∘ (pureT (R.omap Y) ∘ f) ≡⟨ ℂ.isAssociative ⟩
joinT Y ∘ pureT (R.omap Y) ∘ f ≡⟨ cong (λ φ → φ ∘ f) (proj₁ isInverse) ⟩
𝟙 ∘ f ≡⟨ proj₂ ℂ.isIdentity ⟩
f ∎
𝟙 ∘ f ≡⟨ ℂ.leftIdentity ⟩
f ∎
isDistributive : IsDistributive
isDistributive {X} {Y} {Z} g f = sym aux

4
src/Cat/Category/NaturalTransformation.agda
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@ -72,8 +72,8 @@ module NaturalTransformation {ℓc ℓc' ℓd ℓd' : Level}
identityNatural : (F : Functor ℂ 𝔻) → Natural F F (identityTrans F)
identityNatural F {A = A} {B = B} f = begin
𝔻 [ identityTrans F B ∘ F→ f ] ≡⟨⟩
𝔻 [ 𝟙 𝔻 ∘ F→ f ] ≡⟨ proj₂ 𝔻.isIdentity ⟩
F→ f ≡⟨ sym (proj₁ 𝔻.isIdentity) ⟩
𝔻 [ 𝟙 𝔻 ∘ F→ f ] ≡⟨ 𝔻.leftIdentity ⟩
F→ f ≡⟨ sym 𝔻.rightIdentity ⟩
𝔻 [ F→ f ∘ 𝟙 𝔻 ] ≡⟨⟩
𝔻 [ F→ f ∘ identityTrans F A ] ∎
where

2
src/Cat/Category/Yoneda.agda
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@ -54,7 +54,7 @@ module _ {ℓ : Level} {ℂ : Category ℓ ℓ} where
isIdentity {c} = lemSig (naturalIsProp {F = presheaf c} {presheaf c}) _ _ eq
where
eq : (λ C x → ℂ [ ℂ.𝟙 ∘ x ]) ≡ identityTrans (presheaf c)
eq = funExt λ A → funExt λ B → proj₂ ℂ.isIdentity
eq = funExt λ A → funExt λ B → ℂ.leftIdentity
isDistributive : IsDistributive
isDistributive {A} {B} {C} {f = f} {g}