fredefox/cat
1
0
Fork 0
Code Issues Pull requests Releases Wiki Activity

Rename assoc to isAssociative

Browse source
This commit is contained in:
Frederik Hanghøj Iversen 2018-02-23 12:43:49 +01:00
parent 852056cc44
commit 5cbc409770
8 changed files with 39 additions and 39 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

14
src/Cat/Categories/Cat.agda
View file

@ -43,8 +43,8 @@ module _ (ℓ ℓ' : Level) where
}
private
open RawCategory RawCat
assoc : IsAssociative
assoc {f = F} {G} {H} = assc {F = F} {G = G} {H = H}
isAssociative : IsAssociative
isAssociative {f = F} {G} {H} = assc {F = F} {G = G} {H = H}
-- TODO: Rename `ident'` to `ident` after changing how names are exposed in Functor.
ident' : IsIdentity identity
ident' = ident-r , ident-l
@ -96,7 +96,7 @@ module _ {ℓ ℓ' : Level} (unprovable : IsCategory (RawCat ℓ ℓ')) where
postulate univalent : Univalence.Univalent :rawProduct: ident'
instance
:isCategory: : IsCategory :rawProduct:
IsCategory.assoc :isCategory: = Σ≡ C.assoc D.assoc
IsCategory.isAssociative :isCategory: = Σ≡ C.isAssociative D.isAssociative
IsCategory.ident :isCategory: = ident'
IsCategory.arrowIsSet :isCategory: = issSet
IsCategory.univalent :isCategory: = univalent
@ -288,15 +288,15 @@ module _ (ℓ : Level) (unprovable : IsCategory (RawCat ℓ ℓ)) where
𝔻 [ func→ H (ℂ [ g ∘ f ]) ∘ (ηθ A) ]
≡⟨ cong (λ φ → 𝔻 [ φ ∘ ηθ A ]) (H.distrib) ⟩
𝔻 [ 𝔻 [ func→ H g ∘ func→ H f ] ∘ (ηθ A) ]
≡⟨ sym assoc ⟩
≡⟨ sym isAssociative ⟩
𝔻 [ func→ H g ∘ 𝔻 [ func→ H f ∘ ηθ A ] ]
≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) assoc ⟩
≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) isAssociative ⟩
𝔻 [ func→ H g ∘ 𝔻 [ 𝔻 [ func→ H f ∘ η A ] ∘ θ A ] ]
≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) (cong (λ φ → 𝔻 [ φ ∘ θ A ]) (sym (ηNat f))) ⟩
𝔻 [ func→ H g ∘ 𝔻 [ 𝔻 [ η B ∘ func→ G f ] ∘ θ A ] ]
≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) (sym assoc) ⟩
≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) (sym isAssociative) ⟩
𝔻 [ func→ H g ∘ 𝔻 [ η B ∘ 𝔻 [ func→ G f ∘ θ A ] ] ]
≡⟨ assoc ⟩
≡⟨ isAssociative ⟩
𝔻 [ 𝔻 [ func→ H g ∘ η B ] ∘ 𝔻 [ func→ G f ∘ θ A ] ]
≡⟨ cong (λ φ → 𝔻 [ φ ∘ 𝔻 [ func→ G f ∘ θ A ] ]) (sym (ηNat g)) ⟩
𝔻 [ 𝔻 [ η C ∘ func→ G g ] ∘ 𝔻 [ func→ G f ∘ θ A ] ]

6
src/Cat/Categories/Fam.agda
View file

@ -25,8 +25,8 @@ module _ (ℓa ℓb : Level) where
c ⟨ g ∘ f ⟩ = _∘_ {c = c} g f
module _ {A B C D : Obj'} {f : Arr A B} {g : Arr B C} {h : Arr C D} where
assoc : (D ⟨ h ∘ C ⟨ g ∘ f ⟩ ⟩) ≡ D ⟨ D ⟨ h ∘ g ⟩ ∘ f ⟩
assoc = Σ≡ refl refl
isAssociative : (D ⟨ h ∘ C ⟨ g ∘ f ⟩ ⟩) ≡ D ⟨ D ⟨ h ∘ g ⟩ ∘ f ⟩
isAssociative = Σ≡ refl refl
module _ {A B : Obj'} {f : Arr A B} where
ident : B ⟨ f ∘ one ⟩ ≡ f × B ⟨ one {B} ∘ f ⟩ ≡ f
@ -44,7 +44,7 @@ module _ (ℓa ℓb : Level) where
instance
isCategory : IsCategory RawFam
isCategory = record
{ assoc = λ {A} {B} {C} {D} {f} {g} {h} → assoc {D = D} {f} {g} {h}
{ isAssociative = λ {A} {B} {C} {D} {f} {g} {h} → isAssociative {D = D} {f} {g} {h}
; ident = λ {A} {B} {f} → ident {A} {B} {f = f}
; arrowIsSet = {!!}
; univalent = {!!}

10
src/Cat/Categories/Free.agda
View file

@ -26,16 +26,16 @@ module _ {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') where
open Category ℂ
private
p-assoc : {A B C D : Object} {r : Path Arrow A B} {q : Path Arrow B C} {p : Path Arrow C D}
p-isAssociative : {A B C D : Object} {r : Path Arrow A B} {q : Path Arrow B C} {p : Path Arrow C D}
→ p ++ (q ++ r) ≡ (p ++ q) ++ r
p-assoc {r = r} {q} {empty} = refl
p-assoc {A} {B} {C} {D} {r = r} {q} {cons x p} = begin
p-isAssociative {r = r} {q} {empty} = refl
p-isAssociative {A} {B} {C} {D} {r = r} {q} {cons x p} = begin
cons x p ++ (q ++ r) ≡⟨ cong (cons x) lem ⟩
cons x ((p ++ q) ++ r) ≡⟨⟩
(cons x p ++ q) ++ r ∎
where
lem : p ++ (q ++ r) ≡ ((p ++ q) ++ r)
lem = p-assoc {r = r} {q} {p}
lem = p-isAssociative {r = r} {q} {p}
ident-r : ∀ {A} {B} {p : Path Arrow A B} → concatenate p empty ≡ p
ident-r {p = empty} = refl
@ -57,7 +57,7 @@ module _ {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') where
}
RawIsCategoryFree : IsCategory RawFree
RawIsCategoryFree = record
{ assoc = λ { {f = f} {g} {h} → p-assoc {r = f} {g} {h}}
{ isAssociative = λ { {f = f} {g} {h} → p-isAssociative {r = f} {g} {h}}
; ident = ident-r , ident-l
; arrowIsSet = {!!}
; univalent = {!!}

14
src/Cat/Categories/Fun.agda
View file

@ -84,11 +84,11 @@ module _ {ℓc ℓc' ℓd ℓd' : Level} {ℂ : Category ℓc ℓc'} {𝔻 : Cat
proj₁ ((θ , _) :⊕: (η , _)) = θ ∘nt η
proj₂ ((θ , θNat) :⊕: (η , ηNat)) {A} {B} f = begin
𝔻 [ (θ ∘nt η) B ∘ F.func→ f ] ≡⟨⟩
𝔻 [ 𝔻 [ θ B ∘ η B ] ∘ F.func→ f ] ≡⟨ sym assoc ⟩
𝔻 [ 𝔻 [ θ B ∘ η B ] ∘ F.func→ f ] ≡⟨ sym isAssociative ⟩
𝔻 [ θ B ∘ 𝔻 [ η B ∘ F.func→ f ] ] ≡⟨ cong (λ φ → 𝔻 [ θ B ∘ φ ]) (ηNat f) ⟩
𝔻 [ θ B ∘ 𝔻 [ G.func→ f ∘ η A ] ] ≡⟨ assoc ⟩
𝔻 [ θ B ∘ 𝔻 [ G.func→ f ∘ η A ] ] ≡⟨ isAssociative ⟩
𝔻 [ 𝔻 [ θ B ∘ G.func→ f ] ∘ η A ] ≡⟨ cong (λ φ → 𝔻 [ φ ∘ η A ]) (θNat f) ⟩
𝔻 [ 𝔻 [ H.func→ f ∘ θ A ] ∘ η A ] ≡⟨ sym assoc ⟩
𝔻 [ 𝔻 [ H.func→ f ∘ θ A ] ∘ η A ] ≡⟨ sym isAssociative ⟩
𝔻 [ H.func→ f ∘ 𝔻 [ θ A ∘ η A ] ] ≡⟨⟩
𝔻 [ H.func→ f ∘ (θ ∘nt η) A ] ∎
where
@ -130,9 +130,9 @@ module _ {ℓc ℓc' ℓd ℓd' : Level} {ℂ : Category ℓc ℓc'} {𝔻 : Cat
R = (_:⊕:_ {A} {B} {D} (_:⊕:_ {B} {C} {D} ζ' η') θ')
_g⊕f_ = _:⊕:_ {A} {B} {C}
_h⊕g_ = _:⊕:_ {B} {C} {D}
:assoc: : L ≡ R
:assoc: = lemSig (naturalIsProp {F = A} {D})
L R (funExt (λ x → assoc))
:isAssociative: : L ≡ R
:isAssociative: = lemSig (naturalIsProp {F = A} {D})
L R (funExt (λ x → isAssociative))
where
open Category 𝔻
@ -168,7 +168,7 @@ module _ {ℓc ℓc' ℓd ℓd' : Level} {ℂ : Category ℓc ℓc'} {𝔻 : Cat
instance
:isCategory: : IsCategory RawFun
:isCategory: = record
{ assoc = λ {A B C D} → :assoc: {A} {B} {C} {D}
{ isAssociative = λ {A B C D} → :isAssociative: {A} {B} {C} {D}
; ident = λ {A B} → :ident: {A} {B}
; arrowIsSet = λ {F} {G} → naturalTransformationIsSets {F} {G}
; univalent = {!!}

8
src/Cat/Categories/Rel.agda
View file

@ -149,10 +149,10 @@ module _ {A B C D : Set} {S : Subset (A × B)} {R : Subset (B × C)} {Q : Subset
≃ (Σ[ b ∈ B ] (a , b) ∈ S × (Σ[ c ∈ C ] (b , c) ∈ R × (c , d) ∈ Q))
equi = fwd Cubical.FromStdLib., isequiv
-- assocc : Q + (R + S) ≡ (Q + R) + S
is-assoc : (Σ[ c ∈ C ] (Σ[ b ∈ B ] (a , b) ∈ S × (b , c) ∈ R) × (c , d) ∈ Q)
-- isAssociativec : Q + (R + S) ≡ (Q + R) + S
is-isAssociative : (Σ[ c ∈ C ] (Σ[ b ∈ B ] (a , b) ∈ S × (b , c) ∈ R) × (c , d) ∈ Q)
≡ (Σ[ b ∈ B ] (a , b) ∈ S × (Σ[ c ∈ C ] (b , c) ∈ R × (c , d) ∈ Q))
is-assoc = equivToPath equi
is-isAssociative = equivToPath equi
RawRel : RawCategory (lsuc lzero) (lsuc lzero)
RawRel = record
@ -164,7 +164,7 @@ RawRel = record
RawIsCategoryRel : IsCategory RawRel
RawIsCategoryRel = record
{ assoc = funExt is-assoc
{ isAssociative = funExt is-isAssociative
; ident = funExt ident-l , funExt ident-r
; arrowIsSet = {!!}
; univalent = {!!}

6
src/Cat/Categories/Sets.agda
View file

@ -25,7 +25,7 @@ module _ (ℓ : Level) where
_∘_ SetsRaw = Function._∘′_
SetsIsCategory : IsCategory SetsRaw
assoc SetsIsCategory = refl
isAssociative SetsIsCategory = refl
proj₁ (ident SetsIsCategory) = funExt λ _ → refl
proj₂ (ident SetsIsCategory) = funExt λ _ → refl
arrowIsSet SetsIsCategory {B = (_ , s)} = setPi λ _ → s
@ -99,7 +99,7 @@ module _ {ℓa ℓb : Level} where
}
; isFunctor = record
{ ident = funExt λ _ → proj₂ ident
; distrib = funExt λ x → sym assoc
; distrib = funExt λ x → sym isAssociative
}
}
where
@ -114,7 +114,7 @@ module _ {ℓa ℓb : Level} where
}
; isFunctor = record
{ ident = funExt λ x → proj₁ ident
; distrib = funExt λ x → assoc
; distrib = funExt λ x → isAssociative
}
}
where

10
src/Cat/Category.agda
View file

@ -101,7 +101,7 @@ record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc
open RawCategory ℂ
open Univalence ℂ public
field
assoc : IsAssociative
isAssociative : IsAssociative
ident : IsIdentity 𝟙
arrowIsSet : ArrowsAreSets
univalent : Univalent ident
@ -144,7 +144,7 @@ module _ {ℓa ℓb : Level} {C : RawCategory ℓa ℓb} where
geq = begin
g ≡⟨ sym (fst ident) ⟩
g ∘ 𝟙 ≡⟨ cong (λ φ → g ∘ φ) (sym ε') ⟩
g ∘ (f ∘ g') ≡⟨ assoc ⟩
g ∘ (f ∘ g') ≡⟨ isAssociative ⟩
(g ∘ f) ∘ g' ≡⟨ cong (λ φ → φ ∘ g') η ⟩
𝟙 ∘ g' ≡⟨ snd ident ⟩
g' ∎
@ -181,7 +181,7 @@ module _ {ℓa ℓb : Level} {C : RawCategory ℓa ℓb} where
foo = pathJ P helper Y.ident ident
eqUni : U ident Y.univalent
eqUni = foo Y.univalent
IC.assoc (done i) = propIsAssociative x X.assoc Y.assoc i
IC.isAssociative (done i) = propIsAssociative x X.isAssociative Y.isAssociative i
IC.ident (done i) = ident i
IC.arrowIsSet (done i) = propArrowIsSet x X.arrowIsSet Y.arrowIsSet i
IC.univalent (done i) = eqUni i
@ -216,7 +216,7 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
RawCategory._∘_ OpRaw = Function.flip _∘_
OpIsCategory : IsCategory OpRaw
IsCategory.assoc OpIsCategory = sym assoc
IsCategory.isAssociative OpIsCategory = sym isAssociative
IsCategory.ident OpIsCategory = swap ident
IsCategory.arrowIsSet OpIsCategory = arrowIsSet
IsCategory.univalent OpIsCategory = {!!}
@ -242,7 +242,7 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
open IsCategory
module IsCat = IsCategory (ℂ .isCategory)
rawIsCat : (i : I) → IsCategory (rawOp i)
assoc (rawIsCat i) = IsCat.assoc
isAssociative (rawIsCat i) = IsCat.isAssociative
ident (rawIsCat i) = IsCat.ident
arrowIsSet (rawIsCat i) = IsCat.arrowIsSet
univalent (rawIsCat i) = IsCat.univalent

10
src/Cat/Category/Properties.agda
View file

@ -19,9 +19,9 @@ module _ {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} { A B : Category.Object
iso-is-epi (f- , left-inv , right-inv) g₀ g₁ eq = begin
g₀ ≡⟨ sym (proj₁ ident) ⟩
g₀ ∘ 𝟙 ≡⟨ cong (_∘_ g₀) (sym right-inv) ⟩
g₀ ∘ (f ∘ f-) ≡⟨ assoc ⟩
g₀ ∘ (f ∘ f-) ≡⟨ isAssociative ⟩
(g₀ ∘ f) ∘ f- ≡⟨ cong (λ φ → φ ∘ f-) eq ⟩
(g₁ ∘ f) ∘ f- ≡⟨ sym assoc ⟩
(g₁ ∘ f) ∘ f- ≡⟨ sym isAssociative ⟩
g₁ ∘ (f ∘ f-) ≡⟨ cong (_∘_ g₁) right-inv ⟩
g₁ ∘ 𝟙 ≡⟨ proj₁ ident ⟩
g₁ ∎
@ -31,9 +31,9 @@ module _ {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} { A B : Category.Object
begin
g₀ ≡⟨ sym (proj₂ ident) ⟩
𝟙 ∘ g₀ ≡⟨ cong (λ φ → φ ∘ g₀) (sym left-inv) ⟩
(f- ∘ f) ∘ g₀ ≡⟨ sym assoc ⟩
(f- ∘ f) ∘ g₀ ≡⟨ sym isAssociative ⟩
f- ∘ (f ∘ g₀) ≡⟨ cong (_∘_ f-) eq ⟩
f- ∘ (f ∘ g₁) ≡⟨ assoc ⟩
f- ∘ (f ∘ g₁) ≡⟨ isAssociative ⟩
(f- ∘ f) ∘ g₁ ≡⟨ cong (λ φ → φ ∘ g₁) left-inv ⟩
𝟙 ∘ g₁ ≡⟨ proj₂ ident ⟩
g₁ ∎
@ -65,7 +65,7 @@ module _ {ℓ : Level} {ℂ : Category ℓ ℓ} (unprovable : IsCategory (RawCat
module _ {A B : ℂ.Object} (f : ℂ [ A , B ]) where
:func→: : NaturalTransformation (prshf A) (prshf B)
:func→: = (λ C x → ℂ [ f ∘ x ]) , λ f₁ → funExt λ _ → ℂ.assoc
:func→: = (λ C x → ℂ [ f ∘ x ]) , λ f₁ → funExt λ _ → ℂ.isAssociative
module _ {c : Category.Object ℂ} where
eqTrans : (λ _ → Transformation (prshf c) (prshf c))