Make level-parameters to Category explicit

2018-01-21 01:11:08 +01:00 · 2018-01-21 01:11:08 +01:00 · 07e4269399
parent 0990a3778f
commit 07e4269399
8 changed files with 31 additions and 31 deletions
--- a/src/Cat/Categories/Cat.agda
+++ b/src/Cat/Categories/Cat.agda
@ -20,7 +20,7 @@ snd (lift-eq a b i) = b i

 open Functor
 open Category
-module _ {ℓ ℓ' : Level} {A B : Category {ℓ} {ℓ'}} where
+module _ {ℓ ℓ' : Level} {A B : Category ℓ ℓ'} where
  lift-eq-functors : {f g : Functor A B}
    → (eq* : Functor.func* f ≡ Functor.func* g)
    → (eq→ : PathP (λ i → ∀ {x y} → Arrow A x y → Arrow B (eq* i x) (eq* i y))
@ -41,11 +41,11 @@ module _ {ℓ ℓ' : Level} {A B : Category {ℓ} {ℓ'}} where
 module _ {ℓ ℓ' : Level} where
  private
    _⊛_ = functor-comp
-    module _ {A B C D : Category {ℓ} {ℓ'}} {f : Functor A B} {g : Functor B C} {h : Functor C D} where
+    module _ {A B C D : Category ℓ ℓ'} {f : Functor A B} {g : Functor B C} {h : Functor C D} where
      postulate assc : h ⊛ (g ⊛ f) ≡ (h ⊛ g) ⊛ f
      -- assc = lift-eq-functors refl refl {!refl!} λ i j → {!!}

-    module _ {A B : Category {ℓ} {ℓ'}} {f : Functor A B} where
+    module _ {A B : Category ℓ ℓ'} {f : Functor A B} where
      lem : (func* f) ∘ (func* (identity {C = A})) ≡ func* f
      lem = refl
      -- lemmm : func→ {C = A} {D = B} (f ⊛ identity) ≡ func→ f
@ -62,10 +62,10 @@ module _ {ℓ ℓ' : Level} where
      postulate ident-l : identity ⊛ f ≡ f
      -- ident-l = lift-eq-functors lem lemmm {!refl!} {!!}

-  CatCat : Category {lsuc (ℓ ⊔ ℓ')} {ℓ ⊔ ℓ'}
+  CatCat : Category (lsuc (ℓ ⊔ ℓ')) (ℓ ⊔ ℓ')
  CatCat =
    record
-      { Object = Category {ℓ} {ℓ'}
+      { Object = Category ℓ ℓ'
      ; Arrow = Functor
      ; 𝟙 = identity
      ; _⊕_ = functor-comp
@ -74,7 +74,7 @@ module _ {ℓ ℓ' : Level} where
      ; ident = ident-r , ident-l
      }

-module _  {ℓ : Level} (C D : Category {ℓ} {ℓ}) where
+module _  {ℓ : Level} (C D : Category ℓ ℓ) where
  private
    proj₁ : Arrow CatCat (catProduct C D) C
    proj₁ = record { func* = fst ; func→ = fst ; ident = refl ; distrib = refl }
--- a/src/Cat/Categories/Rel.agda
+++ b/src/Cat/Categories/Rel.agda
@ -154,7 +154,7 @@ 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))
  is-assoc = equivToPath equi

-Rel : Category
+Rel : Category (lsuc lzero) (lsuc lzero)
 Rel = record
  { Object = Set
  ; Arrow = λ S R → Subset (S × R)
--- a/src/Cat/Categories/Sets.agda
+++ b/src/Cat/Categories/Sets.agda
@ -15,7 +15,7 @@ open import Cat.Functor
 Fun : {ℓ : Level} → ( T U : Set ℓ ) → Set ℓ
 Fun T U = T → U

-Sets : {ℓ : Level} → Category {lsuc ℓ} {ℓ}
+Sets : {ℓ : Level} → Category (lsuc ℓ) ℓ
 Sets {ℓ} = record
  { Object = Set ℓ
  ; Arrow = λ T U → Fun {ℓ} T U
@ -26,11 +26,11 @@ Sets {ℓ} = record
  }

 -- Covariant Presheaf
-Representable : {ℓ ℓ' : Level} → (ℂ : Category {ℓ} {ℓ'}) → Set (ℓ ⊔ lsuc ℓ')
+Representable : {ℓ ℓ' : Level} → (ℂ : Category ℓ ℓ') → Set (ℓ ⊔ lsuc ℓ')
 Representable {ℓ' = ℓ'} ℂ = Functor ℂ (Sets {ℓ'})

 -- The "co-yoneda" embedding.
-representable : {ℓ ℓ' : Level} → {ℂ : Category {ℓ} {ℓ'}} → Category.Object ℂ → Representable ℂ
+representable : ∀ {ℓ ℓ'} {ℂ : Category ℓ ℓ'} → Category.Object ℂ → Representable ℂ
 representable {ℂ = ℂ} A = record
  { func* = λ B → ℂ.Arrow A B
  ; func→ = λ f g → f ℂ.⊕ g
@ -41,11 +41,11 @@ representable {ℂ = ℂ} A = record
    open module ℂ = Category ℂ

 -- Contravariant Presheaf
-Presheaf : {ℓ ℓ' : Level} → (ℂ : Category {ℓ} {ℓ'}) → Set (ℓ ⊔ lsuc ℓ')
+Presheaf : ∀ {ℓ ℓ'} (ℂ : Category ℓ ℓ') → Set (ℓ ⊔ lsuc ℓ')
 Presheaf {ℓ' = ℓ'} ℂ = Functor (Opposite ℂ) (Sets {ℓ'})

 -- Alternate name: `yoneda`
-presheaf : {ℓ ℓ' : Level} → {ℂ : Category {ℓ} {ℓ'}} → Category.Object (Opposite ℂ) → Presheaf ℂ
+presheaf : {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} → Category.Object (Opposite ℂ) → Presheaf ℂ
 presheaf {ℂ = ℂ} B = record
  { func* = λ A → ℂ.Arrow A B
  ; func→ = λ f g → g ℂ.⊕ f
--- a/src/Cat/Category.agda
+++ b/src/Cat/Category.agda
@ -24,7 +24,7 @@ syntax ∃!-syntax (λ x → B) = ∃![ x ] B

 postulate undefined : {ℓ : Level} → {A : Set ℓ} → A

-record Category {ℓ ℓ'} : Set (lsuc (ℓ' ⊔ ℓ)) where
+record Category (ℓ ℓ' : Level) : Set (lsuc (ℓ' ⊔ ℓ)) where
  -- adding no-eta-equality can speed up type-checking.
  no-eta-equality
  field
@ -44,7 +44,7 @@ record Category {ℓ ℓ'} : Set (lsuc (ℓ' ⊔ ℓ)) where

 open Category public

-module _ {ℓ ℓ' : Level} {ℂ : Category {ℓ} {ℓ'}} { A B : ℂ .Object } where
+module _ {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} { A B : ℂ .Object } where
  private
    open module ℂ = Category ℂ
    _+_ = ℂ._⊕_
@ -93,10 +93,10 @@ epi-mono-is-not-iso f =
 -}

 -- Isomorphism of objects
-_≅_ : { ℓ ℓ' : Level } → { ℂ : Category {ℓ} {ℓ'} } → ( A B : Object ℂ ) → Set ℓ'
+_≅_ : ∀ {ℓ ℓ'} {ℂ : Category ℓ ℓ'} (A B : Object ℂ) → Set ℓ'
 _≅_ {ℂ = ℂ} A B = Σ[ f ∈ ℂ .Arrow A B ] (Isomorphism {ℂ = ℂ} f)

-IsProduct : ∀ {ℓ ℓ'} (ℂ : Category {ℓ} {ℓ'}) {A B obj : Object ℂ} (π₁ : Arrow ℂ obj A) (π₂ : Arrow ℂ obj B) → Set (ℓ ⊔ ℓ')
+IsProduct : ∀ {ℓ ℓ'} (ℂ : Category ℓ ℓ') {A B obj : Object ℂ} (π₁ : Arrow ℂ obj A) (π₂ : Arrow ℂ obj B) → Set (ℓ ⊔ ℓ')
 IsProduct ℂ {A = A} {B = B} π₁ π₂
  = ∀ {X : ℂ.Object} (x₁ : ℂ.Arrow X A) (x₂ : ℂ.Arrow X B)
  → ∃![ x ] (π₁ ℂ.⊕ x ≡ x₁ × π₂ ℂ.⊕ x ≡ x₂)
@ -110,7 +110,7 @@ IsProduct ℂ {A = A} {B = B} π₁ π₂
 --      isProduct : ∀ {X : ℂ .Object} (x₁ : ℂ .Arrow X A) (x₂ : ℂ .Arrow X B)
 --        → ∃![ x ] (ℂ ._⊕_ π₁ x ≡ x₁ × ℂ. _⊕_ π₂ x ≡ x₂)

-record Product {ℓ ℓ' : Level} {ℂ : Category {ℓ} {ℓ'}} (A B : ℂ .Object) : Set (ℓ ⊔ ℓ') where
+record Product {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} (A B : ℂ .Object) : Set (ℓ ⊔ ℓ') where
  no-eta-equality
  field
    obj : ℂ .Object
@ -119,7 +119,7 @@ record Product {ℓ ℓ' : Level} {ℂ : Category {ℓ} {ℓ'}} (A B : ℂ .Obje
    {{isProduct}} : IsProduct ℂ proj₁ proj₂

 mutual
-  catProduct : {ℓ : Level} → (C D : Category {ℓ} {ℓ}) → Category {ℓ} {ℓ}
+  catProduct : ∀ {ℓ} (C D : Category ℓ ℓ) → Category ℓ ℓ
  catProduct C D =
    record
      { Object = C.Object × D.Object
@ -146,10 +146,10 @@ mutual
  -- arrowProduct = {!!}

  -- Arrows in the product-category
-  arrowProduct : ∀ {ℓ} {C D : Category {ℓ} {ℓ}} (c d : Object (catProduct C D)) → Set ℓ
+  arrowProduct : ∀ {ℓ} {C D : Category ℓ ℓ} (c d : Object (catProduct C D)) → Set ℓ
  arrowProduct {C = C} {D = D} (c , d) (c' , d') = Arrow C c c' × Arrow D d d'

-Opposite : ∀ {ℓ ℓ'} → Category {ℓ} {ℓ'} → Category {ℓ} {ℓ'}
+Opposite : ∀ {ℓ ℓ'} → Category ℓ ℓ' → Category ℓ ℓ'
 Opposite ℂ =
  record
    { Object = ℂ.Object
@ -173,10 +173,10 @@ Opposite ℂ =
 -- assoc (Opposite-is-involution i) = {!!}
 -- ident (Opposite-is-involution i) = {!!}

-Hom : {ℓ ℓ' : Level} → (ℂ : Category {ℓ} {ℓ'}) → (A B : Object ℂ) → Set ℓ'
+Hom : {ℓ ℓ' : Level} → (ℂ : Category ℓ ℓ') → (A B : Object ℂ) → Set ℓ'
 Hom ℂ A B = Arrow ℂ A B

-module _ {ℓ ℓ' : Level} {ℂ : Category {ℓ} {ℓ'}} where
+module _ {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} where
  HomFromArrow : (A : ℂ .Object) → {B B' : ℂ .Object} → (g : ℂ .Arrow B B')
    → Hom ℂ A B → Hom ℂ A B'
  HomFromArrow _A = _⊕_ ℂ
--- a/src/Cat/Category/Free.agda
+++ b/src/Cat/Category/Free.agda
@ -6,12 +6,12 @@ open import Data.Product

 open import Cat.Category as C

-module _ {ℓ ℓ' : Level} (ℂ : Category {ℓ} {ℓ'}) where
+module _ {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') where
  private
    open module ℂ = Category ℂ
    Obj = ℂ.Object

-  Path : ( a b : Obj ) → Set
+  Path : ( a b : Obj ) → Set ℓ'
  Path a b = undefined

  postulate emptyPath : (o : Obj) → Path o o
@ -28,7 +28,7 @@ module _ {ℓ ℓ' : Level} (ℂ : Category {ℓ} {ℓ'}) where
        ident-r : concatenate {A} {A} {B} p (emptyPath A) ≡ p
        ident-l : concatenate {A} {B} {B} (emptyPath B) p ≡ p

-  Free : Category
+  Free : Category ℓ ℓ'
  Free = record
    { Object = Obj
    ; Arrow = Path
--- a/src/Cat/Category/Properties.agda
+++ b/src/Cat/Category/Properties.agda
@ -7,7 +7,7 @@ open import Cat.Functor
 open import Cat.Categories.Sets

 module _ {ℓa ℓa' ℓb ℓb'} where
-  Exponential : Category {ℓa} {ℓa'} → Category {ℓb} {ℓb'} → Category {{!!}} {{!!}}
+  Exponential : Category ℓa ℓa' → Category ℓb ℓb' → Category ? ?
  Exponential A B = record
    { Object = {!!}
    ; Arrow = {!!}
@ -19,5 +19,5 @@ module _ {ℓa ℓa' ℓb ℓb'} where

 _⇑_ = Exponential

-yoneda : ∀ {ℓ ℓ'} → {ℂ : Category {ℓ} {ℓ'}} → Functor ℂ (Sets ⇑ (Opposite ℂ))
+yoneda : ∀ {ℓ ℓ'} → {ℂ : Category ℓ ℓ'} → Functor ℂ (Sets ⇑ (Opposite ℂ))
 yoneda = {!!}
--- a/src/Cat/Cubical.agda
+++ b/src/Cat/Cubical.agda
@ -42,7 +42,7 @@ module _ {ℓ ℓ' : Level} (Ns : Set ℓ) where
    Mor = Σ themap rules

  -- The category of names and substitutions
-  ℂ : Category -- {ℓo} {lsuc lzero ⊔ ℓo}
+  ℂ : Category ℓ ℓ -- ℓo (lsuc lzero ⊔ ℓo)
  ℂ = record
    -- { Object = FiniteDecidableSubset
    { Object = Ns → Bool
--- a/src/Cat/Functor.agda
+++ b/src/Cat/Functor.agda
@ -6,7 +6,7 @@ open import Function

 open import Cat.Category

-record Functor {ℓc ℓc' ℓd ℓd'} (C : Category {ℓc} {ℓc'}) (D : Category {ℓd} {ℓd'})
+record Functor {ℓc ℓc' ℓd ℓd'} (C : Category ℓc ℓc') (D : Category ℓd ℓd')
  : Set (ℓc ⊔ ℓc' ⊔ ℓd ⊔ ℓd') where
  private
    open module C = Category C
@ -21,7 +21,7 @@ record Functor {ℓc ℓc' ℓd ℓd'} (C : Category {ℓc} {ℓc'}) (D : Catego
    distrib : { c c' c'' : C.Object} {a : C.Arrow c c'} {a' : C.Arrow c' c''}
      → func→ (a' C.⊕ a) ≡ func→ a' D.⊕ func→ a

-module _ {ℓ ℓ' : Level} {A B C : Category {ℓ} {ℓ'}} (F : Functor B C) (G : Functor A B) where
+module _ {ℓ ℓ' : Level} {A B C : Category ℓ ℓ'} (F : Functor B C) (G : Functor A B) where
  private
    open module F = Functor F
    open module G = Functor G
@ -56,7 +56,7 @@ module _ {ℓ ℓ' : Level} {A B C : Category {ℓ} {ℓ'}} (F : Functor B C) (G
      }

 -- The identity functor
-identity : {ℓ ℓ' : Level} → {C : Category {ℓ} {ℓ'}} → Functor C C
+identity : ∀ {ℓ ℓ'} → {C : Category ℓ ℓ'} → Functor C C
 -- Identity = record { F* = λ x → x ; F→ = λ x → x ; ident = refl ; distrib = refl }
 identity = record
  { func* = λ x → x