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Tidy up groups (#64)

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Patrick Stevens
2019-11-03 17:12:48 +00:00
committed by GitHub
parent e4daab7153
commit d95f510cdd
42 changed files with 1438 additions and 1038 deletions
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19
Categories/Category.agda
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@@ -2,26 +2,19 @@
open import LogicalFormulae
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Numbers.Naturals
open import Numbers.Naturals.Naturals
open import Numbers.Naturals.Order
open import Vectors
open import Semirings.Definition
open import Categories.Definition
module Categories.Category where
postulate
extensionality : {a b : _} {S : Set a}{T : S → Set b} {f g : (x : S) → T x} → ((x : S) → f x ≡ g x) → f ≡ g
record Category {a b : _} : Set (lsuc (a ⊔ b)) where
field
objects : Set a
arrows : objects → objects → Set b
id : (x : objects) → arrows x x
_∘_ : {x y z : objects} → arrows y z → arrows x y → arrows x z
rightId : {x y : objects} → (f : arrows x y) → (id y) ∘ f ≡ f
leftId : {x y : objects} → (f : arrows x y) → f ∘ (id x) ≡ f
associative : {x y z w : objects} → (f : arrows z w) → (g : arrows y z) → (h : arrows x y) → (f ∘ (g ∘ h)) ≡ (f ∘ g) ∘ h
dual : {a b : _} → Category {a} {b} → Category {a} {b}
dual record { objects = objects ; arrows = arrows ; id = id ; _∘_ = _∘_ ; rightId = rightId ; leftId = leftId ; associative = associative } = record { objects = objects ; arrows = λ i j → arrows j i ; id = id ; _∘_ = λ {x y z} g f → f ∘ g ; rightId = λ {x y} f → leftId f ; leftId = λ {x y} f → rightId f ; associative = λ {x y z w} f g h → equalityCommutative (associative h g f) }
dual record { objects = objects ; arrows = arrows ; id = id ; _∘_ = _∘_ ; rightId = rightId ; leftId = leftId ; compositionAssociative = associative } = record { objects = objects ; arrows = λ i j → arrows j i ; id = id ; _∘_ = λ {x y z} g f → f ∘ g ; rightId = λ {x y} f → leftId f ; leftId = λ {x y} f → rightId f ; associative = λ {x y z w} f g h → equalityCommutative (associative h g f) }
SET : {a : _} → Category {lsuc a} {a}
SET {a} = record { objects = Set a ; arrows = λ i j → (i → j) ; id = λ X x → x ; _∘_ = λ g f x → g (f x) ; rightId = λ f → refl ; leftId = λ f → refl ; associative = λ f g h → refl }
@@ -55,7 +48,7 @@ NatPreorder = record { objects = ℕ ; arrows = λ m n → m ≤N n ; id = λ x
leqUnique (inr a=b1) (inr a=b2) rewrite a=b1 | a=b2 = refl
NatMonoid : Category {lzero} {lzero}
NatMonoid = record { objects = True ; arrows = λ _ _ → ℕ ; id = λ x → 0 ; _∘_ = λ f g → f +N g ; rightId = λ f → refl ; leftId = λ f → addZeroRight f ; associative = λ a b c → equalityCommutative (additionNIsAssociative a b c) }
NatMonoid = record { objects = True ; arrows = λ _ _ → ℕ ; id = λ x → 0 ; _∘_ = λ f g → f +N g ; rightId = λ f → refl ; leftId = λ f → Semiring.sumZeroRight ℕSemiring f ; associative = λ a b c → Semiring.+Associative ℕSemiring a b c }
DISCRETE : {a : _} (X : Set a) → Category {a} {a}
DISCRETE X = record { objects = X ; arrows = _≡_ ; id = λ x → refl ; _∘_ = λ f g → transitivity g f ; rightId = λ f → ≡Unique (transitivity f refl) f ; leftId = λ f → ≡Unique (transitivity refl f) f ; associative = λ f g h → ≡Unique (transitivity (transitivity h g) f) (transitivity h (transitivity g f)) }

20
Categories/Definition.agda Normal file
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{-# OPTIONS --warning=error --safe --without-K #-}
open import LogicalFormulae
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Numbers.Naturals.Naturals
open import Numbers.Naturals.Order
open import Vectors
open import Semirings.Definition
module Categories.Definition where
record Category {a b : _} : Set (lsuc (a ⊔ b)) where
field
objects : Set a
arrows : objects → objects → Set b
id : (x : objects) → arrows x x
_∘_ : {x y z : objects} → arrows y z → arrows x y → arrows x z
rightId : {x y : objects} → (f : arrows x y) → (id y) ∘ f ≡ f
leftId : {x y : objects} → (f : arrows x y) → f ∘ (id x) ≡ f
compositionAssociative : {x y z w : objects} → (f : arrows z w) → (g : arrows y z) → (h : arrows x y) → (f ∘ (g ∘ h)) ≡ (f ∘ g) ∘ h

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Categories/Examples.agda Normal file
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{-# OPTIONS --warning=error --safe --without-K #-}
open import LogicalFormulae
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Numbers.Naturals.Naturals
open import Numbers.Naturals.Order
open import Vectors
open import Semirings.Definition
open import Categories.Definition
open import Groups.Definition
module Categories.Examples where
SET : {a : _} → Category {lsuc a} {a}
Category.objects (SET {a}) = Set a
Category.arrows (SET {a}) = λ a b → (a → b)
Category.id (SET {a}) = λ x → λ y → y
Category._∘_ (SET {a}) = λ f g → λ x → f (g x)
Category.rightId (SET {a}) = λ f → refl
Category.leftId (SET {a}) = λ f → refl
Category.compositionAssociative (SET {a}) = λ f g h → refl
GROUP : {a : _} → Category {lsuc a} {a}
Category.objects (GROUP {a}) = Group {! !} {! !}
Category.arrows (GROUP {a}) = {! !}
Category.id (GROUP {a}) = {! !}
Category._∘_ (GROUP {a}) = {! !}
Category.rightId (GROUP {a}) = {! !}
Category.leftId (GROUP {a}) = {! !}
Category.compositionAssociative (GROUP {a}) = {! !}