cat/src/Cat/Category/Monad/Monoidal.agda

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{---
Monoidal formulation of monads
---}
{-# OPTIONS --cubical --allow-unsolved-metas #-}
open import Agda.Primitive
open import Data.Product
open import Cubical
open import Cubical.NType.Properties using (lemPropF ; lemSig ; lemSigP)
open import Cubical.GradLemma using (gradLemma)
open import Cat.Category
open import Cat.Category.Functor as F
open import Cat.Category.NaturalTransformation
open import Cat.Categories.Fun
module Cat.Category.Monad.Monoidal {a b : Level} ( : Category a b) where
-- "A monad in the monoidal form" [voe]
private
= a b
open Category using (Object ; Arrow ; 𝟙 ; _∘_)
open NaturalTransformation
record RawMonad : Set where
field
R : EndoFunctor
pureNT : NaturalTransformation F.identity R
joinNT : NaturalTransformation F[ R R ] R
-- Note that `pureT` and `joinT` differs from their definition in the
-- kleisli formulation only by having an explicit parameter.
pureT : Transformation F.identity R
pureT = proj₁ pureNT
pureN : Natural F.identity R pureT
pureN = proj₂ pureNT
joinT : Transformation F[ R R ] R
joinT = proj₁ joinNT
joinN : Natural F[ R R ] R joinT
joinN = proj₂ joinNT
Romap = Functor.omap R
Rfmap = Functor.fmap R
bind : {X Y : Object} [ X , Romap Y ] [ Romap X , Romap Y ]
bind {X} {Y} f = joinT Y Rfmap f
IsAssociative : Set _
IsAssociative = {X : Object}
joinT X Rfmap (joinT X) joinT X joinT (Romap X)
IsInverse : Set _
IsInverse = {X : Object}
joinT X pureT (Romap X) 𝟙
× joinT X Rfmap (pureT X) 𝟙
IsNatural = {X Y} f joinT Y Rfmap f pureT X f
IsDistributive = {X Y Z} (g : Arrow Y (Romap Z)) (f : Arrow X (Romap Y))
joinT Z Rfmap g (joinT Y Rfmap f)
joinT Z Rfmap (joinT Z Rfmap g f)
record IsMonad (raw : RawMonad) : Set where
open RawMonad raw public
field
isAssociative : IsAssociative
isInverse : IsInverse
private
module R = Functor R
module = Category
isNatural : IsNatural
isNatural {X} {Y} f = begin
joinT Y R.fmap f pureT X ≡⟨ sym .isAssociative
joinT Y (R.fmap f pureT X) ≡⟨ cong (λ φ joinT Y φ) (sym (pureN f))
joinT Y (pureT (R.omap Y) f) ≡⟨ .isAssociative
joinT Y pureT (R.omap Y) f ≡⟨ cong (λ φ φ f) (proj₁ isInverse)
𝟙 f ≡⟨ proj₂ .isIdentity
f
isDistributive : IsDistributive
isDistributive {X} {Y} {Z} g f = sym aux
where
module R² = Functor F[ R R ]
distrib3 : {A B C D} {a : Arrow C D} {b : Arrow B C} {c : Arrow A B}
R.fmap (a b c)
R.fmap a R.fmap b R.fmap c
distrib3 {a = a} {b} {c} = begin
R.fmap (a b c) ≡⟨ R.isDistributive
R.fmap (a b) R.fmap c ≡⟨ cong (_∘ _) R.isDistributive
R.fmap a R.fmap b R.fmap c
aux = begin
joinT Z R.fmap (joinT Z R.fmap g f)
≡⟨ cong (λ φ joinT Z φ) distrib3
joinT Z (R.fmap (joinT Z) R.fmap (R.fmap g) R.fmap f)
≡⟨⟩
joinT Z (R.fmap (joinT Z) R².fmap g R.fmap f)
≡⟨ cong (_∘_ (joinT Z)) (sym .isAssociative)
joinT Z (R.fmap (joinT Z) (R².fmap g R.fmap f))
≡⟨ .isAssociative
(joinT Z R.fmap (joinT Z)) (R².fmap g R.fmap f)
≡⟨ cong (λ φ φ (R².fmap g R.fmap f)) isAssociative
(joinT Z joinT (R.omap Z)) (R².fmap g R.fmap f)
≡⟨ .isAssociative
joinT Z joinT (R.omap Z) R².fmap g R.fmap f
≡⟨⟩
((joinT Z joinT (R.omap Z)) R².fmap g) R.fmap f
≡⟨ cong (_∘ R.fmap f) (sym .isAssociative)
(joinT Z (joinT (R.omap Z) R².fmap g)) R.fmap f
≡⟨ cong (λ φ φ R.fmap f) (cong (_∘_ (joinT Z)) (joinN g))
(joinT Z (R.fmap g joinT Y)) R.fmap f
≡⟨ cong (_∘ R.fmap f) .isAssociative
joinT Z R.fmap g joinT Y R.fmap f
≡⟨ sym (Category.isAssociative )
joinT Z R.fmap g (joinT Y R.fmap f)
record Monad : Set where
field
raw : RawMonad
isMonad : IsMonad raw
open IsMonad isMonad public
private
module _ {m : RawMonad} where
open RawMonad m
propIsAssociative : isProp IsAssociative
propIsAssociative x y i {X}
= Category.arrowsAreSets _ _ (x {X}) (y {X}) i
propIsInverse : isProp IsInverse
propIsInverse x y i {X} = e1 i , e2 i
where
xX = x {X}
yX = y {X}
e1 = Category.arrowsAreSets _ _ (proj₁ xX) (proj₁ yX)
e2 = Category.arrowsAreSets _ _ (proj₂ xX) (proj₂ yX)
open IsMonad
propIsMonad : (raw : _) isProp (IsMonad raw)
IsMonad.isAssociative (propIsMonad raw a b i) j
= propIsAssociative {raw}
(isAssociative a) (isAssociative b) i j
IsMonad.isInverse (propIsMonad raw a b i)
= propIsInverse {raw}
(isInverse a) (isInverse b) i
module _ {m n : Monad} (eq : Monad.raw m Monad.raw n) where
private
eqIsMonad : (λ i IsMonad (eq i)) [ Monad.isMonad m Monad.isMonad n ]
eqIsMonad = lemPropF propIsMonad eq
Monad≡ : m n
Monad.raw (Monad≡ i) = eq i
Monad.isMonad (Monad≡ i) = eqIsMonad i