cat/src/Cat/Category/Monad.agda

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{-# OPTIONS --cubical --allow-unsolved-metas #-}
module Cat.Category.Monad where
open import Agda.Primitive
open import Data.Product
open import Cubical
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open import Cubical.NType.Properties using (lemPropF ; lemSig)
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open import Cubical.GradLemma using (gradLemma)
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open import Cat.Category
open import Cat.Category.Functor as F
open import Cat.Category.NaturalTransformation
open import Cat.Categories.Fun
-- "A monad in the monoidal form" [voe]
module Monoidal {a b : Level} ( : Category a b) where
private
= a b
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open Category using (Object ; Arrow ; 𝟙 ; _∘_)
open NaturalTransformation
record RawMonad : Set where
field
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R : EndoFunctor
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pureNT : NaturalTransformation F.identity R
joinNT : NaturalTransformation F[ R R ] R
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-- Note that `pureT` and `joinT` differs from their definition in the
-- kleisli formulation only by having an explicit parameter.
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pureT : Transformation F.identity R
pureT = proj₁ pureNT
pureN : Natural F.identity R pureT
pureN = proj₂ pureNT
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joinT : Transformation F[ R R ] R
joinT = proj₁ joinNT
joinN : Natural F[ R R ] R joinT
joinN = proj₂ joinNT
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Romap = Functor.func* R
Rfmap = Functor.func→ R
bind : {X Y : Object} [ X , Romap Y ] [ Romap X , Romap Y ]
bind {X} {Y} f = joinT Y Rfmap f
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IsAssociative : Set _
IsAssociative = {X : Object}
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joinT X Rfmap (joinT X) joinT X joinT (Romap X)
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IsInverse : Set _
IsInverse = {X : Object}
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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
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private
module R = Functor R
module = Category
isNatural : IsNatural
isNatural {X} {Y} f = begin
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joinT Y R.func→ f pureT X ≡⟨ sym .isAssociative
joinT Y (R.func→ f pureT X) ≡⟨ cong (λ φ joinT Y φ) (sym (pureN f))
joinT Y (pureT (R.func* Y) f) ≡⟨ .isAssociative
joinT Y pureT (R.func* Y) f ≡⟨ cong (λ φ φ f) (proj₁ isInverse)
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𝟙 f ≡⟨ proj₂ .isIdentity
f
isDistributive : IsDistributive
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isDistributive {X} {Y} {Z} g f = sym aux
where
module R² = Functor F[ R R ]
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distrib3 : {A B C D} {a : Arrow C D} {b : Arrow B C} {c : Arrow A B}
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R.func→ (a b c)
R.func→ a R.func→ b R.func→ c
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distrib3 {a = a} {b} {c} = begin
R.func→ (a b c) ≡⟨ R.isDistributive
R.func→ (a b) R.func→ c ≡⟨ cong (_∘ _) R.isDistributive
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R.func→ a R.func→ b R.func→ c
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aux = begin
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joinT Z R.func→ (joinT Z R.func→ g f)
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≡⟨ cong (λ φ joinT Z φ) distrib3
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joinT Z (R.func→ (joinT Z) R.func→ (R.func→ g) R.func→ f)
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≡⟨⟩
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joinT Z (R.func→ (joinT Z) R².func→ g R.func→ f)
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≡⟨ cong (_∘_ (joinT Z)) (sym .isAssociative)
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joinT Z (R.func→ (joinT Z) (R².func→ g R.func→ f))
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≡⟨ .isAssociative
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(joinT Z R.func→ (joinT Z)) (R².func→ g R.func→ f)
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≡⟨ cong (λ φ φ (R².func→ g R.func→ f)) isAssociative
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(joinT Z joinT (R.func* Z)) (R².func→ g R.func→ f)
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≡⟨ .isAssociative
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joinT Z joinT (R.func* Z) R².func→ g R.func→ f
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≡⟨⟩
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((joinT Z joinT (R.func* Z)) R².func→ g) R.func→ f
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≡⟨ cong (_∘ R.func→ f) (sym .isAssociative)
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(joinT Z (joinT (R.func* Z) R².func→ g)) R.func→ f
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≡⟨ cong (λ φ φ R.func→ f) (cong (_∘_ (joinT Z)) (joinN g))
(joinT Z (R.func→ g joinT Y)) R.func→ f
≡⟨ cong (_∘ R.func→ f) .isAssociative
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joinT Z R.func→ g joinT Y R.func→ f
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≡⟨ sym (Category.isAssociative )
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joinT Z R.func→ g (joinT Y R.func→ f)
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record Monad : Set where
field
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raw : RawMonad
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isMonad : IsMonad raw
open IsMonad isMonad public
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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)
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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
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private
eqIsMonad : (λ i IsMonad (eq i)) [ Monad.isMonad m Monad.isMonad n ]
eqIsMonad = lemPropF propIsMonad eq
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Monad≡ : m n
Monad.raw (Monad≡ i) = eq i
Monad.isMonad (Monad≡ i) = eqIsMonad i
-- "A monad in the Kleisli form" [voe]
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module Kleisli {a b : Level} ( : Category a b) where
private
= a b
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module = Category
open using (Arrow ; 𝟙 ; Object ; _∘_ ; _>>>_)
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-- | Data for a monad.
--
-- Note that (>>=) is not expressible in a general category because objects
-- are not generally types.
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record RawMonad : Set where
field
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omap : Object Object
pure : {X : Object} [ X , omap X ]
bind : {X Y : Object} [ X , omap Y ] [ omap X , omap Y ]
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-- | functor map
--
-- This should perhaps be defined in a "Klesli-version" of functors as well?
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fmap : {A B} [ A , B ] [ omap A , omap B ]
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fmap f = bind (pure f)
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-- | Composition of monads aka. the kleisli-arrow.
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_>=>_ : {A B C : Object} [ A , omap B ] [ B , omap C ] [ A , omap C ]
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f >=> g = f >>> (bind g)
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-- | Flattening nested monads.
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join : {A : Object} [ omap (omap A) , omap A ]
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join = bind 𝟙
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------------------
-- * Monad laws --
------------------
-- There may be better names than what I've chosen here.
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IsIdentity = {X : Object}
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bind pure 𝟙 {omap X}
IsNatural = {X Y : Object} (f : [ X , omap Y ])
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pure >>> (bind f) f
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IsDistributive = {X Y Z : Object} (g : [ Y , omap Z ]) (f : [ X , omap Y ])
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(bind f) >>> (bind g) bind (f >=> g)
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-- | Functor map fusion.
--
-- This is really a functor law. Should we have a kleisli-representation of
-- functors as well and make them a super-class?
Fusion = {X Y Z : Object} {g : [ Y , Z ]} {f : [ X , Y ]}
fmap (g f) fmap g fmap f
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-- In the ("foreign") formulation of a monad `IsNatural`'s analogue here would be:
IsNaturalForeign : Set _
IsNaturalForeign = {X : Object} join {X} fmap join join join
IsInverse : Set _
IsInverse = {X : Object} join {X} pure 𝟙 × join {X} fmap pure 𝟙
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record IsMonad (raw : RawMonad) : Set where
open RawMonad raw public
field
isIdentity : IsIdentity
isNatural : IsNatural
isDistributive : IsDistributive
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-- | Map fusion is admissable.
fusion : Fusion
fusion {g = g} {f} = begin
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fmap (g f) ≡⟨⟩
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bind ((f >>> g) >>> pure) ≡⟨ cong bind .isAssociative
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bind (f >>> (g >>> pure)) ≡⟨ cong (λ φ bind (f >>> φ)) (sym (isNatural _))
bind (f >>> (pure >>> (bind (g >>> pure)))) ≡⟨⟩
bind (f >>> (pure >>> fmap g)) ≡⟨⟩
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bind ((fmap g pure) f) ≡⟨ cong bind (sym .isAssociative)
bind (fmap g (pure f)) ≡⟨ sym distrib
bind (pure g) bind (pure f) ≡⟨⟩
fmap g fmap f
where
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distrib : fmap g fmap f bind (fmap g (pure f))
distrib = isDistributive (pure g) (pure f)
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-- | This formulation gives rise to the following endo-functor.
private
rawR : RawFunctor
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RawFunctor.func* rawR = omap
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RawFunctor.func→ rawR = fmap
isFunctorR : IsFunctor rawR
IsFunctor.isIdentity isFunctorR = begin
bind (pure 𝟙) ≡⟨ cong bind (proj₁ .isIdentity)
bind pure ≡⟨ isIdentity
𝟙
IsFunctor.isDistributive isFunctorR {f = f} {g} = begin
bind (pure (g f)) ≡⟨⟩
fmap (g f) ≡⟨ fusion
fmap g fmap f ≡⟨⟩
bind (pure g) bind (pure f)
-- TODO: Naming!
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R : EndoFunctor
Functor.raw R = rawR
Functor.isFunctor R = isFunctorR
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private
open NaturalTransformation
R⁰ : EndoFunctor
R⁰ = F.identity
: EndoFunctor
= F[ R R ]
module R = Functor R
module R = Functor R⁰
module R² = Functor
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pureT : Transformation R⁰ R
pureT A = pure
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pureN : Natural R⁰ R pureT
pureN {A} {B} f = begin
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pureT B R⁰.func→ f ≡⟨⟩
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pure f ≡⟨ sym (isNatural _)
bind (pure f) pure ≡⟨⟩
fmap f pure ≡⟨⟩
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R.func→ f pureT A
joinT : Transformation R
joinT C = join
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joinN : Natural R joinT
joinN f = begin
join R².func→ f ≡⟨⟩
bind 𝟙 R².func→ f ≡⟨⟩
R².func→ f >>> bind 𝟙 ≡⟨⟩
fmap (fmap f) >>> bind 𝟙 ≡⟨⟩
fmap (bind (f >>> pure)) >>> bind 𝟙 ≡⟨⟩
bind (bind (f >>> pure) >>> pure) >>> bind 𝟙
≡⟨ isDistributive _ _
bind ((bind (f >>> pure) >>> pure) >=> 𝟙)
≡⟨⟩
bind ((bind (f >>> pure) >>> pure) >>> bind 𝟙)
≡⟨ cong bind .isAssociative
bind (bind (f >>> pure) >>> (pure >>> bind 𝟙))
≡⟨ cong (λ φ bind (bind (f >>> pure) >>> φ)) (isNatural _)
bind (bind (f >>> pure) >>> 𝟙)
≡⟨ cong bind (proj₂ .isIdentity)
bind (bind (f >>> pure))
≡⟨ cong bind (sym (proj₁ .isIdentity))
bind (𝟙 >>> bind (f >>> pure)) ≡⟨⟩
bind (𝟙 >=> (f >>> pure))
≡⟨ sym (isDistributive _ _)
bind 𝟙 >>> bind (f >>> pure) ≡⟨⟩
bind 𝟙 >>> fmap f ≡⟨⟩
bind 𝟙 >>> R.func→ f ≡⟨⟩
R.func→ f bind 𝟙 ≡⟨⟩
R.func→ f join
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pureNT : NaturalTransformation R⁰ R
proj₁ pureNT = pureT
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proj₂ pureNT = pureN
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joinNT : NaturalTransformation R
proj₁ joinNT = joinT
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proj₂ joinNT = joinN
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isNaturalForeign : IsNaturalForeign
isNaturalForeign = begin
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fmap join >>> join ≡⟨⟩
bind (join >>> pure) >>> bind 𝟙
≡⟨ isDistributive _ _
bind ((join >>> pure) >>> bind 𝟙)
≡⟨ cong bind .isAssociative
bind (join >>> (pure >>> bind 𝟙))
≡⟨ cong (λ φ bind (join >>> φ)) (isNatural _)
bind (join >>> 𝟙)
≡⟨ cong bind (proj₂ .isIdentity)
bind join ≡⟨⟩
bind (bind 𝟙)
≡⟨ cong bind (sym (proj₁ .isIdentity))
bind (𝟙 >>> bind 𝟙) ≡⟨⟩
bind (𝟙 >=> 𝟙) ≡⟨ sym (isDistributive _ _)
bind 𝟙 >>> bind 𝟙 ≡⟨⟩
join >>> join
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isInverse : IsInverse
isInverse = inv-l , inv-r
where
inv-l = begin
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pure >>> join ≡⟨⟩
pure >>> bind 𝟙 ≡⟨ isNatural _
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𝟙
inv-r = begin
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fmap pure >>> join ≡⟨⟩
bind (pure >>> pure) >>> bind 𝟙
≡⟨ isDistributive _ _
bind ((pure >>> pure) >=> 𝟙) ≡⟨⟩
bind ((pure >>> pure) >>> bind 𝟙)
≡⟨ cong bind .isAssociative
bind (pure >>> (pure >>> bind 𝟙))
≡⟨ cong (λ φ bind (pure >>> φ)) (isNatural _)
bind (pure >>> 𝟙)
≡⟨ cong bind (proj₂ .isIdentity)
bind pure ≡⟨ isIdentity
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𝟙
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record Monad : Set where
field
raw : RawMonad
isMonad : IsMonad raw
open IsMonad isMonad public
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private
module _ (raw : RawMonad) where
open RawMonad raw
propIsIdentity : isProp IsIdentity
propIsIdentity x y i = .arrowsAreSets _ _ x y i
propIsNatural : isProp IsNatural
propIsNatural x y i = λ f
.arrowsAreSets _ _ (x f) (y f) i
propIsDistributive : isProp IsDistributive
propIsDistributive x y i = λ g f
.arrowsAreSets _ _ (x g f) (y g f) i
open IsMonad
propIsMonad : (raw : _) isProp (IsMonad raw)
IsMonad.isIdentity (propIsMonad raw x y i)
= propIsIdentity raw (isIdentity x) (isIdentity y) i
IsMonad.isNatural (propIsMonad raw x y i)
= propIsNatural raw (isNatural x) (isNatural y) i
IsMonad.isDistributive (propIsMonad raw x y i)
= propIsDistributive raw (isDistributive x) (isDistributive y) i
module _ {m n : Monad} (eq : Monad.raw m Monad.raw n) where
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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
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-- | The monoidal- and kleisli presentation of monads are equivalent.
--
-- This is problem 2.3 in [voe].
module _ {a b : Level} { : Category a b} where
private
module = Category
open using (Object ; Arrow ; 𝟙 ; _∘_ ; _>>>_)
open Functor using (func* ; func→)
module M = Monoidal
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module K = Kleisli
module _ (m : M.RawMonad) where
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open M.RawMonad m
forthRaw : K.RawMonad
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K.RawMonad.omap forthRaw = Romap
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K.RawMonad.pure forthRaw = pureT _
K.RawMonad.bind forthRaw = bind
module _ {raw : M.RawMonad} (m : M.IsMonad raw) where
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private
module MI = M.IsMonad m
forthIsMonad : K.IsMonad (forthRaw raw)
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K.IsMonad.isIdentity forthIsMonad = proj₂ MI.isInverse
K.IsMonad.isNatural forthIsMonad = MI.isNatural
K.IsMonad.isDistributive forthIsMonad = MI.isDistributive
forth : M.Monad K.Monad
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Kleisli.Monad.raw (forth m) = forthRaw (M.Monad.raw m)
Kleisli.Monad.isMonad (forth m) = forthIsMonad (M.Monad.isMonad m)
module _ (m : K.Monad) where
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open K.Monad m
backRaw : M.RawMonad
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M.RawMonad.R backRaw = R
M.RawMonad.pureNT backRaw = pureNT
M.RawMonad.joinNT backRaw = joinNT
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private
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open M.RawMonad backRaw
module R = Functor (M.RawMonad.R backRaw)
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backIsMonad : M.IsMonad backRaw
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M.IsMonad.isAssociative backIsMonad {X} = begin
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joinT X R.func→ (joinT X) ≡⟨⟩
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join fmap (joinT X) ≡⟨⟩
join fmap join ≡⟨ isNaturalForeign
join join ≡⟨⟩
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joinT X joinT (R.func* X)
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M.IsMonad.isInverse backIsMonad {X} = inv-l , inv-r
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where
inv-l = begin
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joinT X pureT (R.func* X) ≡⟨⟩
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join pure ≡⟨ proj₁ isInverse
𝟙
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inv-r = begin
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joinT X R.func→ (pureT X) ≡⟨⟩
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join fmap pure ≡⟨ proj₂ isInverse
𝟙
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back : K.Monad M.Monad
Monoidal.Monad.raw (back m) = backRaw m
Monoidal.Monad.isMonad (back m) = backIsMonad m
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module _ (m : K.Monad) where
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private
open K.Monad m
bindEq : {X Y}
K.RawMonad.bind (forthRaw (backRaw m)) {X} {Y}
K.RawMonad.bind (K.Monad.raw m)
bindEq {X} {Y} = begin
K.RawMonad.bind (forthRaw (backRaw m)) ≡⟨⟩
(λ f join fmap f) ≡⟨⟩
(λ f bind (f >>> pure) >>> bind 𝟙) ≡⟨ funExt lem
(λ f bind f) ≡⟨⟩
bind
where
lem : (f : Arrow X (omap Y))
bind (f >>> pure) >>> bind 𝟙
bind f
lem f = begin
bind (f >>> pure) >>> bind 𝟙
≡⟨ isDistributive _ _
bind ((f >>> pure) >>> bind 𝟙)
≡⟨ cong bind .isAssociative
bind (f >>> (pure >>> bind 𝟙))
≡⟨ cong (λ φ bind (f >>> φ)) (isNatural _)
bind (f >>> 𝟙)
≡⟨ cong bind (proj₂ .isIdentity)
bind f
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forthRawEq : forthRaw (backRaw m) K.Monad.raw m
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K.RawMonad.omap (forthRawEq _) = omap
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K.RawMonad.pure (forthRawEq _) = pure
K.RawMonad.bind (forthRawEq i) = bindEq i
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fortheq : (m : K.Monad) forth (back m) m
fortheq m = K.Monad≡ (forthRawEq m)
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module _ (m : M.Monad) where
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private
open M.Monad m
module KM = K.Monad (forth m)
module R = Functor R
omapEq : KM.omap Romap
omapEq = refl
bindEq : {X Y} {f : Arrow X (Romap Y)} KM.bind f bind f
bindEq {X} {Y} {f} = begin
KM.bind f ≡⟨⟩
joinT Y Rfmap f ≡⟨⟩
bind f
joinEq : {X} KM.join joinT X
joinEq {X} = begin
KM.join ≡⟨⟩
KM.bind 𝟙 ≡⟨⟩
bind 𝟙 ≡⟨⟩
joinT X Rfmap 𝟙 ≡⟨ cong (λ φ _ φ) R.isIdentity
joinT X 𝟙 ≡⟨ proj₁ .isIdentity
joinT X
fmapEq : {A B} KM.fmap {A} {B} Rfmap
fmapEq {A} {B} = funExt (λ f begin
KM.fmap f ≡⟨⟩
KM.bind (f >>> KM.pure) ≡⟨⟩
bind (f >>> pureT _) ≡⟨⟩
Rfmap (f >>> pureT B) >>> joinT B ≡⟨⟩
Rfmap (f >>> pureT B) >>> joinT B ≡⟨ cong (λ φ φ >>> joinT B) R.isDistributive
Rfmap f >>> Rfmap (pureT B) >>> joinT B ≡⟨ .isAssociative
joinT B Rfmap (pureT B) Rfmap f ≡⟨ cong (λ φ φ Rfmap f) (proj₂ isInverse)
𝟙 Rfmap f ≡⟨ proj₂ .isIdentity
Rfmap f
)
rawEq : Functor.raw KM.R Functor.raw R
RawFunctor.func* (rawEq i) = omapEq i
RawFunctor.func→ (rawEq i) = fmapEq i
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Req : M.RawMonad.R (backRaw (forth m)) R
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Req = Functor≡ rawEq
open NaturalTransformation
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postulate
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pureNTEq : (λ i NaturalTransformation F.identity (Req i))
[ M.RawMonad.pureNT (backRaw (forth m)) pureNT ]
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joinNTEq : (λ i NaturalTransformation F[ Req i Req i ] (Req i))
[ M.RawMonad.joinNT (backRaw (forth m)) joinNT ]
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backRawEq : backRaw (forth m) M.Monad.raw m
-- stuck
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M.RawMonad.R (backRawEq i) = Req i
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M.RawMonad.pureNT (backRawEq i) = pureNTEq i -- pureNTEq i
M.RawMonad.joinNT (backRawEq i) = joinNTEq i
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backeq : (m : M.Monad) back (forth m) m
backeq m = M.Monad≡ (backRawEq m)
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eqv : isEquiv M.Monad K.Monad forth
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eqv = gradLemma forth back fortheq backeq
Monoidal≃Kleisli : M.Monad K.Monad
Monoidal≃Kleisli = forth , eqv