cat/src/Cat/Category/Monad.agda

{-# OPTIONS --cubical #-}
module Cat.Category.Monad where

open import Agda.Primitive

open import Data.Product

open import Cubical

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" [vlad]
module Monoidal {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
  private
    ℓ = ℓa ⊔ ℓb

  open Category ℂ hiding (IsAssociative)
  open NaturalTransformation ℂ ℂ
  record RawMonad : Set ℓ where
    field
      R : Functor ℂ ℂ
      -- pure
      ηNat : NaturalTransformation F.identity R
      -- (>=>)
      μNat : NaturalTransformation F[ R ∘ R ] R


    private
      module R  = Functor R
      module RR = Functor F[ R ∘ R ]
      module _ {X : Object} where
        -- module IdRX = Functor (F.identity {C = RX})

        η : Transformation F.identity R
        η = proj₁ ηNat
        ηX  : ℂ [ X                    , R.func* X ]
        ηX = η X
        RηX : ℂ [ R.func* X            , R.func* (R.func* X) ] -- ℂ [ R.func* X , {!R.func* (R.func* X))!} ]
        RηX = R.func→ ηX
        ηRX = η (R.func* X)
        IdRX : Arrow (R.func* X) (R.func* X)
        IdRX = 𝟙 {R.func* X}

        μ : Transformation F[ R ∘ R ] R
        μ = proj₁ μNat
        μX  : ℂ [ RR.func* X           , R.func* X ]
        μX = μ X
        RμX : ℂ [ R.func* (RR.func* X) , RR.func* X ]
        RμX = R.func→ μX
        μRX : ℂ [ RR.func* (R.func* X) , R.func* (R.func* X) ]
        μRX = μ (R.func* X)

        IsAssociative' : Set _
        IsAssociative' = ℂ [ μX ∘ RμX ] ≡ ℂ [ μX ∘ μRX ]
        IsInverse' : Set _
        IsInverse'
          = ℂ [ μX ∘ ηRX ] ≡ IdRX
          × ℂ [ μX ∘ RηX ] ≡ IdRX

    -- We don't want the objects to be indexes of the type, but rather just
    -- universally quantify over *all* objects of the category.
    IsAssociative = {X : Object} → IsAssociative' {X}
    IsInverse = {X : Object} → IsInverse' {X}

  record IsMonad (raw : RawMonad) : Set ℓ where
    open RawMonad raw public
    field
      isAssociative : IsAssociative
      isInverse : IsInverse

  record Monad : Set ℓ where
    field
      raw : RawMonad
      isMonad : IsMonad raw
    open IsMonad isMonad public

-- "A monad in the Kleisli form" [vlad]
module Kleisli {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
  private
    ℓ = ℓa ⊔ ℓb

  open Category ℂ hiding (IsIdentity)
  record RawMonad : Set ℓ where
    field
      RR : Object → Object
      η : {X : Object} → ℂ [ X , RR X ]
      rr : {X Y : Object} → ℂ [ X , RR Y ] → ℂ [ RR X , RR Y ]
    -- Name suggestions are welcome!
    IsIdentity     = {X : Object}
      → rr η ≡ 𝟙 {RR X}
    IsNatural      = {X Y : Object}   (f : ℂ [ X , RR Y ])
      → (ℂ [ rr f ∘ η ]) ≡ f
    IsDistributive = {X Y Z : Object} (g : ℂ [ Y , RR Z ]) (f : ℂ [ X , RR Y ])
      → ℂ [ rr g ∘ rr f ] ≡ rr (ℂ [ rr g ∘ f ])

  record IsMonad (raw : RawMonad) : Set ℓ where
    open RawMonad raw public
    field
      isIdentity     : IsIdentity
      isNatural      : IsNatural
      isDistributive : IsDistributive

  record Monad : Set ℓ where
    field
      raw : RawMonad
      isMonad : IsMonad raw
    open IsMonad isMonad public