open import Cat.Functor.Adjoint
open import Cat.Functor.Base
open import Cat.Prelude

open Functor
open _=>_

module Cat.Diagram.Monad {o h : _} (C : Precategory o h) where

import Cat.Reasoning C as C

Monads🔗

A monad on a category $\ca{C}$ is one way of categorifying the concept of monoid. Specifically, rather than living in a monoidal category, a monad lives in a bicategory. Here, we concern ourselves with the case of monads in the bicategory of categories, so that we may say: A monad is an endofunctor $M$, equipped with a unit natural transformation $\id{Id} \To M$, and a multiplication $(M \circ M) \To M$.

record Monad : Type (o ⊔ h) where
  no-eta-equality
  field
    M    : Functor C C
    unit : Id => M
    mult : (M F∘ M) => M

  module unit = _=>_ unit
  module mult = _=>_ mult

  M₀ = F₀ M
  M₁ = F₁ M
  M-id = F-id M
  M-∘ = F-∘ M

Furthermore, these natural transformations must satisfy identity and associativity laws exactly analogous to those of a monoid.

  field
    left-ident  : ∀ {x} → mult.η x C.∘ M₁ (unit.η x) ≡ C.id
    right-ident : ∀ {x} → mult.η x C.∘ unit.η (M₀ x) ≡ C.id

  field
    mult-assoc : ∀ {x} → mult.η x C.∘ M₁ (mult.η x) ≡ mult.η x C.∘ mult.η (M₀ x)

Algebras over a monad🔗

One way of interpreting a monad $M$ is as giving a signature for an algebraic theory. For instance, the free monoid monad describes the signature for the theory of monoids, and the free group monad describes the theory of groups.

Under this light, an algebra over a monad is a way of evaluating the abstract operations given by a monadic expression to a concrete value. Formally, an algebra for $M$ is given by a choice of object $A$ and a morphism $\nu : M(A) \to A$.

record Algebra-on (M : Monad) (ob : C.Ob) : Type (o ⊔ h) where
  no-eta-equality
  open Monad M

  field
    ν : C.Hom (M₀ ob) ob

This morphism must satisfy equations categorifying those which define a monoid action. If we think of $M$ as specifying a signature of effects, then v-unit says that the unit has no effects, and v-mult says that, given two layers $M(M(A))$, it doesn’t matter whether you first join then evaluate, or evaluate twice.

    ν-unit : ν C.∘ unit.η ob ≡ C.id
    ν-mult : ν C.∘ M₁ ν ≡ ν C.∘ mult.η ob

Algebra : Monad → Type (o ⊔ h)
Algebra M = Σ (Algebra-on M)

Eilenberg-Moore Category🔗

If we take a monad $M$ as the signature of an (algebraic) theory, and $M$-algebras as giving models of that theory, then we can ask (like with everything in category theory): Are there maps between interpretations? The answer (as always!) is yes: An algebra homomorphism is a map of the underlying objects which “commutes with the algebras”.

record Algebra-hom (M : Monad) (X Y : Algebra M) : Type (o ⊔ h) where
  no-eta-equality
  private
    module X = Algebra-on (X .snd)
    module Y = Algebra-on (Y .snd)

  open Monad M

  field
    morphism : C.Hom (X .fst) (Y .fst)
    commutes : morphism C.∘ X.ν ≡ Y.ν C.∘ M₁ morphism

open Algebra-hom

We can be more specific about “commuting with the algebras” by drawing a square: A map $m : X \to Y$ in the ambient category is a homomorphism of $M$-algebras when the square below commutes.

Since commutes is an identification between morphisms, it inhabits a proposition (because Hom-sets are sets), equality of algebra homomorphisms only depends on an equality of their underlying morphisms.

Algebra-hom-path
  : {M : Monad} {X Y : Algebra M} {F G : Algebra-hom M X Y}
  → morphism F ≡ morphism G
  → F ≡ G
Algebra-hom-path x i .morphism = x i
Algebra-hom-path {M = M} {X} {Y} {F} {G} x i .commutes =
  is-prop→pathp (λ i → C.Hom-set _ _ (x i C.∘ X .snd .Algebra-on.ν)
                                     (Y .snd .Algebra-on.ν C.∘ Monad.M₁ M (x i)))
    (F .commutes) (G .commutes) i

Since the square we drew above commutes for the identity morphism, and we can show that the composite of two algebra homomorphisms is an algebra homomorphism, they assemble into a category: The Eilenberg-Moore category of $M$.

module _ (M : Monad) where
  private
    module M = Monad M
  open M hiding (M)
  open Precategory
  open Algebra-on

  Eilenberg-Moore : Precategory _ _
  Eilenberg-Moore .Ob = Algebra M
  Eilenberg-Moore .Hom X Y = Algebra-hom M X Y

Defining the identity and composition maps is mostly an exercise in categorical yoga:

  Eilenberg-Moore .id {o , x} = record
    { morphism = C.id
    ; commutes = C.id C.∘ ν x     ≡⟨ C.id-comm-sym ⟩≡
                 ν x C.∘ C.id     ≡⟨ ap (C._∘_ _) (sym M-id) ⟩≡
                 ν x C.∘ M₁ C.id  ∎
    }

  Eilenberg-Moore ._∘_ {_ , x} {_ , y} {_ , z} F G = record
    { morphism = morphism F C.∘ morphism G
    ; commutes = (morphism F C.∘ morphism G) C.∘ ν x            ≡⟨ C.extendr (commutes G) ⟩≡
                  (morphism F C.∘ ν y) C.∘ M₁ (morphism G)       ≡⟨ ap₂ C._∘_ (commutes F) refl ⟩≡
                  (ν z C.∘ M₁ (morphism F)) C.∘ M₁ (morphism G)  ≡⟨ C.pullr (sym (M-∘ _ _)) ⟩≡
                  ν z C.∘ M₁ (morphism F C.∘ morphism G)         ∎
    }

  Eilenberg-Moore .idr f = Algebra-hom-path (C.idr (morphism f))
  Eilenberg-Moore .idl f = Algebra-hom-path (C.idl (morphism f))
  Eilenberg-Moore .assoc f g h = Algebra-hom-path (C.assoc _ _ _)
  Eilenberg-Moore .Hom-set X Y = is-hlevel≃  (Iso→Equiv eqv e⁻¹) (hlevel )
    where open C.HLevel-instance

By projecting the underlying object of the algebras, and the underlying morphisms of the homomorphisms between them, we can define a functor from Eilenberg-Moore back to the underlying category:

  Forget : Functor Eilenberg-Moore C
  Forget .F₀ = fst
  Forget .F₁ = Algebra-hom.morphism
  Forget .F-id = refl
  Forget .F-∘ f g = refl

The lemma Algebra-hom-path says exactly that this functor is faithful.

  Forget-is-faithful : is-faithful Forget
  Forget-is-faithful = Algebra-hom-path

Free Algebras🔗

In exactly the same way that we may construct a free group by taking the inhabitants of some set $X$ as generating the “words” of a group, we can, given an object $A$ of the underlying category, build a free $M$-algebra on $A$. Keeping with our interpretation of monads as logical signatures, this is the syntactic model of $M$, with a set of “neutrals” chosen from the object $A$.

This construction is a lot simpler to do in generality than in any specific case: We can always turn $A$ into an $M$-algebra by taking the underlying object to be $M(A)$, and the algebra map to be the monadic multiplication; The associativity and unit laws of the monad itself become those of the $M$-action.

  Free : Functor C Eilenberg-Moore
  Free .F₀ A = M₀ A ,
    record
      { ν      = mult.η A
      ; ν-mult = mult-assoc
      ; ν-unit = right-ident
      }

The construction of free $M$-algebras is furthermore functorial on the underlying objects; Since the monadic multiplication is a natural transformation $M\circ M \To M$, the naturality condition (drawn below) doubles as showing that the functorial action of $M$ can be taken as an algebraic action:

  Free .F₁ f = record
    { morphism = M₁ f
    ; commutes = sym (mult.is-natural _ _ _)
    }
  Free .F-id = Algebra-hom-path M-id
  Free .F-∘ f g = Algebra-hom-path (M-∘ f g)

This is a free construction in the precise sense of the word: it’s the left adjoint to the functor Forget, so in particular it provides a systematic, universal way of mapping from $\ca{C}$ to $\ca{C}^M$.

  open _⊣_

  Free⊣Forget : Free ⊣ Forget
  Free⊣Forget .unit = NT M.unit.η M.unit.is-natural
  Free⊣Forget .counit .η x =
    record { morphism = x .snd .ν
           ; commutes = sym (x .snd .ν-mult)
           }
  Free⊣Forget .counit .is-natural x y f =
    Algebra-hom-path (sym (commutes f))
  Free⊣Forget .zig = Algebra-hom-path left-ident
  Free⊣Forget .zag {x} = x .snd .ν-unit