open import Algebra.Group.Cat.Base
open import Algebra.Group.Free
open import Algebra.Prelude
open import Algebra.Monoid
open import Algebra.Group

open import Cat.Functor.Adjoint.Monadic
open import Cat.Functor.Adjoint.Monad
open import Cat.Functor.Equivalence
open import Cat.Functor.Properties
open import Cat.Diagram.Monad

import Algebra.Group.Cat.Base as Grp

import Cat.Reasoning

module Algebra.Group.Cat.Monadic {ℓ} where

private
  F : Functor (Sets ℓ) (Groups ℓ)
  F = free-objects→functor make-free-group

  F⊣U : F ⊣ _
  F⊣U = free-objects→left-adjoint make-free-group

  K = Comparison F⊣U

  T : Monad (Sets ℓ)
  T = Adjunction→Monad F⊣U
  module F = Functor F
  module T = Monad T
  module K = Functor K
  module Sets^T = Cat.Reasoning (Eilenberg-Moore (Sets ℓ) T)

Monadicity of the category of groups🔗

We prove that the category ${\mathbf{Groups}}_{\kappa }$ is monadic over ${\mathbf{Sets}}_{\kappa }\text{,}$ or more specifically that the free group adjunction $F⊣U$ is monadic. Rather than appealing to a monadicity theorem, we show this directly by calculation. This actually gives us a slightly sharper result, too: rather than showing that the comparison functor is an equivalence, we show directly that it is an isomorphism of categories. This doesn’t exactly matter since ${\mathbf{Sets}}_{\kappa }$ and ${\mathbf{Groups}}_{\kappa }$ are both univalent categories, but it’s interesting that it’s easier to construct an isomorphism than it is to construct an equivalence.

Let us abbreviate the monad induced by the free group adjunction by $T\text{.}$ What we must show is that any $T\text{-algebra}$ structure on a set $G$ gives rise to a group structure on $G\text{,}$ and that this process is reversible: If $\nu$ is our original algebra, applying our process then applying the comparison functor has to give $\nu$ back.

Algebra-on→group-on : {G : Set ℓ} → Algebra-on (Sets ℓ) T G → Group-on ∣ G ∣
Algebra-on→group-on {G = G} alg = grp where
  open Algebra-on alg

  mult : ∣ G ∣ → ∣ G ∣ → ∣ G ∣
  mult x y = ν (inc x ◆ inc y)

The thing to keep in mind is that a $T\text{-algebra}$ structure is a way of “folding” a word built up from generators in the set $G\text{,}$ so that if we build a word from two generators $x,y\text{,}$ folding that should be the same thing as a binary multiplication $xy\text{.}$ That’s the definition of the induced multiplication structure! We now must show that this definition is indeed a group structure, which is an incredibly boring calculation.

  abstract
    assoc : ∀ x y z → mult x (mult y z) ≡ mult (mult x y) z
    assoc x y z = sym $
      ν (inc (ν (inc x ◆ inc y)) ◆ inc z)                ≡⟨ (λ i → ν (inc (ν (inc x ◆ inc y)) ◆ inc (ν-unit (~ i) z))) ⟩≡
      ν (inc (ν (inc x ◆ inc y)) ◆ inc (ν (inc z)))      ≡⟨ happly ν-mult (inc _ ◆ inc _) ⟩≡
      ν (T.mult.η G (inc (inc x ◆ inc y) ◆ inc (inc z))) ≡˘⟨ ap ν (f-assoc _ _ _) ⟩≡˘
      ν (T.mult.η G (inc (inc x) ◆ inc (inc y ◆ inc z))) ≡˘⟨ happly ν-mult (inc _ ◆ inc _) ⟩≡˘
      ν (inc (ν (inc x)) ◆ inc (ν (inc y ◆ inc z)))      ≡⟨ (λ i → ν (inc (ν-unit i x) ◆ inc (ν (inc y ◆ inc z)))) ⟩≡
      ν (inc x ◆ inc (ν (inc y ◆ inc z)))                ∎

    invl : ∀ x → mult (ν (inv (inc x))) x ≡ ν nil
    invl x =
      ν (inc (ν (inv (inc x))) ◆ inc x)                ≡⟨ (λ i → ν (inc (ν (inv (inc x))) ◆ inc (ν-unit (~ i) x))) ⟩≡
      ν (inc (ν (inv (inc x))) ◆ inc (ν (inc x)))      ≡⟨ happly ν-mult (inc _ ◆ inc _) ⟩≡
      ν (T.mult.η G (inc (inv (inc x)) ◆ inc (inc x))) ≡⟨ ap ν (f-invl _) ⟩≡
      ν (T.mult.η G (inc nil))                         ≡⟨⟩
      ν nil                                            ∎

    idl' : ∀ x → mult (ν nil) x ≡ x
    idl' x =
      ν (inc (ν nil) ◆ inc x)            ≡⟨ (λ i → ν (inc (ν nil) ◆ inc (ν-unit (~ i) x))) ⟩≡
      ν (inc (ν nil) ◆ inc (ν (inc x)))  ≡⟨ happly ν-mult (inc _ ◆ inc _) ⟩≡
      ν (T.mult.η G (nil ◆ inc (inc x))) ≡⟨ ap ν (f-idl _) ⟩≡
      ν (inc x)                          ≡⟨ happly ν-unit x ⟩≡
      x                                  ∎

  grp : Group-on ∣ G ∣
  grp .Group-on._⋆_ = mult
  grp .Group-on.has-is-group = to-group-on fg .Group-on.has-is-group where
    fg : make-group ∣ G ∣
    fg .make-group.group-is-set = G .is-tr
    fg .make-group.unit = ν nil
    fg .make-group.mul = mult
    fg .make-group.inv x = ν (inv (inc x))
    fg .make-group.assoc = assoc
    fg .make-group.invl = invl
    fg .make-group.idl = idl'

We now show that this construction fits into defining an inverse (on the nose!) to the comparison functor. This is slightly easier than it sounds like: We must show that the functor is fully faithful, and that our mapping above is indeed invertible.

Fully faithfulness is almost immediate: a homomorphism of $T\text{-algebras}$ $K(G)\to K(H)$ preserves underlying multiplication because the algebra structure of $K(G)$ (resp. $K(H)\text{)}$ is defined in terms of that multiplication. Since we leave the underlying map intact, and being a homomorphism (either kind) is a property, the comparison functor is fully faithful.

Group-is-monadic : is-monadic F⊣U
Group-is-monadic = is-precat-iso→is-equivalence
  record { has-is-ff = ff ; has-is-iso = is-iso→is-equiv isom } where
  open Algebra-hom
  open Algebra-on

  k₁inv : ∀ {G H} → Algebra-hom (Sets ℓ) T (K.₀ G) (K.₀ H) → Groups.Hom G H
  k₁inv hom .hom = hom .morphism
  k₁inv hom .preserves .is-group-hom.pres-⋆ x y = happly (hom .commutes) (inc x ◆ inc y)

  ff : is-fully-faithful K
  ff = is-iso→is-equiv $ iso k₁inv (λ x → Algebra-hom-path (Sets ℓ) refl)
                                   (λ x → Grp.Grp↪Sets-is-faithful refl)

To show that the object mapping of the comparison functor is invertible, we appeal to univalence. Since the types are kept the same, it suffices to show that passing between a multiplication and an algebra map doesn’t lose any information. This is immediate in one direction, but the other direction is by induction on “words”.

  isom : is-iso K.₀
  isom .is-iso.inv (A , alg) = A , Algebra-on→group-on alg
  isom .is-iso.linv x = ∫-Path Groups-equational
    (total-hom (λ x → x) (record { pres-⋆ = λ x y → refl }))
    id-equiv
  isom .is-iso.rinv x = Σ-pathp refl (Algebra-on-pathp _ _ go) where
    alg = x .snd
    grp = Algebra-on→group-on alg
    rec = fold-free-group {G = x .fst , grp} (λ x → x)
    module G = Group-on grp

    alg-gh : is-group-hom (Free-Group ⌞ x ⌟ .snd) grp (x .snd .ν)
    alg-gh .is-group-hom.pres-⋆ x y = sym (happly (alg .ν-mult) (inc _ ◆ inc _))

    go : rec .hom ≡ x .snd .ν
    go = funext $ Free-elim-prop _ (λ _ → hlevel )
      (λ x → sym (happly (alg .ν-unit) x))
      (λ x p y q → rec .preserves .is-group-hom.pres-⋆ x y
                ·· ap₂ G._⋆_ p q
                ·· happly (alg .ν-mult) (inc _ ◆ inc _))
      (λ x p → is-group-hom.pres-inv (rec .preserves) {x = x}
              ·· ap G.inverse p
              ·· sym (is-group-hom.pres-inv alg-gh {x = x}))
      refl