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.Equivalence
open import Cat.Functor.Properties

import Algebra.Group.Cat.Base as Grp

import Cat.Reasoning

module Algebra.Group.Cat.Monadic {β} where

private
F : Functor (Sets β) (Groups β)
F = Free-groups.to-functor

Fβ£U : F β£ _

K = Comparison Fβ£U

module F = Functor F
module K = Functor K
module Sets^T = Cat.Reasoning (Eilenberg-Moore (Sets β) T)


# Monadicity of the category of groupsπ

We prove that the category $\mathbf{Groups}_\kappa$ is monadic over $\mathbf{Sets}_\kappa$, or more specifically that the free group adjunction $F \dashv U$ 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 doesnβt exactly matter since $\mathbf{Sets}_\kappa$ and $\mathbf{Groups}_\kappa$ are both univalent categories, but itβs interesting that itβ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 $T$. What we must show is that any $T$-algebra structure on a set $G$ gives rise to a group structure on $G$, 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-onβgroup-on : {G : Set β} β Algebra-on (Sets β) T G β Group-on β£ G β£
Algebra-onβgroup-on {G = G} alg = grp where
open Algebra-on alg

mult : β£ G β£ β β£ G β£ β β£ G β£
mult x y = Ξ½ (inc x β inc y)


The thing to keep in mind is that a $T$-algebra structure is a way of βfoldingβ a word built up from generators in the set $G$, so that if we build a word from two generators $x, y$, folding that should be the same thing as a binary multiplication $xy$. Thatβ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.

Iβm not exaggerating, itβs super boring.
  abstract
assoc : β x y z β mult x (mult y z) β‘ mult (mult x y) z
assoc x y z = sym $Ξ½ (inc (Ξ½ (inc x β inc y)) β inc z) β‘β¨ (Ξ» i β Ξ½ (inc (Ξ½ (inc x β inc y)) β inc (Ξ½-unit (~ i) z))) β©β‘ Ξ½ (inc (Ξ½ (inc x β inc y)) β inc (Ξ½ (inc z))) β‘β¨ happly Ξ½-mult (inc _ β inc _) β©β‘ Ξ½ (T.mult.Ξ· G (inc (inc x β inc y) β inc (inc z))) β‘Λβ¨ ap Ξ½ (f-assoc _ _ _) β©β‘Λ Ξ½ (T.mult.Ξ· G (inc (inc x) β inc (inc y β inc z))) β‘Λβ¨ happly Ξ½-mult (inc _ β inc _) β©β‘Λ Ξ½ (inc (Ξ½ (inc x)) β inc (Ξ½ (inc y β inc z))) β‘β¨ (Ξ» i β Ξ½ (inc (Ξ½-unit i x) β inc (Ξ½ (inc y β inc z)))) β©β‘ Ξ½ (inc x β inc (Ξ½ (inc y β inc z))) β invl : β x β mult (Ξ½ (inv (inc x))) x β‘ Ξ½ nil invl x = Ξ½ (inc (Ξ½ (inv (inc x))) β inc x) β‘β¨ (Ξ» i β Ξ½ (inc (Ξ½ (inv (inc x))) β inc (Ξ½-unit (~ i) x))) β©β‘ Ξ½ (inc (Ξ½ (inv (inc x))) β inc (Ξ½ (inc x))) β‘β¨ happly Ξ½-mult (inc _ β inc _) β©β‘ Ξ½ (T.mult.Ξ· G (inc (inv (inc x)) β inc (inc x))) β‘β¨ ap Ξ½ (f-invl _) β©β‘ Ξ½ (T.mult.Ξ· G (inc nil)) β‘β¨β© Ξ½ nil β idlβ² : β x β mult (Ξ½ nil) x β‘ x idlβ² x = Ξ½ (inc (Ξ½ nil) β inc x) β‘β¨ (Ξ» i β Ξ½ (inc (Ξ½ nil) β inc (Ξ½-unit (~ i) x))) β©β‘ Ξ½ (inc (Ξ½ nil) β inc (Ξ½ (inc x))) β‘β¨ happly Ξ½-mult (inc _ β inc _) β©β‘ Ξ½ (T.mult.Ξ· G (nil β inc (inc x))) β‘β¨ ap Ξ½ (f-idl _) β©β‘ Ξ½ (inc x) β‘β¨ happly Ξ½-unit x β©β‘ x β grp : Group-on β£ G β£ grp .Group-on._β_ = mult grp .Group-on.has-is-group = to-group-on fg .Group-on.has-is-group where fg : make-group β£ G β£ fg .make-group.group-is-set = G .is-tr fg .make-group.unit = Ξ½ nil fg .make-group.mul = mult fg .make-group.inv x = Ξ½ (inv (inc x)) fg .make-group.assoc = assoc fg .make-group.invl = invl fg .make-group.idl = idlβ²  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 $T$-algebras $K(G) \to K(H)$ preserves underlying multiplication because the algebra structure of $K(G)$ (resp. $K(H)$) 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 Fβ£U Group-is-monadic = is-precat-isoβis-equivalence record { has-is-ff = ff ; has-is-iso = is-isoβis-equiv isom } where open Algebra-hom open Algebra-on kβinv : β {G H} β Algebra-hom (Sets β) T (K.β G) (K.β H) β Groups.Hom G H kβinv hom .hom = hom .morphism kβinv hom .preserves .is-group-hom.pres-β x y = happly (hom .commutes) (inc x β inc y) ff : is-fully-faithful K ff = is-isoβis-equiv$ iso kβinv (Ξ» x β Algebra-hom-path (Sets β) refl)
(Ξ» x β Grp.Forget-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 doesnβt lose any information. This is immediate in one direction, but the other direction is by induction on βwordsβ.

  isom : is-iso K.β
isom .is-iso.inv (A , alg) = A , Algebra-onβgroup-on alg
isom .is-iso.linv x = β«-Path Groups-equational
(total-hom (Ξ» x β x) (record { pres-β = Ξ» x y β refl }))
id-equiv
isom .is-iso.rinv x = Ξ£-pathp refl (Algebra-on-pathp _ _ go) where
alg = x .snd
grp = Algebra-onβgroup-on alg
rec = fold-free-group {G = x .fst , grp} (Ξ» x β x)
module G = Group-on grp

alg-gh : is-group-hom (Free-Group β x β .snd) grp (x .snd .Ξ½)
alg-gh .is-group-hom.pres-β x y = sym (happly (alg .Ξ½-mult) (inc _ β inc _))

go : rec .hom β‘ x .snd .Ξ½
go = funext \$ Free-elim-prop _ (Ξ» _ β hlevel 1)
(Ξ» x β sym (happly (alg .Ξ½-unit) x))
(Ξ» x y p q β rec .preserves .is-group-hom.pres-β x y
Β·Β· apβ G._β_ p q
Β·Β· happly (alg .Ξ½-mult) (inc _ β inc _))
(Ξ» x p β is-group-hom.pres-inv (rec .preserves) {x = x}
Β·Β· ap G.inverse p
Β·Β· sym (is-group-hom.pres-inv alg-gh {x = x}))
refl