open import Algebra.Group.Cat.Base
open import Algebra.Group

open import Cat.Instances.Sets.Complete as SL
open import Cat.Prelude

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


# Finite limits of groups🔗

We present explicit computations of finite limits in the category of groups, though do note that the terminal group is also initial (i.e. it is a zero object). Knowing that the category of groups admits a right adjoint functor into the category of sets (the underlying set functor) drives us in computing limits of groups as limits of sets, and equipping those with a group structure: we are forced to do this since right adjoints preserve limits.

## The zero group🔗

open import Cat.Diagram.Terminal (Groups ℓ)
open import Cat.Diagram.Initial (Groups ℓ)
open import Cat.Diagram.Zero (Groups ℓ)


The zero object in the category of groups is given by the unit type, equipped with its unique group structure. Correspondingly, we may refer to this group in prose either as $0$ or as $\{\star\}$.

Zero-group : ∀ {ℓ} → Group ℓ
Zero-group = to-group zg where
zg : make-group (Lift _ ⊤)
zg .make-group.group-is-set x y p q i j = lift tt
zg .make-group.unit = lift tt
zg .make-group.mul = λ x y → lift tt
zg .make-group.inv x = lift tt
zg .make-group.assoc x y z = refl
zg .make-group.invl x = refl
zg .make-group.invr x = refl
zg .make-group.idl x = refl

Zero-group-is-initial : is-initial Zero-group
Zero-group-is-initial (_ , G) .centre = total-hom (λ x → G.unit) gh where
module G = Group-on G
gh : Group-hom _ _ (λ x → G.unit)
gh .pres-⋆ x y =
G.unit            ≡˘⟨ G.idl ⟩≡˘
G.unit G.⋆ G.unit ∎
Zero-group-is-initial (_ , G) .paths x =
Homomorphism-path λ _ → sym (Group-hom.pres-id (x .preserves))

Zero-group-is-terminal : is-terminal Zero-group
Zero-group-is-terminal _ .centre =
total-hom (λ _ → lift tt) record { pres-⋆ = λ _ _ _ → lift tt }
Zero-group-is-terminal _ .paths x = Homomorphism-path λ _ → refl

Zero-group-is-zero : is-zero Zero-group
Zero-group-is-zero = record
{ has-is-initial = Zero-group-is-initial
; has-is-terminal = Zero-group-is-terminal
}

∅ᴳ : Zero
∅ᴳ .Zero.∅ = Zero-group
∅ᴳ .Zero.has-is-zero = Zero-group-is-zero


## Direct products🔗

open import Cat.Diagram.Product (Groups ℓ)


We compute the product of two groups $G \times H$ as the product of their underlying sets, equipped with the operation of “pointwise addition”.

Direct-product : Group ℓ → Group ℓ → Group ℓ
Direct-product (G , Gg) (H , Hg) = to-group G×Hg where
module G = Group-on Gg
module H = Group-on Hg

G×Hg : make-group (∣ G ∣ × ∣ H ∣)
G×Hg .make-group.group-is-set = hlevel!
G×Hg .make-group.unit = G.unit , H.unit
G×Hg .make-group.mul (a , x) (b , y) = a G.⋆ b , x H.⋆ y
G×Hg .make-group.inv (a , x) = a G.⁻¹ , x H.⁻¹
G×Hg .make-group.assoc x y z = ap₂ _,_ (sym G.associative) (sym H.associative)
G×Hg .make-group.invl x = ap₂ _,_ G.inversel H.inversel
G×Hg .make-group.invr x = ap₂ _,_ G.inverser H.inverser
G×Hg .make-group.idl x = ap₂ _,_ G.idl H.idl


The projection maps and universal factoring are all given exactly as for the category of sets.

proj₁ : Groups.Hom (Direct-product G H) G
proj₁ .hom = fst
proj₁ .preserves .pres-⋆ x y = refl

proj₂ : Groups.Hom (Direct-product G H) H
proj₂ .hom = snd
proj₂ .preserves .pres-⋆ x y = refl

factor : Groups.Hom G H → Groups.Hom G K → Groups.Hom G (Direct-product H K)
factor f g .hom x = f .hom x , g .hom x
factor f g .preserves .pres-⋆ x y = ap₂ _,_ (f .preserves .pres-⋆ _ _) (g .preserves .pres-⋆ _ _)

Direct-product-is-product : is-product {G} {H} proj₁ proj₂
Direct-product-is-product {G} {H} = p where
open is-product
p : is-product _ _
p .⟨_,_⟩ = factor
p .π₁∘factor = Forget-is-faithful refl
p .π₂∘factor = Forget-is-faithful refl
p .unique other p q = Forget-is-faithful (funext λ x →
ap₂ _,_ (happly (ap hom p) x) (happly (ap hom q) x))


What sets the direct product of groups apart from (e.g.) the cartesian product of sets is that both “factors” embed into the direct product, by taking the identity as the other coordinate: $x \hookrightarrow (x, 1)$. Indeed, in the category of abelian groups, the direct product is also a coproduct.

inj₁ : G Groups.↪ Direct-product G H
inj₁ {G} {H} .mor .hom x = x , H .snd .unit
inj₁ {G} {H} .mor .preserves .pres-⋆ x y = ap (_ ,_) (sym (H .snd .idl))
inj₁ {G} {H} .monic g h x = Forget-is-faithful (funext λ e i → x i .hom e .fst)

inj₂ : H Groups.↪ Direct-product G H
inj₂ {H} {G} .mor .hom x = G .snd .unit , x
inj₂ {H} {G} .mor .preserves .pres-⋆ x y = ap (_, _) (sym (G .snd .idl))
inj₂ {H} {G} .monic g h x = Forget-is-faithful (funext λ e i → x i .hom e .snd)


## Equalisers🔗

open import Cat.Diagram.Equaliser


The equaliser of two group homomorphisms $f, g : G \to H$ is given by their equaliser as Set-morphisms, equipped with the evident group structure. Indeed, we go ahead and compute the actual Equaliser in sets, and re-use all of its infrastructure to make an equaliser in Groups.

module _ {G H : Group ℓ} (f g : Groups.Hom G H) where
private
module G = Group-on (G .snd)
module H = Group-on (H .snd)

module f = Group-hom (f .preserves)
module g = Group-hom (g .preserves)
module seq = Equaliser
(SL.Sets-equalisers
{A = G.underlying-set}
{B = H.underlying-set}
(f .hom) (g .hom))


Recall that points there are elements of the domain (here, a point $x : G$) together with a proof that $f(x) = g(x)$. To “lift” the group structure from $G$ to ${\mathrm{equ}}(f,g)$, we must prove that, if $f(x) = g(x)$ and $f(y) = g(y)$, then $f(x\star y) = g(x\star y)$. But this follows from $f$ and $g$ being group homomorphisms:

  Equaliser-group : Group ℓ
Equaliser-group = to-group equ-group where
equ-⋆ : ∣ seq.apex ∣ → ∣ seq.apex ∣ → ∣ seq.apex ∣
equ-⋆ (a , p) (b , q) = (a G.⋆ b) , r where abstract
r : f .hom (G .snd ._⋆_ a b) ≡ g .hom (G .snd ._⋆_ a b)
r = f.pres-⋆ a b ·· ap₂ H._⋆_ p q ·· sym (g.pres-⋆ _ _)

equ-inv : ∣ seq.apex ∣ → ∣ seq.apex ∣
equ-inv (x , p) = x G.⁻¹ , q where abstract
q : f .hom (G.inverse x) ≡ g .hom (G.inverse x)
q = f.pres-inv ·· ap H._⁻¹ p ·· sym g.pres-inv

abstract
invs : f .hom G.unit ≡ g .hom G.unit
invs = f.pres-id ∙ sym g.pres-id


Similar yoga must be done for the inverse maps and the group unit.

    equ-group : make-group ∣ seq.apex ∣
equ-group .make-group.group-is-set = seq.apex .is-tr
equ-group .make-group.unit = G.unit , invs
equ-group .make-group.mul = equ-⋆
equ-group .make-group.inv = equ-inv
equ-group .make-group.assoc x y z = Σ-prop-path (λ _ → H.has-is-set _ _) (sym G.associative)
equ-group .make-group.invl x = Σ-prop-path (λ _ → H.has-is-set _ _) G.inversel
equ-group .make-group.invr x = Σ-prop-path (λ _ → H.has-is-set _ _) G.inverser
equ-group .make-group.idl x = Σ-prop-path (λ _ → H.has-is-set _ _) G.idl

open is-equaliser
open Equaliser


We can then, pretty effortlessly, prove that the Equaliser-group, together with the canonical injection ${\mathrm{equ}}(f,g) {\hookrightarrow}G$, equalise the group homomorphisms $f$ and $g$.

  Groups-equalisers : Equaliser (Groups ℓ) f g
Groups-equalisers .apex = Equaliser-group
Groups-equalisers .equ = total-hom fst record { pres-⋆ = λ x y → refl }
Groups-equalisers .has-is-eq .equal = Forget-is-faithful seq.equal
Groups-equalisers .has-is-eq .limiting {F = F} {e′} p = total-hom map lim-gh where
map = seq.limiting {F = underlying-set (F .snd)} (ap hom p)

lim-gh : Group-hom _ _ map
lim-gh .pres-⋆ x y = Σ-prop-path (λ _ → H.has-is-set _ _) (e′ .preserves .pres-⋆ _ _)

Groups-equalisers .has-is-eq .universal {F = F} {p = p} = Forget-is-faithful
(seq.universal {F = underlying-set (F .snd)} {p = ap hom p})

Groups-equalisers .has-is-eq .unique {F = F} {p = p} q = Forget-is-faithful
(seq.unique {F = underlying-set (F .snd)} {p = ap hom p} (ap hom q))


Putting all of these constructions together, we get the proof that Groups is finitely complete, since we can compute pullbacks as certain equalisers.

open import Cat.Diagram.Limit.Finite

Groups-finitely-complete : Finitely-complete (Groups ℓ)
Groups-finitely-complete = with-equalisers (Groups ℓ) top prod Groups-equalisers
where
top : Terminal
top .Terminal.top = Zero-group
top .Terminal.has⊤ = Zero-group-is-terminal

prod : ∀ A B → Product A B
prod A B .Product.apex = Direct-product A B
prod A B .Product.π₁ = proj₁
prod A B .Product.π₂ = proj₂
prod A B .Product.has-is-product = Direct-product-is-product