open import 1Lab.Path.Reasoning
open import 1Lab.Prelude

open import Algebra.Group.Cat.Base
open import Algebra.Group.Concrete
open import Algebra.Group.Ab

open import Cat.Functor.Equivalence
open import Cat.Morphism

open import Data.Set.Truncation

open import Homotopy.Space.Delooping
open import Homotopy.Connectedness
open import Homotopy.Space.Circle

open ConcreteGroup

module Algebra.Group.Concrete.Abelian where

Concrete abelian groups🔗

A concrete abelian group is, unsurprisingly, a concrete group in which concatenation of loops at the base point is commutative.

module _ {ℓ} (G : ConcreteGroup ℓ) where
  is-concrete-abelian : Type ℓ
  is-concrete-abelian = ∀ (p q : pt G ≡ pt G) → p ∙ q ≡ q ∙ p

This is a property:

  open ConcreteGroup G using (H-Level-B)

  is-concrete-abelian-is-prop : is-prop is-concrete-abelian
  is-concrete-abelian-is-prop = hlevel 

As an easy consequence, in an abelian group $\mathbf{B}G\text{,}$ we can fill any square that has equal opposite sides.

  concrete-abelian→square
    : is-concrete-abelian
    → {x : ⌞ B G ⌟} → (p q : x ≡ x) → Square p q q p
  concrete-abelian→square ab {x} = connected∙-elim-prop
    {P = λ x → (p q : x ≡ x) → Square p q q p}
    (G .has-is-connected)
    (hlevel )
    (λ p q → commutes→square (ab p q))
    x

Still unsurprisingly, the delooping of an abelian group is abelian.

Deloop-abelian
  : ∀ {ℓ} {G : Group ℓ}
  → is-commutative-group G → is-concrete-abelian (Concrete G)
Deloop-abelian G-ab = ∙-comm _ G-ab

The circle is another example, being the delooping of $\mathbb{Z}\text{.}$

π₁-abelian→abelian
  : ∀ {ℓ} {G : ConcreteGroup ℓ}
  → is-commutative-group (π₁B G) → is-concrete-abelian G
π₁-abelian→abelian {G = G} π₁G-ab = subst is-concrete-abelian
  (Concrete≃Abstract.η G)
  (Deloop-abelian π₁G-ab)

S¹-concrete-abelian : is-concrete-abelian S¹-concrete
S¹-concrete-abelian = π₁-abelian→abelian {G = S¹-concrete}
  (subst is-commutative-group
    (sym π₁S¹≡ℤ)
    (Abelian→Group-on-abelian (ℤ-ab .snd)))

First abelian group cohomology🔗

When $H$ is a concrete abelian group, something interesting happens: for any other concrete group $G\text{,}$ the set of pointed maps $\mathbf{B}G{\to }^{\bullet }\mathbf{B}H$ (i.e. group homomorphisms from $G$ to $H\text{)}$ turns out to be equivalent to the set truncation of the type of unpointed maps, $\Vert \mathbf{B}G\to \mathbf{B}H{\Vert }_0\text{.}$

This is a special case of a theorem that relates this set truncation with the set of orbits of the action of the inner automorphism group of $H$ on the set of group homomorphisms $\mathbf{B}G{\to }^{\bullet }\mathbf{B}H\text{.}$ We do not prove this here, but see (Bezem et al. 2023, theorem 5.12.2). In the special case that $H$ is abelian, its inner automorphism group is trivial, and we can avoid quotienting.

In even fancier language, it is also a computation of the first cohomology group of $G$ with coefficients in $H\text{,}$ hence we adopt the notation $H^1(G,H)=\Vert \mathbf{B}G\to \mathbf{B}H{\Vert }_0\text{.}$

H¹[_,_] : ∀ {ℓ ℓ'} → ConcreteGroup ℓ → ConcreteGroup ℓ' → Type (ℓ ⊔ ℓ')
H¹[ G , H ] = ∥ (⌞ B G ⌟ → ⌞ B H ⌟) ∥₀

First, note that there is always a natural map $(\mathbf{B}G{\to }^{\bullet }\mathbf{B}H)\to H^1(G,H)$ that just forgets the pointing path.

class
  : ∀ {ℓ ℓ'} (G : ConcreteGroup ℓ) (H : ConcreteGroup ℓ')
  → (B G →∙ B H) → H¹[ G , H ]
class G H (f , _) = inc f

Now assume $H$ is abelian; we will show that this map is injective and surjective, so that it is an equivalence of sets.

module _ {ℓ ℓ'}
  (G : ConcreteGroup ℓ)
  (H : ConcreteGroup ℓ') (H-ab : is-concrete-abelian H)
  where
  open ConcreteGroup H using (H-Level-B)

Surjectivity is the easy part: since $H$ is connected, every map is merely homotopic to a pointed map.

  class-surjective : is-surjective (class G H)
  class-surjective = ∥-∥₀-elim (λ _ → hlevel ) λ f → do
    ptf ← H .has-is-connected (f (pt G))
    inc ((f , ptf) , refl)

For injectivity, we restrict our attention to the case of two pointed maps whose underlying maps are definitionally equal. In other words, we prove that any two pointings of $f$ (say $p,q:f({\bullet }_G)\equiv {\bullet }_H\text{)}$ yield equal pointed maps.

In order to define our underlying homotopy, we proceed by induction: since $G$ is a pointed connected type, it suffices to give a path $\alpha :f({\bullet }_G)\equiv f({\bullet }_G)$ and to show that every loop $l:{\bullet }_G\equiv {\bullet }_G$ respects this identification, in the sense of the following square:

Since our homotopy additionally has to be pointed, we know that we must have $\alpha =p\bullet q^{-1}\text{.}$ This is where the fact that $H$ is abelian comes in: the above square has equal opposite sides, so it automatically commutes.

  class-injective
    : ∀ f → (p q : f (pt G) ≡ pt H)
    → Path (B G →∙ B H) (f , p) (f , q)
  class-injective f p q = Σ-pathp
    (funext E.elim)
    (transpose (symP (∙→square'' E.elim-β-point)))
    where
      α : f (pt G) ≡ f (pt G)
      α = p ∙ sym q

      coh : ∀ (l : pt G ≡ pt G) → Square (ap f l) α α (ap f l)
      coh l = concrete-abelian→square H H-ab (ap f l) α

      module E = connected∙-elim-set {P = λ g → f g ≡ f g}
        (G .has-is-connected) (hlevel ) α coh

By a quick path induction, we can conclude that class is an equivalence between sets.

  class-is-equiv : is-equiv (class G H)
  class-is-equiv = injective-surjective→is-equiv! (inj _ _) class-surjective
    where
      inj : ∀ f g → class G H f ≡ class G H g → f ≡ g
      inj (f , ptf) (g , ptg) f≡g = ∥-∥-rec
        (ConcreteGroups-Hom-set G H _ _)
        (λ f≡g → J (λ g _ → ∀ ptg → (f , ptf) ≡ (g , ptg))
                   (class-injective f ptf)
                   f≡g ptg)
        (∥-∥₀-path.to f≡g)

  first-concrete-abelian-group-cohomology
    : (B G →∙ B H) ≃ H¹[ G , H ]
  first-concrete-abelian-group-cohomology
    = class G H , class-is-equiv

Translating this across our equivalence of categories gives a similar statement for abstract groups.

module _ {ℓ}
  (G : Group ℓ)
  (H : Group ℓ) (H-ab : is-commutative-group H)
  where

  Deloop-hom-equiv : (Deloop∙ G →∙ Deloop∙ H) ≃ Hom (Groups ℓ) G H
  Deloop-hom-equiv = ff+split-eso→hom-equiv π₁F
    (λ {G} {H} → π₁F-is-ff {_} {G} {H})
    π₁F-is-split-eso
    G H .snd .snd

  first-abelian-group-cohomology
    : H¹[ Concrete G , Concrete H ] ≃ Hom (Groups ℓ) G H
  first-abelian-group-cohomology =
    first-concrete-abelian-group-cohomology
      (Concrete G) (Concrete H) (Deloop-abelian H-ab) e⁻¹
    ∙e Deloop-hom-equiv