open import 1Lab.Path.Reasoning

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

open import Cat.Functor.Equivalence
open import Cat.Functor.Properties
open import Cat.Morphism
open import Cat.Prelude

open import Data.Int

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

open is-group-hom
open Precategory
open Functor

module Algebra.Group.Concrete where

private variable
  ℓ ℓ' : Level

Concrete groups🔗

In homotopy type theory, we can give an alternative definition of groups: they are the pointed, connected groupoids. The idea is that those groupoids contain exactly the same information as their fundamental group.

Such groups are dubbed concrete, because they represent the groups of symmetries of a given object (the base point); by opposition, “algebraic” Groups are called abstract.

record ConcreteGroup ℓ : Type (lsuc ℓ) where
  no-eta-equality
  constructor concrete-group
  field
    B                : Type∙ ℓ
    has-is-connected : is-connected∙ B
    has-is-groupoid  : is-groupoid ⌞ B ⌟

  pt : ⌞ B ⌟
  pt = B .snd

Given a concrete group $G,$ the underlying pointed type is denoted $B G,$ by analogy with the delooping of an abstract group; note that, here, the delooping is the chosen representation of $G,$ so B is just a record field. We write $∙_{G}$ for the base point.

Concrete groups are pointed connected types, so they enjoy the following elimination principle, which will be useful later:

  B-elim-contr : {P : ⌞ B ⌟ → Type ℓ'}
               → is-contr (P pt)
               → ∀ x → P x
  B-elim-contr {P = P} c = connected∙-elim-prop {P = P} has-is-connected
    (is-contr→is-prop c) (c .centre)

  instance
    H-Level-B : ∀ {n} → H-Level ⌞ B ⌟ (3 + n)
    H-Level-B = basic-instance 3 has-is-groupoid

open ConcreteGroup

instance
  Underlying-ConcreteGroup : Underlying (ConcreteGroup ℓ)
  Underlying-ConcreteGroup {ℓ} .Underlying.ℓ-underlying = ℓ
  Underlying-ConcreteGroup .⌞_⌟ G = ⌞ B G ⌟

ConcreteGroup-path : {G H : ConcreteGroup ℓ} → B G ≡ B H → G ≡ H
ConcreteGroup-path {G = G} {H} p = go prop! prop! where
  go : PathP (λ i → is-connected∙ (p i)) (G .has-is-connected) (H .has-is-connected)
     → PathP (λ i → is-groupoid ⌞ p i ⌟) (G .has-is-groupoid) (H .has-is-groupoid)
     → G ≡ H
  go c g i .B = p i
  go c g i .has-is-connected = c i
  go c g i .has-is-groupoid = g i

The delooping of a group is a concrete group. In fact, we will prove later that all concrete groups arise as deloopings.

Concrete : ∀ {ℓ} → Group ℓ → ConcreteGroup ℓ
Concrete G .B = Deloop∙ G
Concrete G .has-is-connected = Deloop-is-connected
Concrete G .has-is-groupoid = squash

Another important example of a concrete group is the circle: the delooping of the integers.

opaque
  S¹-is-connected : is-connected∙ S¹∙
  S¹-is-connected = S¹-elim (inc refl) prop!

S¹-concrete : ConcreteGroup lzero
S¹-concrete .B = S¹∙
S¹-concrete .has-is-connected = S¹-is-connected
S¹-concrete .has-is-groupoid = hlevel 3

The category of concrete groups🔗

The notion of group homomorphism between two groups $G$ and $H$ gets translated to, on the “concrete” side, pointed maps $B G \to^{∙} B H .$

The pointedness condition ensures that these maps behave like abstract group homomorphisms; in particular, that they form a set.

ConcreteGroups-Hom-set
  : (G : ConcreteGroup ℓ) (H : ConcreteGroup ℓ')
  → is-set (B G →∙ B H)
ConcreteGroups-Hom-set G H (f , ptf) (g , ptg) p q =
  Σ-set-square (λ _ → hlevel 2) (funext-square (B-elim-contr G square))
  where
    open ConcreteGroup H using (H-Level-B)
    square : is-contr ((λ j → p j .fst (pt G)) ≡ (λ j → q j .fst (pt G)))
    square .centre i j = hcomp (∂ i ∨ ∂ j) λ where
      k (k = i0) → pt H
      k (i = i0) → p j .snd (~ k)
      k (i = i1) → q j .snd (~ k)
      k (j = i0) → ptf (~ k)
      k (j = i1) → ptg (~ k)
    square .paths _ = H .has-is-groupoid _ _ _ _ _ _

These naturally assemble into a category.

ConcreteGroups : ∀ ℓ → Precategory (lsuc ℓ) ℓ
ConcreteGroups ℓ .Ob = ConcreteGroup ℓ
ConcreteGroups _ .Hom G H = B G →∙ B H
ConcreteGroups _ .Hom-set = ConcreteGroups-Hom-set

The rest of the categorical structure is inherited from pointed functions, and univalence follows from the univalence of the universe of groupoids.

ConcreteGroups _ .id = id∙
ConcreteGroups _ ._∘_ = _∘∙_
ConcreteGroups _ .idr f = Σ-pathp refl (∙-idl _)
ConcreteGroups _ .idl f = Σ-pathp refl (∙-idr _)
ConcreteGroups _ .assoc (f , ptf) (g , ptg) (h , pth) = Σ-pathp refl $
  ⌜ ap f (ap g pth ∙ ptg) ⌝ ∙ ptf   ≡⟨ ap! (ap-∙ f _ _) ⟩≡
  (ap (f ⊙ g) pth ∙ ap f ptg) ∙ ptf ≡⟨ sym (∙-assoc _ _ _) ⟩≡
  ap (f ⊙ g) pth ∙ ap f ptg ∙ ptf   ∎

private
  iso→equiv : ∀ {a b} → Isomorphism (ConcreteGroups ℓ) a b → ⌞ a ⌟ ≃ ⌞ b ⌟
  iso→equiv im = Iso→Equiv (im .to .fst ,
    iso (im .from .fst) (happly (ap fst (im .invl))) (happly (ap fst (im .invr))))

ConcreteGroups-is-category : is-category (ConcreteGroups ℓ)
ConcreteGroups-is-category .to-path im = ConcreteGroup-path $
  Σ-pathp (ua (iso→equiv im)) (path→ua-pathp _ (im .to .snd))
ConcreteGroups-is-category {ℓ} .to-path-over im = ≅-pathp (ConcreteGroups ℓ) _ _ $
  Σ-pathp (funextP λ _ → path→ua-pathp _ refl)
    (λ i j → path→ua-pathp (iso→equiv im) (λ i → im .to .snd (i ∧ j)) i)

Concrete vs. abstract🔗

Our goal is now to prove that concrete groups and abstract groups are equivalent, in the sense that there is an equivalence of categories between ConcreteGroups and Groups.

Since we’re dealing with groupoids, we can use the alternative definition of the fundamental group that avoids set truncations.

module _ (G : ConcreteGroup ℓ) where
  open π₁Groupoid (B G) (G .has-is-groupoid)
    renaming (π₁ to π₁B; π₁≡π₀₊₁ to π₁B≡π₀₊₁)
    public

We define a functor from concrete groups to abstract groups. The object mapping is given by taking the fundamental group. Given a pointed map $f : B G \to^{∙} B H,$ we can apply it to a loop on $∙_{G}$ to get a loop on $f (∙_{G});$ then, we use the fact that $f$ is pointed to get a loop on $∙_{H}$ by conjugation.

π₁F : Functor (ConcreteGroups ℓ) (Groups ℓ)
π₁F .F₀ = π₁B
π₁F .F₁ (f , ptf) .hom x = conj ptf (ap f x)

By some simple path yoga, this preserves multiplication, and the construction is functorial:

π₁F .F₁ (f , ptf) .preserves .pres-⋆ x y =
  conj ptf ⌜ ap f (x ∙ y) ⌝             ≡⟨ ap! (ap-∙ f _ _) ⟩≡
  conj ptf (ap f x ∙ ap f y)            ≡⟨ conj-of-∙ _ _ _ ⟩≡
  conj ptf (ap f x) ∙ conj ptf (ap f y) ∎

π₁F .F-id = ext conj-refl
π₁F .F-∘ (f , ptf) (g , ptg) = ext λ x →
  conj (ap f ptg ∙ ptf) (ap (f ⊙ g) x)        ≡˘⟨ conj-∙ _ _ _ ⟩≡˘
  conj ptf ⌜ conj (ap f ptg) (ap (f ⊙ g) x) ⌝ ≡˘⟨ ap¡ (ap-conj f _ _) ⟩≡˘
  conj ptf (ap f (conj ptg (ap g x)))         ∎

We start by showing that π₁F is split essentially surjective. This is the easy part: to build a concrete group out of an abstract group, we simply take its Delooping, and use the fact that the fundamental group of the delooping recovers the original group.

_ = Deloop

π₁F-is-split-eso : is-split-eso (π₁F {ℓ})
π₁F-is-split-eso G .fst = Concrete G
π₁F-is-split-eso G .snd = path→iso (π₁B≡π₀₊₁ (Concrete G) ∙ sym (G≡π₁B G))

We now tackle the hard part: to prove that π₁F is fully faithful. In order to show that the action on morphisms is an equivalence, we need a way of “delooping” a group homomorphism $f : π_{1} (B G) \to π_{1} (B H)$ into a pointed map $B G \to^{∙} B H .$

module Deloop-Hom {G H : ConcreteGroup ℓ} (f : Groups ℓ .Hom (π₁B G) (π₁B H)) where
  open ConcreteGroup H using (H-Level-B)

How can we build such a map? All we know about $B G$ is that it is a pointed connected groupoid, and thus that it comes with the elimination principle B-elim-contr. This suggests that we need to define a type family $C : B G \to Type$ such that $C (∙_{G})$ is contractible, conclude that $\forall x . C (x)$ holds and extract a map $B G \to^{∙} B H$ from that. The following construction is adapted from (Bezem et al. 2023, sec. 4.10):

  record C (x : ⌞ G ⌟) : Type ℓ where
    constructor mk
    field
      y : ⌞ H ⌟
      p : pt G ≡ x → pt H ≡ y
      f-p : (ω : pt G ≡ pt G) (α : pt G ≡ x) → p (ω ∙ α) ≡ f # ω ∙ p α

Our family sends a point $x : B G$ to a point $y : B H$ with a function $p$ that sends based paths ending at $x$ to based paths ending at $y,$ with the additional constraint that $p$ must “extend” $f,$ in the sense that a loop on the left can be factored out using $f .$

For the centre of contraction, we simply pick $p$ to be $f,$ sending loops on $∙_{G}$ to loops on $∙_{H} .$

  C-contr : is-contr (C (pt G))
  C-contr .centre .C.y = pt H
  C-contr .centre .C.p = f .hom
  C-contr .centre .C.f-p = f .preserves .pres-⋆

As it turns out, such a structure is entirely determined by the pair $(y, p (refl) : ∙_{H} \equiv y),$ which means it is contractible.

  C-contr .paths (mk y p f-p) i = mk (pt≡y i) (funextP f≡p i) (□≡□ i) where
    pt≡y : pt H ≡ y
    pt≡y = p refl

    f≡p : ∀ ω → Square refl (f # ω) (p ω) (p refl)
    f≡p ω = ∙-filler (f # ω) (p refl) ▷ (sym (f-p ω refl) ∙ ap p (∙-idr ω))

    □≡□ : PathP (λ i → ∀ ω α → f≡p (ω ∙ α) i ≡ f # ω ∙ f≡p α i) (f .preserves .pres-⋆) f-p
    □≡□ = is-prop→pathp (λ i → hlevel 1) _ _

We can now apply the elimination principle and unpack our data:

  c : ∀ x → C x
  c = B-elim-contr G C-contr

  g : B G →∙ B H
  p : {x : ⌞ G ⌟} → pt G ≡ x → pt H ≡ g .fst x

  g .fst x = c x .C.y
  g .snd = sym (p refl)

  p {x} = c x .C.p

  f-p : (ω : pt G ≡ pt G) (α : pt G ≡ pt G) → p (ω ∙ α) ≡ f # ω ∙ p α
  f-p = c (pt G) .C.f-p

In order to show that this is a delooping of $f$ (i.e. that $Π_{1} (g) \equiv f),$ we need one more thing: that $p$ extends $g$ on the right. We get this essentially for free, by path induction, because $p (α)$ ends at $g (x)$ by definition.

  p-g : (α : pt G ≡ pt G) {x' : ⌞ G ⌟} (l : pt G ≡ x')
      → p (α ∙ l) ≡ p α ∙ ap (g .fst) l
  p-g α = J (λ _ l → p (α ∙ l) ≡ p α ∙ ap (g .fst) l)
    (ap p (∙-idr _) ∙ sym (∙-idr _))

Since $g$ is pointed by $p (refl),$ this lets us conclude that we have found a right inverse to $Π_{1} :$

  f≡apg : (ω : pt G ≡ pt G) → Square (p refl) (f # ω) (ap (g .fst) ω) (p refl)
  f≡apg ω = commutes→square $
    p refl ∙ ap (g .fst) ω ≡˘⟨ p-g refl ω ⟩≡˘
    p (refl ∙ ω)           ≡˘⟨ ap p ∙-id-comm ⟩≡˘
    p (ω ∙ refl)           ≡⟨ f-p ω refl ⟩≡
    f # ω ∙ p refl         ∎

  rinv : π₁F .F₁ {G} {H} g ≡ f
  rinv = ext λ ω → pathp→conj (symP (f≡apg ω))

We are most of the way there. In order to get a proper equivalence, we must check that delooping $Π_{1} (f)$ gives us back the same pointed map $f .$

We use the same trick, building upon what we’ve done before: start by defining a family that asserts that $p_{x}$ agrees with $f$ all the way, not just on loops:

module Deloop-Hom-π₁F {G H : ConcreteGroup ℓ} (f : B G →∙ B H) where
  open Deloop-Hom {G = G} {H} (π₁F .F₁ {G} {H} f) public
  open ConcreteGroup H using (H-Level-B)

  C' : ∀ x → Type _
  C' x = Σ (f .fst x ≡ g .fst x) λ eq
       → (α : pt G ≡ x) → Square (f .snd) (ap (f .fst) α) (p α) eq

This is a property, and $∙_{G}$ has it:

  C'-contr : is-contr (C' (pt G))
  C'-contr .centre .fst = f .snd ∙ sym (g .snd)
  C'-contr .centre .snd α = commutes→square $
    f .snd ∙ p ⌜ α ⌝                                ≡˘⟨ ap¡ (∙-idr _) ⟩≡˘
    f .snd ∙ ⌜ p (α ∙ refl) ⌝                       ≡⟨ ap! (f-p α refl) ⟩≡
    f .snd ∙ conj (f .snd) (ap (f .fst) α) ∙ p refl ≡˘⟨ ∙-extendl (∙-swapl (sym (conj-defn _ _))) ⟩≡˘
    ap (f .fst) α ∙ f .snd ∙ p refl                 ∎
  C'-contr .paths (eq , eq-paths) = Σ-prop-path! $
    sym (∙-unique _ (transpose (eq-paths refl)))

Using the elimination principle again, we get enough information about g to conclude that it is equal to f, so that we have a left inverse.

  c' : ∀ x → C' x
  c' = B-elim-contr G C'-contr

  g≡f : ∀ x → g .fst x ≡ f .fst x
  g≡f x = sym (c' x .fst)

The homotopy g≡f is pointed by definition, but we need to bend the path into a Square:

  β : g≡f (pt G) ≡ sym (f .snd ∙ sym (g .snd))
  β = ap (sym ⊙ fst) (sym (C'-contr .paths (c' (pt G))))

  ptg≡ptf : Square (g≡f (pt G)) (g .snd) (f .snd) refl
  ptg≡ptf i j = hcomp (∂ i ∨ ∂ j) λ where
    k (k = i0) → ∙-filler (f .snd) (sym (g .snd)) (~ j) (~ i)
    k (i = i0) → g .snd j
    k (i = i1) → f .snd (j ∧ k)
    k (j = i0) → β (~ k) i
    k (j = i1) → f .snd (~ i ∨ k)

  linv : g ≡ f
  linv = funext∙ g≡f ptg≡ptf

Phew. At last, π₁F is fully faithful.

π₁F-is-ff : is-fully-faithful (π₁F {ℓ})
π₁F-is-ff {ℓ} {G} {H} = is-iso→is-equiv $ iso
  (Deloop-Hom.g {G = G} {H})
  (Deloop-Hom.rinv {G = G} {H})
  (Deloop-Hom-π₁F.linv {G = G} {H})

We’ve shown that π₁F is fully faithful and essentially surjective; this lets us conclude with the desired equivalence.

π₁F-is-equivalence : is-equivalence (π₁F {ℓ})
π₁F-is-equivalence = ff+split-eso→is-equivalence
  (λ {G} {H} → π₁F-is-ff {_} {G} {H})
  π₁F-is-split-eso

π₁B-is-equiv : is-equiv (π₁B {ℓ})
π₁B-is-equiv = is-cat-equivalence→equiv-on-objects
  ConcreteGroups-is-category
  Groups-is-category
  π₁F-is-equivalence

module Concrete≃Abstract {ℓ} = Equiv (_ , π₁B-is-equiv {ℓ})

References

Bezem, Marc, Ulrik Buchholtz, Pierre Cagne, Bjørn Ian Dundas, and Daniel R. Grayson. 2023. “Symmetry.” https://github.com/UniMath/SymmetryBook.