open import 1Lab.Prelude

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

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

module Homotopy.HSpace where

H-Spaces🔗

record HSpace {ℓ} (A* : Type∙ ℓ) : Type ℓ where
  open Σ A* renaming (fst to A ; snd to a₀)
  field
    μ : A → A → A

    idl    : ∀ x → μ a₀ x ≡ x
    idr    : ∀ x → μ x a₀ ≡ x
    id-coh : idl a₀ ≡ idr a₀

Here, we’re interested in the case where each $\mu (x,-)$ and $\mu (-,x)$ is an equivalence, so we’re really discussing invertible H-spaces. We note that if $A$ is connected then any of its H-space structures is automatically both left- and right-invertible. This is because, since “being an equivalence” is a proposition, it would suffice to check invertibility when $x$ is the basepoint, but in this case $\mu (-,a_0)$ is the identity.

    μ-invl : ∀ y → is-equiv (μ y)
    μ-invr : ∀ y → is-equiv (flip μ y)

  module _ (a b : ⌞ A* ⌟) where
    open Σ (μ-invl a .is-eqv b .centre) renaming (fst to _\\_) public
    open Σ (μ-invr a .is-eqv b .centre) renaming (fst to _//_) public

  μ-\\-l : ∀ a b → μ a (a \\ b) ≡ b
  μ-\\-l a b = Equiv.ε (_ , μ-invl a) b

  μ-\\-r : ∀ a b → a \\ μ a b ≡ b
  μ-\\-r a b = Equiv.η (_ , μ-invl a) b

  μ-zig : ∀ a b → ap (μ a) (μ-\\-r a b) ≡ μ-\\-l a (μ a b)
  μ-zig a b = Equiv.zig (_ , μ-invl a) b

  μ-//-l : ∀ a b → μ (a // b) a ≡ b
  μ-//-l a b = Equiv.ε (_ , μ-invr a) b

  μ-//-r : ∀ a b → a // μ b a ≡ b
  μ-//-r a b = Equiv.η (_ , μ-invr a) b

Using either unit law (we choose $\lambda \text{),}$ we can show that $\mu$ extends to a secondary ‘composition’ operation on the loop space $\Omega A\text{.}$ This operation is also unital on both the left and the right, with the side we didn’t choose having a slightly more complicated argument that involves the coherence ${\lambda }_{a_0}={\rho }_{a_0}\text{.}$

  _⊗_ : a₀ ≡ a₀ → a₀ ≡ a₀ → a₀ ≡ a₀
  p ⊗ q = conj (idl a₀) (ap₂ μ p q)

  ⊗-idl : ∀ p → refl ⊗ p ≡ p
  ⊗-idl p = square→conj (ap (λ e → ap e p) (funext idl))

  ⊗-idr : ∀ p → p ⊗ refl ≡ p
  ⊗-idr p =
    conj (idl a₀) (ap₂ μ p refl) ≡⟨ ap (λ e → conj e (ap₂ μ p refl)) id-coh ⟩≡
    conj (idr a₀) (ap₂ μ p refl) ≡⟨ square→conj (ap (λ e → ap e p) (funext idr)) ⟩≡
    p                            ∎

Moreover, this new operation satisfies an ‘interchange’ law, in that it preserves path composition in both variables. By an incoherent version of the Eckmann-Hilton argument, this means that composition of loops in $\Omega A$ is commutative. This implies that the only connected groupoids with H-space structures are the deloopings of abelian groups.

  ⊗-interchange : ∀ a b c d → (a ∙ b) ⊗ (c ∙ d) ≡ (a ⊗ c) ∙ (b ⊗ d)
  ⊗-interchange a b c d =
    conj (idl a₀) (ap₂ μ (a ∙ b) (c ∙ d))
      ≡⟨ ap (conj (idl a₀)) (ap₂-∙ μ a b c d) ⟩≡
    conj (idl a₀) (ap₂ μ a c ∙ ap₂ μ b d)
      ≡⟨ conj-of-∙ (idl a₀) _ _ ⟩≡
    (a ⊗ c) ∙ (b ⊗ d)
      ∎

  private
    ∙-is-flip-⊗ : (p q : a₀ ≡ a₀) → p ∙ q ≡ q ⊗ p
    ∙-is-flip-⊗ p q =
      p ∙ q                   ≡˘⟨ ap₂ _∙_ (⊗-idl p) (⊗-idr q) ⟩≡˘
      (refl ⊗ p) ∙ (q ⊗ refl) ≡⟨ sym (⊗-interchange refl q p refl) ⟩≡
      (refl ∙ q) ⊗ (p ∙ refl) ≡⟨ ap₂ _⊗_ (∙-idl q) (∙-idr p) ⟩≡
      q ⊗ p                   ∎

    ∙-is-⊗ : (p q : a₀ ≡ a₀) → p ∙ q ≡ p ⊗ q
    ∙-is-⊗ p q =
      p ∙ q                   ≡˘⟨ ap₂ _∙_ (⊗-idr p) (⊗-idl q) ⟩≡˘
      (p ⊗ refl) ∙ (refl ⊗ q) ≡⟨ sym (⊗-interchange p refl refl q) ⟩≡
      (p ∙ refl) ⊗ (refl ∙ q) ≡⟨ ap₂ _⊗_ (∙-idr p) (∙-idl q) ⟩≡
      (p ⊗ q)                 ∎

  ∙-comm : (p q : a₀ ≡ a₀) → p ∙ q ≡ q ∙ p
  ∙-comm p q = ∙-is-flip-⊗ p q ∙ sym (∙-is-⊗ q p)

open HSpace

module _ {ℓ} (G : Group ℓ) (ab : is-commutative-group G) where
  open Group-on (G .snd)

  private

H-Space structures on deloopings🔗

We can define an H-space structure on the delooping $\mathbf{B}G$ of an abelian group by recursion. First, we fix an element $g:G$ and define the “path” case, by elimination into sets — so it suffices to turn a $g:G$ into a $\mathrm{base}\equiv \mathrm{base}\text{,}$ and to show that this is commutative.

    mul₀ : (g : ⌞ G ⌟) (y : Deloop G) → y ≡ y
    mul₀ g = Deloop-elim-set G _ (λ d → hlevel )
      (path g)
      (λ z → commutes→square (Deloop-ab.∙-comm G ab _ _))

Now we extend this to when $x$ is an arbitrary element of $\mathbf{B}G\text{.}$ At the basepoint, we can simply return the other operand; this ensures identity on the left is definitional. For the generating paths, we can use the helper defined above. To show that this respects multiplication, we can lift the generating triangle pathᵀ to mul₀ by elimination into propositions. The truncation case is handled automatically.

    coh : (x y : ⌞ G ⌟) (z : Deloop G) → Square refl (mul₀ x z) (mul₀ (x ⋆ y) z) (mul₀ y z)
    coh x y = Deloop-elim-prop G _ (λ _ → hlevel ) λ i j → pathᵀ x y i j

    mul : Deloop G → Deloop G → Deloop G
    mul base            x = x
    mul (path x i)      y = mul₀ x y i
    mul (pathᵀ x y i j) z = coh x y z i j
    mul (squash x y p q α β i j k) z = squash (mul x z) (mul y z)
      (λ i → mul (p i) z) (λ i → mul (q i) z)
      (λ i j → mul (α i j) z)
      (λ i j → mul (β i j) z) i j k

By elimination into sets we can prove that mul is also unital on the right. For the base case this is once again definitional, and for the path case we’re left with filling a degenerate square which is path in one direction.

    mul-idr : ∀ x → mul x base ≡ x
    mul-idr = Deloop-elim-set G _ (λ _ → hlevel ) refl (λ z i j → path z i)

Finally, for invertibility, it suffices to check mul is an equivalence at the basepoint, in which case our proof above reduces this to showing the identity function is an equivalence.

  HSpace-BG : HSpace (Deloop∙ G)
  HSpace-BG .μ      = mul
  HSpace-BG .idl x  = refl
  HSpace-BG .idr    = mul-idr
  HSpace-BG .id-coh = refl
  HSpace-BG .μ-invl = Deloop-elim-prop _ _ (λ _ → hlevel )
    id-equiv
  HSpace-BG .μ-invr = Deloop-elim-prop _ _ (λ _ → hlevel )
    (subst is-equiv (ext (sym ∘ mul-idr)) id-equiv)

On the circle🔗

We can specialise the discussion above to the circle, in which case we already have many of the components we need. Note that always-loop gives us, for every point $y:S^1\text{,}$ a loop at $y\text{;}$ but loops are nothing but maps from $S^1\text{,}$ so we get a ‘multiplication’ on the circle. We also define the ‘inverse’ map on $S^1\text{.}$¹

mulS¹ : S¹ → S¹ → S¹
mulS¹ base     y = y
mulS¹ (loop i) y = always-loop y i

invS¹ : S¹ → S¹
invS¹ base     = base
invS¹ (loop i) = loop (~ i)

mulS¹-idr : ∀ x → mulS¹ x base ≡ x
mulS¹-idr = S¹-elim refl (λ i j → loop i)

mulS¹-comm : ∀ x y → mulS¹ x y ≡ mulS¹ y x
mulS¹-comm = S¹-elim (λ y → sym (mulS¹-idr y)) (funextP (S¹-elim (λ i j → loop i) prop!))

mulS¹-invl : ∀ x → mulS¹ (invS¹ x) x ≡ base
mulS¹-invl = S¹-elim refl λ i j → hcomp {A = S¹} (∂ i ∨ ∂ j) λ where
  k (k = i0) → base
  k (i = i0) → base
  k (i = i1) → base
  k (j = i0) → hfill (∂ i) k (λ { k (k = i0) → base ; k (i = i0) → loop (~ i ∨ k) ; k (i = i1) → loop (~ i ∧ k) })
  k (j = i1) → base

mulS¹-invr : ∀ x → mulS¹ x (invS¹ x) ≡ base
mulS¹-invr x = mulS¹-comm x (invS¹ x) ∙ mulS¹-invl x

mulS¹-assoc : ∀ x y z → mulS¹ x (mulS¹ y z) ≡ mulS¹ (mulS¹ x y) z
mulS¹-assoc = S¹-elim (λ y z → refl) (funextP (S¹-elim (funextP (S¹-elim (λ i j → loop i) prop!)) prop!))

HSpace-S¹ : HSpace (S¹ , base)
HSpace-S¹ .μ      = mulS¹
HSpace-S¹ .idl x  = refl
HSpace-S¹ .idr    = mulS¹-idr
HSpace-S¹ .id-coh = refl
HSpace-S¹ .μ-invr x =
  is-iso→is-equiv λ where
    .is-iso.from y → mulS¹ (invS¹ x) y
    .is-iso.rinv y → ap₂ mulS¹ (mulS¹-comm (invS¹ x) y) refl ∙ sym (mulS¹-assoc y (invS¹ x) x) ∙ ap (mulS¹ y) (mulS¹-invl x) ∙ mulS¹-idr y
    .is-iso.linv y → ap (mulS¹ (invS¹ x)) (mulS¹-comm y x) ∙ mulS¹-assoc (invS¹ x) x y ∙ ap (flip mulS¹ y) (mulS¹-invl x)
HSpace-S¹ .μ-invl x = is-iso→is-equiv λ where
  .is-iso.from y → mulS¹ (invS¹ x) y
  .is-iso.rinv y → mulS¹-assoc x (invS¹ x) y ∙ ap (flip mulS¹ y) (mulS¹-invr x)
  .is-iso.linv y → mulS¹-assoc (invS¹ x) x y ∙ ap (flip mulS¹ y) (mulS¹-invl x)