{-# OPTIONS -vtactic.hlevel:10 #-}
open import 1Lab.Prelude

open import Algebra.Group.Homotopy

open import Data.List using (_∷_ ; [])

open import Homotopy.Space.Suspension
open import Homotopy.Space.Sphere

module Homotopy.Base where

Synthetic homotopy theory🔗

This module contains the basic definitions for the study of synthetic homotopy theory. Synthetic homotopy theory is the name given to studying $\infty$ -groupoids in their own terms, i.e., the application of homotopy type theory to computing homotopy invariants of spaces. Central to the theory is the concept of pointed type and pointed map. After all, homotopy groups are no more than the set-truncations of n-fold iterated loop spaces, and loop spaces are always relative to a basepoint.

A helper that will come in handy is Σ∙, which attaches the north pole as the basepoint of the suspended space.

Σ∙ : ∀ {ℓ} → Type∙ ℓ → Type∙ ℓ
Σ∙ A = Susp (A .fst) , N

Ω∙ : ∀ {ℓ} → Type∙ ℓ → Type∙ ℓ
Ω∙ (A , a) = Path A a a , refl

The suspension-loop space adjunction🔗

An important stepping stone in calculating loop spaces of higher types is the suspension-loop space adjunction: basepoint-preserving maps from a suspension are the same thing as basepoint-preserving maps into a loop space. We construct the equivalence in two steps, but both halves are constructed in elementary terms.

First, we’ll prove that

$(\Sigma A \to_* B) \simeq \left(\sum_{b_s : B} A \to b_0 \equiv b_s\right),$

which is slightly involved, but not too much. The actual equivalence is very straightforward to construct, but proving that the two maps Σ-map→loops and loops→Σ-map are inverses involves nontrivial path algebra.

module _ {ℓ ℓ′} {A : Type∙ ℓ} {B : Type∙ ℓ′} where
  Σ-map∙→loops : (Σ∙ A →∙ B) → (Σ _ λ bs → A .fst → B .snd ≡ bs)
  Σ-map∙→loops f .fst = f .fst S
  Σ-map∙→loops f .snd a = sym (f .snd) ∙ ap (f .fst) (merid a)

  loops→Σ-map∙ : (Σ _ λ bs → A .fst → B .snd ≡ bs) → (Σ∙ A →∙ B)
  loops→Σ-map∙ pair .fst N           = B .snd
  loops→Σ-map∙ pair .fst S           = pair .fst
  loops→Σ-map∙ pair .fst (merid x i) = pair .snd x i
  loops→Σ-map∙ pair .snd = refl

The construction for turning a family of loops into a basepoint-preserving map into $\Omega B$ is even simpler, perhaps because these are almost definitionally the same thing.

  loops→map∙-Ω : (Σ _ λ bs → A .fst → B .snd ≡ bs) → (A →∙ Ω∙ B)
  loops→map∙-Ω (b , x) .fst a = x a ∙ sym (x (A .snd))
  loops→map∙-Ω (b , x) .snd   = ∙-invr (x (A .snd))

  map∙-Ω→loops : (A →∙ Ω∙ B) → (Σ _ λ bs → A .fst → B .snd ≡ bs)
  map∙-Ω→loops pair .fst = B .snd
  map∙-Ω→loops pair .snd = pair .fst

The path algebra for showing these are both pairs of inverse equivalences is not very interesting, so I’ve kept it hidden.

  Σ-map∙≃loops : (Σ∙ A →∙ B) ≃ (Σ _ λ b → A .fst → B .snd ≡ b)
  Σ-map∙≃loops = Iso→Equiv (Σ-map∙→loops , iso loops→Σ-map∙ invr invl) where
    invr : is-right-inverse loops→Σ-map∙ Σ-map∙→loops
    invr (p , q) = Σ-pathp refl
      (to-pathp (funext (λ a → subst-path-right _ _ ∙ ∙-idr _ ∙ ∙-idl _ ∙ ap q (transport-refl _))))

    lemma : ∀ (f : Σ∙ A →∙ B) x → Square
      (sym (f .snd) ∙ ap (f .fst) (merid x))
      (sym (f .snd)) refl (ap (f .fst) (merid x))
    lemma f x = transport (sym Square≡double-composite-path) $
      sym (sym (f .snd) ∙ ap (f .fst) (merid x)) ·· sym (f .snd) ·· ap (f .fst) (merid x)    ≡⟨ double-composite (sym _) _ _ ⟩≡
      ⌜ sym (sym (f .snd) ∙ ap (f .fst) (merid x)) ⌝ ∙ sym (f .snd) ∙ ap (f .fst) (merid x)  ≡⟨ ap! (sym-∙ (sym _) _) ⟩≡
      (ap (f .fst) (sym (merid x)) ∙ (f .snd)) ∙ sym (f .snd) ∙ ap (f .fst) (merid x)        ≡˘⟨ ∙-assoc _ _ _ ⟩≡˘
      ap (f .fst) (sym (merid x)) ∙ ⌜ (f .snd) ∙ sym (f .snd) ∙ ap (f .fst) (merid x) ⌝      ≡⟨ ap! (∙-cancell (sym (f .snd)) _) ⟩≡
      ap (f .fst) (sym (merid x)) ∙ ap (f .fst) (merid x)                                    ≡⟨ ∙-invl _ ⟩≡
      refl                                                                                   ∎

    invl : is-left-inverse loops→Σ-map∙ Σ-map∙→loops
    invl f = Σ-pathp (funext (λ { N → sym (f .snd)
                                ; S → refl
                                ; (merid x i) → λ j → lemma f x i j }))
                      (to-pathp (subst-path-left _ _ ∙ ∙-idr _ ∙ refl))

  loops≃map∙-Ω : (Σ _ λ bs → A .fst → B .snd ≡ bs) ≃ (A →∙ Ω∙ B)
  loops≃map∙-Ω = Iso→Equiv (loops→map∙-Ω , iso map∙-Ω→loops invr invl) where
    lemma′ : ∀ {ℓ} {A : Type ℓ} {x : A} (q : x ≡ x) (r : refl ≡ q)
           → ap (λ p → q ∙ sym p) r ∙ ∙-invr q ≡ ∙-idr q ∙ sym r
    lemma′ q r =
      J (λ q′ r → ap (λ p → q′ ∙ sym p) r ∙ ∙-invr q′ ≡ ∙-idr q′ ∙ sym r)
        (∙-idl _ ∙ sym (∙-idr _))
        r

    invr : is-right-inverse map∙-Ω→loops loops→map∙-Ω
    invr (b , x) = Σ-pathp (funext (λ a → ap₂ _∙_ refl (ap sym x) ∙ ∙-idr _))
      (to-pathp (subst-path-left _ _ ∙ lemma))
      where
        lemma =
          ⌜ sym (ap₂ _∙_ refl (ap sym x) ∙ ∙-idr (b (A .snd))) ⌝ ∙ ∙-invr (b (A .snd))               ≡⟨ ap! (sym-∙ (sym _) _) ⟩≡
          (sym (∙-idr (b (A .snd))) ∙ ap (b (A .snd) ∙_) (ap sym (sym x))) ∙ ∙-invr (b (A .snd))     ≡⟨ sym (∙-assoc _ _ _) ⟩≡
          sym (∙-idr (b (A .snd))) ∙ ⌜ ap (λ p → b (A .snd) ∙ sym p) (sym x) ∙ ∙-invr (b (A .snd)) ⌝ ≡⟨ ap! (lemma′ (b (A .snd)) (sym x)) ⟩≡
          sym (∙-idr (b (A .snd))) ∙ ∙-idr (b (A .snd)) ∙ x                                          ≡⟨ ∙-cancell _ _ ⟩≡
          x                                                                                          ∎

    invl : is-left-inverse map∙-Ω→loops loops→map∙-Ω
    invl (f , p) = Σ-pathp (p (A .snd)) $ to-pathp $ funext $ λ x →
        subst-path-right _ _ ∙ sym (∙-assoc _ _ _)
      ∙ ap₂ _∙_ refl (∙-invl (p (A .snd))) ∙ ∙-idr _
      ∙ ap p (transport-refl x)

Composing these equivalences, we get the adjunction:

$(\Sigma A \to_* B) \simeq (A \to* \Omega B).$

  Σ-map∙≃map∙-Ω : (Σ∙ A →∙ B) ≃ (A →∙ Ωⁿ 1 B)
  Σ-map∙≃map∙-Ω = Σ-map∙≃loops ∙e loops≃map∙-Ω

Loop spaces are equivalently based maps out of spheres🔗

Repeatedly applying the equivalence we just built, we can build an equivalence

$(S^n \to_* A) \simeq (\Sigma S^{n - 1} \to_* \Omega A) \simeq ... \simeq \Omega^n A,$

thus characterising $n$ -fold loop spaces as basepoint-preserving maps out of $S^n$ , the $n$ -dimensional sphere.

reassoc-Ω : ∀ {ℓ} {A : Type∙ ℓ} n → Ωⁿ n (Ω∙ A) ≡ Ωⁿ (suc n) A
reassoc-Ω zero = refl
reassoc-Ω {A = A} (suc n) =
  Ωⁿ 1 (Ωⁿ n (Ωⁿ 1 A)) ≡⟨ ap (Ωⁿ 1) (reassoc-Ω n) ⟩≡
  Ωⁿ 1 (Ωⁿ (suc n) A)  ∎

Sⁿ : Nat → Type∙ lzero
Sⁿ n = Sⁿ⁻¹ (suc n) , N

As a special case, in the zeroth dimension, we conclude that $(2 \to_* A) \equiv A$ , i.e., basepoint-preserving maps from the booleans (based at either point) are the same thing as points of $A$ .

Ωⁿ≃Sⁿ-map : ∀ {ℓ} {A : Type∙ ℓ} n → (Sⁿ n →∙ A) ≃ Ωⁿ n A .fst
Ωⁿ≃Sⁿ-map {A = A} zero    = Iso→Equiv (from , iso to (λ _ → refl) invr) where
  to : A .fst → ((Susp ⊥ , N) →∙ A)
  to x .fst N = A .snd
  to x .fst S = x
  to x .snd = refl

  from : ((Susp ⊥ , N) →∙ A) → A .fst
  from f = f .fst S

  invr : is-right-inverse from to
  invr (x , p) =
    Σ-pathp (funext (λ { N → sym p
                       ; S → refl }))
            λ i j → p (~ i ∨ j)
Ωⁿ≃Sⁿ-map {A = A} (suc n) =
  (Σ∙ (Susp _ , N) →∙ A)          ≃⟨ Σ-map∙≃map∙-Ω ⟩≃
  ((Susp (Sⁿ⁻¹ n) , N) →∙ Ωⁿ 1 A) ≃⟨ Ωⁿ≃Sⁿ-map n ⟩≃
  Ωⁿ n (Ωⁿ 1 A) .fst              ≃⟨ path→equiv (ap fst (reassoc-Ω n)) ⟩≃
  Ωⁿ (suc n) A .fst               ≃∎

Hubs and spokes🔗

Inspired by the equivalence built above, although not using it directly, we can characterise h-levels in terms of maps of spheres, too. The idea is that, since a map $f : S^n \to A$ is equivalently some loop in $A$ ¹, we can characterise the trivial loops as the constant functions $S^n \to A$ . Correspondingly, if every function $S^n \to A$ is trivial, this means that all $n$ -loops in $A$ are trivial, so that $A$ is $(n+1)$ -truncated!

We refer to a trivialisation of a map $f : S^n \to A$ as being a “hubs-and-spokes” construction. Geometrically, a trivial loop $f : S^n \to A$ can be understood as a map from the $n$ -disc rather than the $n$ -sphere, where the $n$ -disc is the type generated by attaching a new point to the sphere (the “hub”), and paths connecting the hub to each point along the sphere (the “spokes”). The resulting type is contractible, whence every function out of it is constant.

hlevel→hubs-and-spokes
  : ∀ {ℓ} {A : Type ℓ} (n : Nat) → is-hlevel A (suc n)
  → (sph : Sⁿ n .fst → A)
  → Σ[ hub ∈ A ] (∀ x → sph x ≡ hub)

hubs-and-spokes→hlevel
  : ∀ {ℓ} {A : Type ℓ} (n : Nat)
  → ((sph : Sⁿ n .fst → A) → Σ[ hub ∈ A ] (∀ x → sph x ≡ hub))
  → is-hlevel A (suc n)

hlevel→hubs-and-spokes 0 prop sph = sph N , λ x → prop (sph x) (sph N)
hlevel→hubs-and-spokes {A = A} (suc n) h =
  helper λ x y → hlevel→hubs-and-spokes n (h x y)
  where
  helper
    : ((a b : A) → (sph : Sⁿ⁻¹ (1 + n) → a ≡ b) → Σ _ λ hub → ∀ x → sph x ≡ hub)
    → (sph : Sⁿ⁻¹ (2 + n) → A)
    → Σ _ λ hub → ∀ x → sph x ≡ hub
  helper h f = f N , sym ∘ r where
    r : (x : Sⁿ⁻¹ (2 + n)) → f N ≡ f x
    r N = refl
    r S = h (f N) (f S) (λ x i → f (merid x i)) .fst
    r (merid x i) j = hcomp (∂ i ∨ ∂ j) λ where
       k (i = i0) → f N
       k (i = i1) → h (f N) (f S) (λ x i → f (merid x i)) .snd x k j
       k (j = i0) → f N
       k (j = i1) → f (merid x i)
       k (k = i0) → f (merid x (i ∧ j))

hubs-and-spokes→hlevel {A = A} zero spheres x y
  = spheres map .snd N ∙ sym (spheres map .snd S) where
    map : Sⁿ⁻¹ 1 → A
    map N = x
    map S = y
hubs-and-spokes→hlevel {A = A} (suc n) spheres x y =
  hubs-and-spokes→hlevel n $ helper spheres x y where
  helper
    : ((sph : Sⁿ⁻¹ (2 + n) → A) → Σ _ λ hub → ∀ x → sph x ≡ hub)
    → ∀ a b
    → (sph : Sⁿ⁻¹ (1 + n) → a ≡ b)
    → Σ _ λ hub → ∀ x → sph x ≡ hub
  helper h x y f = _ , r  where
    f' : Sⁿ⁻¹ (2 + n) → A
    f' N = x
    f' S = y
    f' (merid u i) = f u i

    r : (s : Sⁿ⁻¹ (1 + n)) → f s ≡ h f' .snd N ∙ sym (h f' .snd S)
    r s i j = hcomp (∂ i ∨ ∂ j) λ where
      k (k = i0) → h f' .snd N (~ i ∨ j)
      k (i = i0) → h f' .snd (merid s j) (~ k)
      k (i = i1) → hfill (∂ j) k λ where
        l (j = i0) → x
        l (j = i1) → h f' .snd S (~ l)
        l (l = i0) → h f' .snd N j
      k (j = i0) → h f' .snd N (~ i ∧ ~ k)
      k (j = i1) → h f' .snd S (~ k)

Using this idea, we can define a general $n$ -truncation type, as a joint generalisation of the propositional and set truncations. While we can not directly build a type with a constructor saying the type is $n$ -truncated, what we can do is freely generate hubs and spokes for any $n$ -sphere drawn on the $n$ -truncation of $A$ . The result is the universal $n$ -type admitting a map from $A$ .

data n-Tr {ℓ} (A : Type ℓ) n : Type ℓ where
  inc    : A → n-Tr A n
  hub    : (r : Sⁿ⁻¹ n → n-Tr A n) → n-Tr A n
  spokes : (r : Sⁿ⁻¹ n → n-Tr A n) → ∀ x → r x ≡ hub r

For both proving that the $n$ -truncation has the right h-level, and proving that one can eliminate from it to $n$ -types, we use the characterisations of truncation in terms of hubs-and-spokes.

n-Tr-is-hlevel
  : ∀ {ℓ} {A : Type ℓ} n → is-hlevel (n-Tr A (suc n)) (suc n)
n-Tr-is-hlevel n = hubs-and-spokes→hlevel n λ sph → hub sph , spokes sph

instance
  n-tr-decomp : ∀ {ℓ} {A : Type ℓ} {n} → hlevel-decomposition (n-Tr A (suc n))
  n-tr-decomp = decomp (quote n-Tr-is-hlevel) (`level-minus 1 ∷ [])

n-Tr-elim
  : ∀ {ℓ ℓ′} {A : Type ℓ} {n}
  → (P : n-Tr A (suc n) → Type ℓ′)
  → (∀ x → is-hlevel (P x) (suc n))
  → (∀ x → P (inc x))
  → ∀ x → P x
n-Tr-elim {A = A} {n} P ptrunc pbase = go where
  module _ (r : Sⁿ n .fst → n-Tr A (1 + n))
           (w : (x : Sⁿ n .fst) → P (r x))
    where
      circle : Sⁿ⁻¹ (1 + n) → P (hub r)
      circle x = subst P (spokes r x) (w x)

      hub′ = hlevel→hubs-and-spokes n (ptrunc (hub r)) circle .fst

      spokes′ : ∀ x → PathP (λ i → P (spokes r x i)) (w x) hub′
      spokes′ x = to-pathp $
        hlevel→hubs-and-spokes n (ptrunc (hub r)) circle .snd x

  go : ∀ x → P x
  go (inc x)        = pbase x
  go (hub r)        = hub′ r (λ x → go (r x))
  go (spokes r x i) = spokes′ r (λ x → go (r x)) x i

n-Tr-elim!
  : ∀ {ℓ ℓ′} {A : Type ℓ} {n}
  → (P : n-Tr A (suc n) → Type ℓ′)
  → {@(tactic hlevel-tactic-worker) hl : ∀ x → is-hlevel (P x) (suc n)}
  → (∀ x → P (inc x))
  → ∀ x → P x
n-Tr-elim! P {hl} f = n-Tr-elim P hl f

n-Tr-rec!
  : ∀ {ℓ ℓ′} {A : Type ℓ} {B : Type ℓ′} {n}
  → {@(tactic hlevel-tactic-worker) hl : is-hlevel B (suc n)}
  → (A → B) → n-Tr A (suc n) → B
n-Tr-rec! {hl = hl} = n-Tr-elim (λ _ → _) (λ _ → hl)

As a simpler case of n-Tr-elim, we have the recursion principle, where the type we are eliminating into is constant.

n-Tr-rec
  : ∀ {ℓ ℓ′} {A : Type ℓ} {B : Type ℓ′} {n}
  → is-hlevel B (suc n) → (A → B) → n-Tr A (suc n) → B
n-Tr-rec hl = n-Tr-elim (λ _ → _) (λ _ → hl)

An initial application of the induction principle for $n$ -truncation of $A$ is characterising its path spaces, at least for the inclusions of points from $A$ . Identifications between (the images of) points in $A$ in the $(n+1)$ -truncation are equivalently $n$ -truncated identifications in $A$ :

n-Tr-path-equiv
  : ∀ {ℓ} {A : Type ℓ} {n} {x y : A}
  → Path (n-Tr A (2 + n)) (inc x) (inc y) ≃ n-Tr (x ≡ y) (suc n)
n-Tr-path-equiv {A = A} {n} {x = x} {y} = Iso→Equiv isom where

The proof is an application of the encode-decode method. We would like to characterise the space

$\mathrm{inc}(x) \equiv_{\|A\|_{2+n}} y,$

so we will, for every $x : A$ , define a type family $\mathrm{code}(x) : \|A\|_{2+n} \to \mathrm{Type}$ , where the fibre of $\mathrm{code}(x)$ over $\mathrm{inc}(y)$ should be $\|x \equiv y\|_{1+n}$ . However, induction principle for $\|A\|_{2+n}$ only allows us to map into $(2+n)$ -types, while $\mathrm{Type}$ itself is not an $n$ -type for any $n$ . We salvage our definition by instead mapping into $(1+n)\text{-}\mathrm{Type}$ , which is a $(2+n)$ -type.

  code : (x : A) (y′ : n-Tr A (2 + n)) → n-Type _ (suc n)
  code x =
    n-Tr-elim!
      (λ y′ → n-Type _ (suc n))
      (λ y′ → el! (n-Tr (Path A x y′) (suc n)))

The rest of the proof boils down to applications of path induction and the induction principle for $\|A\|_{n+2}$ .

  encode′ : ∀ x y → inc x ≡ y → ∣ code x y ∣
  encode′ x _ = J (λ y _ → ∣ code x y ∣) (inc refl)

  decode′ : ∀ x y → ∣ code x y ∣ → inc x ≡ y
  decode′ x =
    n-Tr-elim! _ λ x → n-Tr-rec hlevel! (ap inc)

  rinv : ∀ x y → is-right-inverse (decode′ x y) (encode′ x y)
  rinv x = n-Tr-elim _
    (λ y → Π-is-hlevel (2 + n)
      (λ c → Path-is-hlevel (2 + n) (is-hlevel-suc (suc n) (code x y .is-tr))))
    λ x → n-Tr-elim! _ λ p → ap n-Tr.inc (subst-path-right _ _ ∙ ∙-idl _)

  linv : ∀ x y → is-left-inverse (decode′ x y) (encode′ x y)
  linv x _ = J (λ y p → decode′ x y (encode′ x y p) ≡ p)
    (ap (decode′ x (inc x)) (transport-refl (inc refl)) ∙ refl)

  isom : Iso (inc x ≡ inc y) (n-Tr (x ≡ y) (suc n))
  isom = encode′ _ (inc _)
       , iso (decode′ _ (inc _)) (rinv _ (inc _)) (linv _ (inc _))

Any map $f : S^n \to A$ can be made basepoint-preserving by letting $A$ be based at $f(N)$ .↩︎