⚠️ This module is not part of this project and is only included for reference.
It is either part of the 1lab, the cubical library, or a built-in Agda module.
<!-- ``` open import 1Lab.Reflection.Induction open import 1Lab.Prelude ``` --> ```agda module Homotopy.Space.Suspension where ``` # Suspension {defines="suspension"} Given a type $A$, its **(reduced) suspension** $\Susp A$ is the higher-inductive type generated by the following constructors: ```agda data Susp {ℓ} (A : Type ℓ) : Type ℓ where N S : Susp A merid : A → N ≡ S ``` The names `N`{.Agda} and `S`{.Agda} are meant to evoke the *n*orth and *s*outh poles of a sphere, respectively, and the name `merid`{.Agda} should evoke the *merid*ians. Indeed, we can picture a suspension like a sphere[^diamond]: [^diamond]: Diagrams are hard, okay?! ~~~{.quiver} \begin{tikzpicture} \node[draw,circle,fill,minimum width=1.2pt,inner sep=0,outer sep=2pt,label={[right,yshift=-2.6pt]\tiny N}] (N) at (0,+1.25) {}; \node[draw,circle,fill,minimum width=1.2pt,inner sep=0,outer sep=2pt,label={[right,yshift=-2.6pt]\tiny S}] (S) at (0,-1.25) {}; \coordinate (L) at (-0.8,0); \coordinate (R) at (+0.8,0); \coordinate (T) at (+0.25,+0.525); \coordinate (B) at (-0.25,-0.525); \filldraw[fill=diagramshaded] (T) -- (R) -- (B) -- (L) -- (T); \draw (N) -- (L); \draw (N) -- (R); \draw (N) -- (B); \draw[dashed] (N) -- (T); \draw (S) -- (L); \draw (S) -- (R); \draw (S) -- (B); \node[color=diagramshadedtext] (A) at (0,0) {\tiny $A$}; \end{tikzpicture} ~~~ We have the north and south poles $N$ and $S$ above and below a copy of the space $A$, which is diagrammatically represented by the <span class=shaded>shaded</span> region. In the type theory, we can't really "see" this copy of $A$: we only see its _ghost_, as something keeping all the meridians from collapsing. By convention, we see the suspension as a [[pointed type]] with the *north* pole as the base point. ```agda Susp∙ : ∀ {ℓ} (A : Type ℓ) → Type∙ ℓ Susp∙ A = Susp A , N ``` ```agda Susp-elim : ∀ {ℓ ℓ'} {A : Type ℓ} (P : Susp A → Type ℓ') → (pN : P N) (pS : P S) → (∀ x → PathP (λ i → P (merid x i)) pN pS) → ∀ x → P x Susp-elim P pN pS pmerid N = pN Susp-elim P pN pS pmerid S = pS Susp-elim P pN pS pmerid (merid x i) = pmerid x i unquoteDecl Susp-elim-prop = make-elim-n 1 Susp-elim-prop (quote Susp) ``` Every suspension admits a surjection from the booleans: ```agda 2→Σ : ∀ {ℓ} {A : Type ℓ} → Bool → Susp A 2→Σ true = N 2→Σ false = S 2→Σ-surjective : ∀ {ℓ} {A : Type ℓ} → is-surjective (2→Σ {A = A}) 2→Σ-surjective = Susp-elim-prop (λ _ → hlevel 1) (inc (true , refl)) (inc (false , refl)) ``` The suspension operation [[increases|connectedness of suspensions]] the [[connectedness]] of spaces: if $A$ is $n$-connected, then $\Susp A$ is $(1+n)$-connected. Unfolding this a bit further, if $A$ is a type whose homotopy groups below $n$ are trivial, then $\Susp A$ will have trivial homotopy groups below $1 + n$.