Suspension (Topology)

Definition

1. $X$ is n-truncated (i.e., all homotopy groups above degree $n$ vanish; equivalently, for all $xy:X$, the identity type $(x=y)$ is $(n-1)$-truncated).
2. For every base point $x_0:X$, the $(n+1)$-fold loop space ${\Omega }^{n+1}(X,x_0)$ is contractible.
3. For every base point $x_0:X$, the type of pointed maps $S^{n+1}{\to }_{\ast }(X,x_0)$ is contractible (every pointed map is homotopic to the constant map at $x_0$).

Examples

• n = -2 (contractible types).
  $X$ is contractible $\iff$ the canonical map $X\to \top$ is an equivalence (i.e., $X\simeq \top$).
• n = -1 (propositions).
  $X$ is a proposition $\iff$ the diagonal map $\Delta :X\to X\times X,x\mapsto (x,x)$ is an equivalence. (For a prop, $X\times X\simeq X$ via either projection, and $\Delta$ is a quasi-inverse.)
• n = 0 (sets). $X$ is a set $\iff$ for all $xy:X$, the identity type $(x=y)$ is a proposition (i.e., ${\pi }_1$ is trivial, no higher paths).

Lemma. $X$ is $n$-truncated $\iff$ for all $xy:X$, the identity type $(x=y)$ is $(n-1)$-truncated.
This gives an inductive way to check truncation: props at level $-1$, sets at level $0$, etc.

Sources

https://ncatlab.org/nlab/show/suspension+type https://chatgpt.com/c/690127a4-60d4-8332-b933-8c6e1d45187d