The loop space of a pointed type is simply the type of paths from a point to itself i.e. loops.

Definition

Given a pointed type $(A,\star_A)$ the loop space is the type $\Omega(A,\star_A)\equiv \star_{A} =_{A} \star_{A}$ and has basepoint $refl_{\star A}$.

The $n$-fold iterated loop space$\Omega^n(A,\star_A)$ can be defined by induction on $n$: * $\Omega^0(A,\star_A)=(A,\star_A)$ * $\Omega^{n+1}(A,\star_A)=\Omega^n(\Omega(A,\star_A))$

The HoTT book defines a ‘loop space’ in definitions 2.1.7 and 2.1.8:

Definition 2.1.7 A pointed type$(A,a)$ is a type $A\,:\,\mathcal{U}$ together with a point $a\,:\,A$, called its basepoint. We write $\mathcal{U}_\bullet :\equiv \sum_{(A:\mathcal{U})}A$ for the type of pointed types in the universe $\mathcal{U}$.

Definition 2.1.8 Given a pointed type $(A,a)$, we define the loop space of $(A,a)$ to be the following pointed type:

$\Omega(A,a) = ((a =_A a),\,refl_a)$.

An element of it will be called a loop at $a$. For $n\,:\,\mathbb{N}$, the n-folditerated loop space$\Omega^n(A,a)$ of a pointed type $(A,a)$ is defined recursively by: