loop space (Rev #2, changes)

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

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

Definition 2.1.8Given a pointed type $(A,a)$, we define theloop spaceof $(A,a)$ to be the following pointed type:$\Omega(A,a) = ((a =_A a),\,refl_a)$.

An element of it will be called a

loopat $a$. For $n\,:\,\mathbb{N}$, then-folditerated loop space$\Omega^n(A,a)$ of a pointed type $(A,a)$ is defined recursively by:$\Omega^0(A,a) = (A,a)$

$\Omega^{n+1}(A,a) = \Omega^n(\Omega(A,a))$.An element of it will be called an

n-loopor ann-dimensionalloopat $a$.