The HoTT book defines a ‘loop space’ in definitions 2.1.7 and 2.1.8:
Definition 2.1.7 A pointed type is a type together with a point , called its basepoint. We write for the type of pointed types in the universe .
Definition 2.1.8 Given a pointed type , we define the loop space of to be the following pointed type:
.
An element of it will be called a loop at . For , the n-fold iterated loop space of a pointed type is defined recursively by:
.An element of it will be called an n-loop or an n-dimensional loop at .