Kerodon

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Example 4.6.1.12 (Loop Spaces). Let $(X,x)$ be a pointed Kan complex. The Kan complex $\operatorname{Hom}_{X}(x,x)$ is often denoted by $\Omega (X)$ and referred to as the based loop space of $X$. Note that it can be identified with the fiber over $x$ of the evaluation map

\[ q: \{ x\} \times _{ \operatorname{Fun}(\{ 0\} , X) } \operatorname{Fun}( \Delta ^1, X) \rightarrow \operatorname{Fun}(\{ 1\} , X) = X. \]

By virtue of Example 3.1.7.10, this map is a Kan fibration whose domain is a contractible Kan complex. It follows that the long exact sequence of Theorem 3.2.5.1 yields isomorphisms $\pi _{n}( \operatorname{Hom}_{X}(x,x), \operatorname{id}_{x} ) \simeq \pi _{n+1}(X,x)$ for $n \geq 0$.