# Kerodon

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Example 4.3.5.10. Let $Y$ be a topological space equipped with a base point $y$. Let $P = \{ p: [0,1] \rightarrow Y \}$ denote the collection of all continuous functions from the unit interval $[0,1]$ to $Y$, and let $P_{y} = \{ p \in P: p(1) = y \}$ denote the subset of $P$ consisting of those continuous paths which end at the point $y$. We regard $P$ as a topological space by equipping it with the compact-open topology, so the singular simplicial set $\operatorname{Sing}_{\bullet }(P)$ can be identified with $\operatorname{Fun}( \Delta ^1, \operatorname{Sing}_{\bullet }(Y) )$ (see Warning 2.4.2.18). Identifying $y$ with a vertex of the singular simplicial set $\operatorname{Sing}_{\bullet }(Y)$, Example 4.3.5.9 supplies an isomorphism of simplicial sets

$\operatorname{Sing}_{\bullet }(Y)_{/y} \simeq \operatorname{Sing}_{\bullet }(P) \times _{ \operatorname{Sing}_{\bullet }(Y) } \{ y\} = \operatorname{Sing}_{\bullet }(P_ y).$

In particular, since the topological space $P_ y$ is contractible, the simplicial set $\operatorname{Sing}_{\bullet }(Y)_{/y}$ is a contractible Kan complex (this is a special case of a general phenomenon: see Corollary 4.3.7.14).