# Kerodon

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Example 3.4.0.5. Let $f_0: X_0 \rightarrow X$ and $f_1: X_1 \rightarrow X$ be continuous functions between topological spaces. We let $X_0 \times ^{\mathrm{h}}_{X} X_1$ denote the set of all triples $(x_0, x_1, p)$ where $x_0$ is a point of $X_0$, $x_1$ is a point of $X_1$, and $p: [0,1] \rightarrow X$ is a continuous function satisfying $p(0) = f_0(x_0)$ and $p(1) = f_1(x_1)$. We will refer to $X_0 \times ^{\mathrm{h}}_{X} X_1$ as the homotopy fiber product of $X_0$ with $X_1$ over $X$. The homotopy fiber product $X_1 \times ^{\mathrm{h}}_{X} X_1$ carries a natural topology, given by viewing it as a subspace of the product $X_0 \times X_1 \times \operatorname{Hom}_{\operatorname{Top}}( [0,1], X)$ (where we endow the path space $\operatorname{Hom}_{\operatorname{Top}}([0,1],X)$ with the compact-open topology). We then have a canonical isomorphism of simplicial sets

$\operatorname{Sing}_{\bullet }( X_0 \times _{X}^{\mathrm{h}} X_1) \simeq \operatorname{Sing}_{\bullet }(X_0) \times ^{\mathrm{h}}_{ \operatorname{Sing}_{\bullet }(X) } \operatorname{Sing}_{\bullet }(X_1)$

where the right hand side is the homotopy fiber product of Kan complexes given in Construction 3.4.0.3.