Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 2.4.1.5 (Topological Spaces). Let $\operatorname{Top}$ denote the category whose objects are topological spaces and whose morphisms are continuous functions. Then $\operatorname{Top}$ can be promoted to a simplicial category $\operatorname{Top}_{\bullet }$: given a pair of topological spaces $X$ and $Y$, we define the simplicial set $\operatorname{Hom}_{\operatorname{Top}}(X,Y)_{\bullet }$ informally by the formula

\[ \operatorname{Hom}_{\operatorname{Top}}(X,Y)_{n} = \operatorname{Hom}_{\operatorname{Top}}( | \Delta ^ n | \times X, Y) \]

In particular, a vertex of $\operatorname{Hom}_{\operatorname{Top}}(X,Y)_{\bullet }$ can be identified with a continuous function $f: X \rightarrow Y$. Moreover, for any topological space $Y$, we have a canonical isomorphism of simplicial sets $\operatorname{Hom}_{\operatorname{Top}}( \ast , Y )_{\bullet } \simeq \operatorname{Sing}_{\bullet }(Y)$, where $\operatorname{Sing}_{\bullet }(Y)$ is the singular simplicial set of Construction 1.1.7.1.