Kerodon

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

Example 1.4.1.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be a topological space. Then we have a canonical bijection

\[ \xymatrix { \{ \text{Functors of $\infty $-categories from $\operatorname{\mathcal{C}}$ to $\operatorname{Sing}_{\bullet }(X)$} \} \ar [d]^{\sim } \\ \{ \text{Continuous functions from $| \operatorname{\mathcal{C}}|$ to $X$} \} . } \]

Here $| \operatorname{\mathcal{C}}|$ denotes the geometric realization of the simplicial set $\operatorname{\mathcal{C}}$ (see Definition 1.1.8.1). Beware that neither side has an obvious interpretation in terms of functors between ordinary categories (even in the special case where $\operatorname{\mathcal{C}}$ is the nerve of a category).