Definition 1.2.3.1. Let $S$ be a simplicial set and let $Y$ be a topological space. We will say that a map of simplicial sets $u: S\rightarrow \operatorname{Sing}_{\bullet }(Y)$ exhibits $Y$ as a geometric realization of $S$ if, for every topological space $X$, the composite map
\[ \operatorname{Hom}_{\operatorname{Top}}( Y, X) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Sing}_{\bullet }( Y ), \operatorname{Sing}_{\bullet }(X) ) \xrightarrow { \circ u} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S, \operatorname{Sing}_{\bullet }(X) ) \]
is a bijection.