Kerodon

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Definition 1.1.8.1. Let $S_{\bullet }$ be a simplicial set and let $Y$ be a topological space. We will say that a map of simplicial sets $u: S_{\bullet } \rightarrow \operatorname{Sing}_{\bullet }(Y)$ exhibits $Y$ as a geometric realization of $S_{\bullet }$ if, for every topological space $X$, the composite map

$\operatorname{Hom}_{\operatorname{Top}}( Y, X) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Sing}_{\bullet }( Y ), \operatorname{Sing}_{\bullet }(X) ) \xrightarrow { \circ u} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S_{\bullet }, \operatorname{Sing}_{\bullet }(X) )$

is bijective.