Definition 3.3.3.1 (Subdivision). Let $X$ and $Y$ be simplicial sets. We will say that a morphism of simplicial sets $u: X \rightarrow \operatorname{Ex}(Y)$ exhibits $Y$ as a subdivision of $X$ if, for every simplicial set $Z$, composition with $u$ induces a bijection $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( Y, Z ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( X, \operatorname{Ex}(Z) )$ (see Construction 3.3.2.5).
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