Theorem 3.6.2.1 (013D)

Theorem 3.6.2.1. The geometric realization functor

\[ \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set}\quad \quad X \mapsto |X| \]

preserves finite limits. In particular, for every diagram of simplicial sets $X \rightarrow Z \leftarrow Y$, the induced map $| X \times _{Z} Y | \rightarrow |X| \times _{|Z|} |Y|$ is a bijection.

Proof of Theorem 3.6.2.1. Let $U: \operatorname{Top}\rightarrow \operatorname{Set}$ denote the forgetful functor. We wish to show that the composite functor

\[ \operatorname{Set_{\Delta }}\xrightarrow { | \bullet | } \operatorname{Top}\xrightarrow {U} \operatorname{Set} \]

preserves finite limits. By virtue of Remark 3.6.2.7, we can write this composite functor as a filtered colimit of functors of the form $X \mapsto |X|_{S}$, where $S$ ranges over all finite subsets of the unit interval $[0,1]$ which contain $0$ and $1$. It will therefore suffice to show that each of the functors $X \mapsto |X|_{S}$ preserves finite limits. Using Proposition 3.6.2.9, see that $X \mapsto |X|_{S}$ can be identified with the evaluation functor $X \mapsto X_{m}$, where $m$ is chosen so that there is an isomorphism of linearly ordered sets $[m] \simeq \pi _0( [0,1] \setminus S)$. $\square$