Remark 3.5.2.7. For each integer $n \geq 0$, the topological $n$-simplex $| \Delta ^ n |$ can be identified with the filtered direct limit $\varinjlim _{S} | \Delta ^ n |_{S}$, where $S$ ranges over the collection of all finite subsets of $[0,1]$ which contain the endpoints $0$ and $1$ (which we regard as a partially ordered set with respect to inclusion). We therefore obtain a canonical isomorphism of cosimplicial sets $\varinjlim _{S} | \Delta ^{\bullet } |_{S} \xrightarrow {\sim } | \Delta ^{\bullet } |$. It follows that, for every simplicial set $X$, the canonical map $\varinjlim _{S} | X|_{S} \rightarrow |X|$ is a bijection.

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