Kerodon

Remark 1.1.8.21. Each of our proofs of Corollary 1.1.8.17 gives additional information that the other does not. Our first proof shows that every simplicial set $S_{\bullet }$ can be built as a colimit of standard simplices in a very specific way: namely, by forming pushouts along boundary inclusions $\operatorname{\partial \Delta }^ n \hookrightarrow \Delta ^ n$ (for a more precise assertion, see the proof of Proposition 1.4.5.13). This extra information was used in the proof of Proposition 1.1.8.4 to show that the geometric realization $| S_{\bullet } |$ is a CW complex (and not merely a topological space which is colimit of disks). On the other hand, our second proof shows that every simplicial set $S_{\bullet }$ can be built in a single step as the colimit of a diagram of standard simplices (which can be chosen in a specific, canonical way).