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Construction (The Category of Simplices of a Simplicial Set). Let $S_{\bullet }$ be a simplicial set. We define a category $\operatorname{{\bf \Delta }}_{S}$ as follows:

  • The objects of $\operatorname{{\bf \Delta }}_{S}$ are pairs $([n], \sigma )$, where $[n]$ is an object of $\operatorname{{\bf \Delta }}$ and $\sigma $ is an $n$-simplex of $S_{\bullet }$.

  • A morphism from $([n], \sigma )$ to $([n'], \sigma ')$ in the category $\operatorname{{\bf \Delta }}_{S}$ is a nondecreasing function $f: [n] \rightarrow [n']$ with the property that the induced map $S_{n'} \rightarrow S_{n}$ carries $\sigma '$ to $\sigma $.

We will refer to $\operatorname{{\bf \Delta }}_{S}$ as the category of simplices of $S_{\bullet }$.