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Construction (Slice Simplicial Sets). Let $f: K \rightarrow X$ be a morphism of simplicial sets. We define a simplicial set $X_{/f}$ as follows:

  • For each $n \geq 0$, an $n$-simplex of $X_{/f}$ is a map of simplicial sets $\overline{f}: \Delta ^ n \star K \rightarrow X$ satisfying $\overline{f}|_{K} = f$.

  • For every nondecreasing function $\alpha : [m] \rightarrow [n]$ in $\operatorname{{\bf \Delta }}$, the associated map

    \[ \alpha ^{\ast }: \{ \textnormal{$n$-simplices of $X_{/f}$} \} \rightarrow \{ \textnormal{$m$-simplices of $X_{/f}$} \} \]

    carries an $n$-simplex $\overline{f}: \Delta ^ n \star K \rightarrow X$ to the composite map

    \[ \Delta ^{m} \star K \xrightarrow { \alpha \star \operatorname{id}_ K} \Delta ^{n} \star K \xrightarrow { \overline{f} } X. \]

We will refer to $X_{/f}$ as the slice simplicial set of $X$ over $f$.