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Variant (Coslice 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 $\overline{f}: K \star \Delta ^ n \rightarrow X$ 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 $\overline{f}: K \star \Delta ^ n \rightarrow X$ to the composite map

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

We will refer to $X_{f/}$ as the coslice simplicial set of $X$ under $f$. As in Remark, it is equipped with a projection map $X_{f/} \rightarrow X$.