Kerodon

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Construction 4.5.8.1. Let $X$ and $Y$ be simplicial sets, let $\pi _{X}: X \times Y \rightarrow X$ and $\pi _{Y}: X \times Y \rightarrow Y$ denote the projection maps, and let $\iota _{X}: X \hookrightarrow X \star Y$ and $\iota _{Y}: Y \hookrightarrow X \star Y$ denote the inclusion maps. Then there is a unique map of simplicial sets $c: X \times \Delta ^1 \times Y \rightarrow X \star Y$ with the property that $c|_{ X \times \{ 0\} \times Y} = \iota _{X} \circ \pi _{X}$ and $c|_{ X \times \{ 1\} \times Y} = \iota _{Y} \circ \pi _{Y}$. Concretely, if $\sigma = ( \sigma _{X}, \sigma _{ \Delta ^1}, \sigma _{Y} )$ is an $n$-simplex of the product $X \times \Delta ^1 \times Y$, then $c(\sigma )$ is the $n$-simplex of $X \star Y$ given by the composition

\[ \Delta ^ n \simeq ( \sigma _{\Delta ^1}^{-1}\{ 0\} ) \star ( \sigma _{\Delta ^1}^{-1} \{ 1\} ) \xrightarrow {\sigma _{X} \star \sigma _{Y} } X \star Y. \]

We will refer to $c: X \times \Delta ^1 \times Y \rightarrow X \star Y$ as the collapse map.