# Kerodon

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Construction 8.1.2.7. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $X$ be a vertex of $\operatorname{\mathcal{C}}$. Let $\sigma$ be an $n$-simplex of the coslice simplicial set $\operatorname{\mathcal{C}}_{X/}$, which we identify with a morphism of simplicial sets $\{ x\} \star \Delta ^{n} \rightarrow \operatorname{\mathcal{C}}$ satisfying $\sigma (x) = X$. Then the composite map

$( \Delta ^{n} )^{\operatorname{op}} \star \Delta ^{n} \twoheadrightarrow \{ x\} \star \Delta ^{n} \xrightarrow {\sigma } \operatorname{\mathcal{C}}$

can be identified with an $n$-simplex of the twisted arrow simplicial set $\operatorname{Tw}(\operatorname{\mathcal{C}})$, which we will denote by $\iota _{X}( \sigma )$. The construction $\sigma \mapsto \iota _{X}( \sigma )$ is compatible with the formation of face and degeneracy operators, and therefore determines a morphism of simplicial sets $\iota _{X}: \operatorname{\mathcal{C}}_{X/} \rightarrow \operatorname{Tw}(\operatorname{\mathcal{C}})$. Moreover, the diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{X/} \ar [r]^-{ \iota _{X} } \ar [d] & \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [d]^{ \lambda _{-} } \\ \{ X\} \ar [r] & \operatorname{\mathcal{C}}^{\operatorname{op}} }$

commutes, where $\lambda _{-}$ is the projection map of Notation 8.1.1.6. It follows that $\iota _{X}$ can be regarded as a morphism of simplicial sets from $\operatorname{\mathcal{C}}_{X/}$ to the fiber $\{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$. We will refer to this morphism as the coslice inclusion.