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Variant 5.4.2.9. Let $\operatorname{\mathcal{C}}$ be a simplicial category. We define a simplicial category $\operatorname{\mathcal{C}}^{\triangleright }$ as follows:

  • The set of objects $\operatorname{Ob}(\operatorname{\mathcal{C}}^{\triangleright })$ is given by the (disjoint) union $\operatorname{Ob}(\operatorname{\mathcal{C}}) \cup \{ Y_0 \} $, where $Y_0$ is an auxiliary symbol.

  • The simplicial morphism sets in $\operatorname{\mathcal{C}}^{\triangleright }$ are given by

    \[ \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\triangleright }}( C,D )_{\bullet } = \begin{cases} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)_{\bullet } & \textnormal{ if } C,D \in \operatorname{Ob}(\operatorname{\mathcal{C}}) \\ \Delta ^{0} & \textnormal{ if } D = Y_0 \\ \emptyset & \textnormal{ otherwise. } \end{cases} \]
  • For objects $C,D,E \in \operatorname{Ob}( \operatorname{\mathcal{C}}^{\triangleright } )$, the composition law

    \[ \circ : \operatorname{Hom}_{ \operatorname{\mathcal{C}}^{\triangleright } }( D, E)_{\bullet } \times \operatorname{Hom}_{ \operatorname{\mathcal{C}}^{\triangleright } }( C, D)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{C}}^{\triangleright } }( C, E)_{\bullet } \]

    is given by the composition law on $\operatorname{\mathcal{C}}$ in the case where $C,D,E \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, and is otherwise uniquely determined.

More informally, the simplicial category $\operatorname{\mathcal{C}}^{\triangleright }$ is obtained from $\operatorname{\mathcal{C}}$ by adjoining a (new) final object $Y_0$. We will refer to $\operatorname{\mathcal{C}}^{\triangleright }$ as the right cone on $\operatorname{\mathcal{C}}$, and to the object $Y_0 \in \operatorname{\mathcal{C}}^{\triangleright }$ as the cone point.