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Notation 5.4.2.8. Let $\operatorname{\mathcal{C}}$ be a simplicial category. We define a simplicial category $\operatorname{\mathcal{C}}^{\triangleleft }$ as follows:

  • The set of objects $\operatorname{Ob}(\operatorname{\mathcal{C}}^{\triangleleft })$ is the (disjoint) union $\operatorname{Ob}(\operatorname{\mathcal{C}}) \cup \{ X_0 \} $, where $X_0$ is an auxiliary symbol.

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

    \[ \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\triangleleft }}( C,D )_{\bullet } = \begin{cases} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)_{\bullet } & \textnormal{ if } C,D \in \operatorname{Ob}(\operatorname{\mathcal{C}}) \\ \Delta ^{0} & \textnormal{ if } C = X_0 \\ \emptyset & \textnormal{ otherwise. } \end{cases} \]
  • For objects $C,D,E \in \operatorname{Ob}( \operatorname{\mathcal{C}}^{\triangleleft } )$, the composition law

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

    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 (since either the left hand side is empty or the right hand side is $\Delta ^0$).

More informally, the simplicial category $\operatorname{\mathcal{C}}^{\triangleleft }$ is obtained from $\operatorname{\mathcal{C}}$ by adjoining a (new) initial object $X_0$. We will refer to $\operatorname{\mathcal{C}}^{\triangleleft }$ as the left cone on $\operatorname{\mathcal{C}}$, and to the object $X_0 \in \operatorname{\mathcal{C}}^{\triangleleft }$ as the cone point.