Kerodon

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Construction 5.4.2.15. Let $\operatorname{\mathcal{C}}$ be a simplicial category and let $Y$ be an object of $\operatorname{\mathcal{C}}$. We define a simplicial functor $V: ( \operatorname{\mathcal{C}}_{/Y })^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ as follows:

  • The functor $V$ carries each object $(C,f) \in \operatorname{\mathcal{C}}_{/Y}$ to the object $C \in \operatorname{\mathcal{C}}$, and carries the cone point $Y_0 \in (\operatorname{\mathcal{C}}_{/Y})^{\triangleright }$ to the object $Y \in \operatorname{\mathcal{C}}$.

  • If $(C,f)$ and $(D,g)$ are objects of $\operatorname{\mathcal{C}}_{/Y}$, then the induced map of simplicial sets

    \[ \operatorname{Hom}_{(\operatorname{\mathcal{C}}_{/Y})^{\triangleright } }( (C,f), (D,g) )_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( V(C,f), V(D,g) )_{\bullet } \]

    is equal to the inclusion map $\operatorname{Hom}_{ \operatorname{\mathcal{C}}_{/Y} }( (C,f), (D,g) )_{\bullet } \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, D)_{\bullet }$.

  • If $(C,f)$ is an object of $\operatorname{\mathcal{C}}_{/Y}$, then the induced map

    \[ \Delta ^{0} = \operatorname{Hom}_{(\operatorname{\mathcal{C}}_{/Y})^{\triangleright } }( (C,f), Y_0 )_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( V(C,f), V(Y_0) )_{\bullet } = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,Y)_{\bullet } \]

    is equal to the vertex $f$.

We will refer to $V$ as the right cone contraction functor. Similarly, to every object $X \in \operatorname{\mathcal{C}}$ we can associate a simplicial functor $V': ( \operatorname{\mathcal{C}}_{X/} )^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ carrying the cone point of $( \operatorname{\mathcal{C}}_{X/})^{\triangleleft }$ to the object $X$, which we will refer to as the left cone contraction functor.