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Construction (Slices of Simplicial-Categories). Let $\operatorname{\mathcal{C}}$ be a simplicial category and let $X$ be an object of $\operatorname{\mathcal{C}}$. We define a simpicial category $\operatorname{\mathcal{C}}_{/X}$ as follows:

  • The objects of $\operatorname{\mathcal{C}}_{/X}$ are pairs $(C,f)$, where $C$ is an object of $\operatorname{\mathcal{C}}$ and $f: C \rightarrow X$ is a vertex of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)_{\bullet }$.

  • Let $(C,f)$ and $(D,g)$ be objects of $\operatorname{\mathcal{C}}_{/X}$. We let $\operatorname{Hom}_{\operatorname{\mathcal{C}}_{/X}}( (C,f), (D,g) )_{\bullet }$ denote the simplicial set given by the fiber product

    \[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)_{\bullet } \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)_{\bullet } } \{ f \} , \]

    which we regard as a simplicial subset of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)_{\bullet }$. More precisely, we let $\operatorname{Hom}_{\operatorname{\mathcal{C}}_{/X}}( (C,f), (D,g) )_{\bullet }$ denote the simplicial subset of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)_{\bullet }$ consisting of those $n$-simplices $\sigma $ for which the composite map is equal to the constant map $\Delta ^{n} \twoheadrightarrow \{ f \} $.

  • Let $(C,f)$, $(D,g)$, and $(E,h)$ be objects of $\operatorname{\mathcal{C}}_{/X}$. Then the composition law

    \[ \circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}_{/X}}( (D,g) , (E,h) )_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}_{/X}}( (C,f) , (D,g) )_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}_{/X}}( (C,f) , (E,h) )_{\bullet } \]

    for the simplicial category $\operatorname{\mathcal{C}}_{/X}$ is given by the restriction of the composition law

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

    for the simplicial category $\operatorname{\mathcal{C}}$.