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Construction 8.1.1.1 (Twisted Arrows in Simplicial Sets). Let $\operatorname{{\bf \Delta }}$ denote the simplex category (Definition 1.1.1.2) and let $\operatorname{\mathcal{C}}$ be a simplicial set. We let $\operatorname{Tw}(\operatorname{\mathcal{C}}): \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$ denote the functor given by the construction

\[ (J \in \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \mapsto \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }( J^{\operatorname{op}} \star J ), \operatorname{\mathcal{C}}). \]

We will refer to $\operatorname{Tw}(\operatorname{\mathcal{C}})$ as the simplicial set of twisted arrows of $\operatorname{\mathcal{C}}$.