Definition 13.1.6 (Twisted arrow category). For \(n \geq 0\), let \(\Tw ^r([n])\) be the partially ordered set of tuples \((i,j)\) with \(0 \leq i \leq j \leq n\), where the partial order is given by \[ (i,j) \leq (k,l) \qquadtext { if and only if} i \leq k \quad \text {and} \quad l \leq j. \] We will generally denote the pair \((i,j)\) by \((i \leq j)\). The poset looks as follows:
Given a morphism of posets \(\phi \colon [n] \to [m]\), we obtain a map \(\Tw ^r(\phi )\colon \Tw ^r([n]) \to \Tw ^r([m])\) by sending \((i \leq j)\) to \((\phi (i) \leq \phi (j))\). This assignment is clearly functorial, resulting in a functor \[ \Tw ^r\colon \simp \to \mathrm {Poset} \subseteq \Cat \subseteq \Cat _{\infty }. \]
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