Warning 8.1.1.9 (03LW)

Warning 8.1.1.9. Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then there is a tautological map $T: \operatorname{\mathcal{C}}\rightarrow \operatorname{Tw}( \operatorname{\mathcal{C}}^{\operatorname{op}} \star \operatorname{\mathcal{C}})$, which carries an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ to the $n$-simplex $T(\sigma )$ of $\operatorname{Tw}( \operatorname{\mathcal{C}}^{\operatorname{op}} \star \operatorname{\mathcal{C}})$ given by the composition

\[ \operatorname{N}_{\bullet }( [n]^{\operatorname{op}} \star [n] ) \xrightarrow { \sigma ^{\operatorname{op}} \star \sigma } \operatorname{\mathcal{C}}^{\operatorname{op}} \star \operatorname{\mathcal{C}}. \]

If $\operatorname{\mathcal{D}}$ is another simplicial set, then precomposition with $T$ induces a comparison map

\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\mathcal{C}}^{\operatorname{op}} \star \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Tw}(\operatorname{\mathcal{C}}^{\operatorname{op}} \star \operatorname{\mathcal{C}}), \operatorname{Tw}(\operatorname{\mathcal{D}}) ) \xrightarrow { \circ T} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\mathcal{C}}, \operatorname{Tw}(\operatorname{\mathcal{D}}) ). \]

Beware that, in general, this map is not a bijection. However, it is a bijection whenever $\operatorname{\mathcal{C}}$ is isomorphic to the nerve of a linearly ordered set $Q$. To prove this, we can write $Q$ as a filtered colimit of its finite subsets and thereby reduce to the case where $Q$ is finite. In this case, the linearly ordered set $Q$ is either empty (in which case the desired result is obvious) or isomorphic to $[n]$ for some integer $n \geq 0$ (in which case the desired result follows from the definition of the $\operatorname{Tw}(\operatorname{\mathcal{D}})$).