Kerodon

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Proposition 8.1.4.7. Let $\operatorname{\mathcal{D}}$ be a simplicial set and let $u: \operatorname{\mathcal{D}}\rightarrow \operatorname{Cospan}( \operatorname{Tw}(\operatorname{\mathcal{D}}) )$ be the unit map of Construction 8.1.4.6. For every simplicial set $\operatorname{\mathcal{C}}$, the composite map

\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}}) & \rightarrow & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Cospan}( \operatorname{Tw}(\operatorname{\mathcal{C}}) ), \operatorname{Cospan}(\operatorname{\mathcal{D}}) ) \\ & \xrightarrow { \circ u} & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\mathcal{D}}, \operatorname{Cospan}(\operatorname{\mathcal{C}}) ) \end{eqnarray*}

is a bijection.

Proof. Let us regard the simplicial set $\operatorname{\mathcal{C}}$ as fixed. For every simplicial set $\operatorname{\mathcal{D}}$, the unit map $u$ of Construction 8.1.4.6 determines a function

\[ \theta _{\operatorname{\mathcal{D}}}: \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{D}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\mathcal{D}}, \operatorname{Cospan}(\operatorname{\mathcal{C}}) ). \]

Using Remark 8.1.1.4, we see that the construction $\operatorname{\mathcal{D}}\mapsto \theta _{\operatorname{\mathcal{D}}}$ carries colimits (in the category of simplicial sets) to limits (in the arrow category $\operatorname{Fun}( [1], \operatorname{Set})$). Consequently, to show that $\theta _{\operatorname{\mathcal{D}}}$ is a bijection, we may assume without loss of generality that $\operatorname{\mathcal{D}}= \Delta ^ n$ is a standard simplex (see Corollary 1.1.8.17). In this case, the desired result follows immediately from the definition of the simplicial set $\operatorname{Cospan}(\operatorname{\mathcal{C}})$. $\square$