Kerodon

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Corollary 3.3.1.8. Let $\operatorname{Set}_{\Delta }^{\mathrm{br}} \subseteq \operatorname{Set_{\Delta }}$ denote the (non-full) subcategory whose objects are braced simplicial sets and whose morphisms are maps $f: X_{\bullet } \rightarrow Y_{\bullet }$ which carry nondegenerate simplices of $X_{\bullet }$ to nondegenerate simplices of $Y_{\bullet }$. Then the construction $X_{\bullet } \mapsto X_{\bullet }^{\mathrm{nd}}$ induces an equivalence of categories $\operatorname{Set}_{\Delta }^{\mathrm{br}} \rightarrow \{ \textnormal{Semisimplicial sets} \}$, with homotopy inverse given by the construction $S_{\bullet } \rightarrow S_{\bullet }^{+}$.

Proof. Let $X_{\bullet }$ and $Y_{\bullet }$ be braced simplicial sets. It follows from Proposition 3.3.1.5 that the restriction functor $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( X_{\bullet }, Y_{\bullet } ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}}, \operatorname{Set})}( X_{\bullet }^{\mathrm{nd}}, Y_{\bullet } )$ is a bijection. Moreover, the image of $\operatorname{Hom}_{\operatorname{Set}_{\Delta }^{\mathrm{br}}}( X_{\bullet }, Y_{\bullet } )$ under this bijection is the collection of morphisms of semisimplicial sets from $X_{\bullet }^{\mathrm{nd}}$ to $Y_{\bullet }^{\mathrm{nd}} \subseteq Y_{\bullet }$. This proves full-faithfulness, and the essential surjectivity follows from Proposition 3.3.1.7. $\square$