Kerodon

Remark 8.5.8.5. In the situation of Theorem 8.5.8.5, we can regard $\iota $ and $V|_{ \operatorname{N}_{\leq n}(\operatorname{Idem}) }$ as morphisms from $\operatorname{N}_{\leq n}(\operatorname{Idem})$ to $\operatorname{\mathcal{E}}$. By construction, these morphisms coincide after composing with the functor $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$. Since $U$ is bijective on simplices of dimensions $< n$, it follows that $\iota $ and $V|_{ \operatorname{N}_{\leq n}( \operatorname{Idem}) }$ coincide when on the $(n-1)$-skeleton of $\operatorname{N}_{\bullet }( \operatorname{Idem})$. Beware that $\iota $ and $V$ do not coincide on the nondegenerate $n$-simplex of $\operatorname{N}_{\bullet }( \operatorname{Idem})$. In fact, we claim that $\iota $ and $V|_{ \operatorname{N}_{\leq n}( \operatorname{Idem}) }$ are not even isomorphic when viewed as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{N}_{\leq n}( \operatorname{Idem}), \operatorname{\mathcal{E}})$. Assume, for a contradiction, that there exists an isomorphism $\alpha $ of $\iota = \operatorname{id}_{\operatorname{\mathcal{E}}} \circ \iota $ with $V|_{ \operatorname{N}_{\leq n}( \operatorname{Idem}) } = (V \circ U) \circ \iota $. Since $\iota $ is inner anodyne, we could then lift $\alpha $ to an isomorphism $\widetilde{\alpha }: \operatorname{id}_{\operatorname{\mathcal{E}}} \rightarrow V \circ U$ in the $\infty $-category $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Idem}), \operatorname{\mathcal{C}})$. It would follow that $U$ is an equivalence of $\infty $-categories (with homotopy inverse given by $V$). Then $(U \circ \iota ): \operatorname{N}_{\leq n}( \operatorname{Idem}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{Idem})$ would be a categorical equivalence of simplicial sets, which contradicts Remark 8.5.4.18.