Kerodon

Remark 5.3.8.12 (Uniqueness of Minimal Models). In the situation of Proposition 5.3.8.11, $\operatorname{\mathcal{E}}^{0}$ is unique up to isomorphism as an object of the slice category $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}}$ (but not unique when regarded as a simplicial subset of $\operatorname{\mathcal{E}}$). More precisely, suppose we are given another simplicial subset $\operatorname{\mathcal{E}}^{1} \subseteq \operatorname{\mathcal{E}}$ which satisfies conditions $(1)$ and $(2)$ of Proposition 5.3.8.11. In particular, the inclusion map $\operatorname{\mathcal{E}}^{1} \hookrightarrow \operatorname{\mathcal{E}}$ is an equivalence of inner fibrations over $\operatorname{\mathcal{C}}$, and therefore admits a homotopy inverse $F: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}^{1}$ The restriction $F|_{ \operatorname{\mathcal{E}}^{0}}: \operatorname{\mathcal{E}}^{0} \rightarrow \operatorname{\mathcal{E}}^{1}$ is then an equivalence of minimal inner fibrations over $\operatorname{\mathcal{C}}$, and therefore an isomorphism of simplicial sets (Remark 5.3.8.6). Moreover, $F$ is isomorphic to the identity map $\operatorname{id}_{\operatorname{\mathcal{E}}^{0}}$ as an object of the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}^{0}, \operatorname{\mathcal{E}})$.