$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 5.3.8.10. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a Kan fibration of simplicial sets. The following conditions are equivalent:
- $(1)$
The Kan fibration $U$ is minimal.
- $(2)$
For every simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$, the fiber product $\operatorname{\mathcal{E}}_{\sigma } = \Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is isomorphic to $\Delta ^ n \times X$, for some minimal Kan complex $X$.
Proof.
We will show that $(1)$ implies $(2)$; the reverse implication is immediate from the definitions. Assume that $U$ is minimal and fix an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$. Applying Corollary 5.3.8.9, we deduce that there is an isomorphism of simplicial sets $\operatorname{\mathcal{E}}_{\sigma } \simeq \operatorname{N}_{\bullet }^{\mathscr {F}}([n])$, for some diagram $\mathscr {F}: [n] \rightarrow \operatorname{Set_{\Delta }}$ which carries each integer $i \in [n]$ to a minimal Kan complex $\mathscr {F}(i)$. For $i \leq j$, the map of Kan complexes $e: \mathscr {F}(i) \rightarrow \mathscr {F}(j)$ is given by covariant transport along the corresponding edge of $\operatorname{\mathcal{C}}$ (Remark 5.3.3.22), and is therefore a homotopy equivalence (Example 5.2.5.9). Since $\mathscr {F}(i)$ and $\mathscr {F}(j)$ are minimal Kan complexes, it follows that $e$ is an isomorphism of simplicial sets (Proposition 4.7.6.13). It follows that $\mathscr {F}$ is isomorphic to the constant diagram $\underline{X}$ (where $X = \mathscr {F}(0)$ is a minimal Kan complex), so that $\operatorname{\mathcal{E}}_{\sigma }$ is isomorphic to $\operatorname{N}_{\bullet }^{\underline{X}}([n]) = \Delta ^ n \times X$.
$\square$