# Kerodon

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Lemma 6.3.2.4. Let $Q$ be a contractible Kan complex, let $e: \Delta ^1 \hookrightarrow Q$ be a monomorphism of simplicial sets, and let $W = \{ \operatorname{id}_{ \Delta ^1} \}$ consist of the single nondegenerate edge of $\Delta ^1$. Then, for any $\infty$-category $\operatorname{\mathcal{E}}$, precomposition with $e$ induces a trivial Kan fibration of simplicial sets

$\theta : \operatorname{Fun}(Q, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}( \Delta ^1[ W^{-1}], \operatorname{\mathcal{E}}) = \operatorname{Isom}(\operatorname{\mathcal{E}}).$

Proof. Since $e$ is a monomorphism, Corollary 4.4.5.3 immediately implies that $\theta$ is an isofibration when regarded as a functor from $\operatorname{Fun}(Q,\operatorname{\mathcal{E}})$ to $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}})$. Using the pullback diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(Q, \operatorname{\mathcal{E}}) \ar [d]^{\theta } \ar [r] & \operatorname{Fun}(Q, \operatorname{\mathcal{E}}) \ar [d]^{\theta } \\ \operatorname{Isom}(\operatorname{\mathcal{E}}) \ar [r] & \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}}), }$

we deduce that $\theta$ is also an isofibration when regarded as a functor from $\operatorname{Fun}(Q, \operatorname{\mathcal{E}})$ to $\operatorname{Isom}(\operatorname{\mathcal{E}})$. Consequently, to show that $\theta$ is a trivial Kan fibration, it will suffice to show that it is an equivalence of $\infty$-categories (Proposition 4.5.5.20). In other words, we are reduced to proving that the morphism $e$ exhibits $Q$ as a localization of $\Delta ^1$ with respect to $W$. Let $q: Q \rightarrow \Delta ^0$ denote the projection map. Since $Q$ is contractible, the morphism $q$ is an equivalence of $\infty$-categories. By virtue of Remark 6.3.1.19, we are reduced to proving that the composite map $\Delta ^1 \xrightarrow {e} Q \xrightarrow {q} \Delta ^0$ exhibits $\Delta ^0$ as a localization of $\Delta ^1$ with respect to $W$, which follows from Example 6.3.1.14. $\square$