Kerodon

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Proposition 4.5.5.4. Let $i: A \hookrightarrow B$ be a monomorphism of simplicial sets. Then $i$ is a categorical equivalence if and only if the following condition is satisfied:

$(\ast )$

Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an isofibration of $\infty$-categories. Then every lifting problem

$\xymatrix@R =50pt@C=50pt{ A \ar [d] \ar [r] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ B \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{D}}}$

has a solution.

Proof. Assume that condition $(\ast )$ is satisfied; we will show that the morphism $i: A \hookrightarrow B$ is a categorical equivalence of simplicial sets (the converse follows from Proposition 4.5.5.1). Fix an $\infty$-category $\operatorname{\mathcal{E}}$; we wish to show that precomposition with $i$ induces a bijection $\theta : \pi _0( \operatorname{Fun}(B, \operatorname{\mathcal{E}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}(A, \operatorname{\mathcal{E}})^{\simeq } )$. The surjectivity of $\theta$ follows by applying condition $(\ast )$ to the isofibration $\operatorname{\mathcal{E}}\rightarrow \Delta ^0$, and the injectivity of $\theta$ follows by applying $\theta$ to the isofibration $\operatorname{Isom}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}\times \operatorname{\mathcal{E}}$ of Corollary 4.4.5.5. $\square$