# Kerodon

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Remark 4.8.8.17. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets, let $n$ be an integer, and let $A \subseteq B$ be simplicial sets. If $B$ has dimension $\leq n+1$, then every lifting problem

$\xymatrix@C =50pt@R=50pt{ A \ar [d] \ar [r] & \operatorname{\mathcal{C}}\ar [d]^{F'} \\ B \ar [r] \ar@ {-->}[ur] & \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})} }$

has a solution. Moreover, if $B$ has dimension $\leq n-1$, then the solution is unique. To prove this, we can assume without loss of generality that $B = \Delta ^ m$ is a standard simplex for some $m \leq n+1$, and that $A = \operatorname{\partial \Delta }^ m$ is its boundary (see Proposition 1.1.4.12). The case $n \leq -2$ is vacuous, and the case $n = -1$ is immediate from the definition. We may therefore assume that $n \geq 0$. Replacing $F$ by the projection map $\Delta ^ m \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\rightarrow \Delta ^ m$, we can reduce to the case where $\operatorname{\mathcal{D}}$ is a standard simplex, so that $U'$ identifies $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ with the homotopy $n$-category $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ (Example 4.8.8.11). In this case, the desired result follows from Corollary 4.8.4.17.