Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 165 / 1 / iv / Solution

Consider a lifting square with ${\Lambda }_1^3\hookrightarrow {\Delta }^3$ on the left, $Q\to N(\mathcal{C})$ on the right, and prescribed bottom simplex $\sigma :{\Delta }^3\to N(\mathcal{C})$. Since $Q$ is a quasicategory, the top horn has some filler $\tilde{\sigma }:{\Delta }^3\to Q$.

The two simplices $p\tilde{\sigma }$ and $\sigma$ of the nerve of a category restrict to the same inner horn. Inner horns in a category's nerve have unique fillers, because composition in a category is defined and unique. Hence $p\tilde{\sigma }=\sigma$, so $\tilde{\sigma }$ is the required lift.