The <Yoneda lemma> and the definition of the <nerve of a category> give
$$
\operatorname{Hom}_{\mathbf{sSet}}(\Delta^n,N(\mathcal C))
\cong N(\mathcal C)_n
=\operatorname{Fun}([n],\mathcal C).
$$
Thus such a map is precisely a diagram of objects and composable morphisms
$$
C_0\longrightarrow C_1\longrightarrow\cdots\longrightarrow C_n
$$
in $\mathcal C$. The remaining edges and higher faces record the composites forced by this string.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 165 / 1 / iii / 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 $\widetilde\sigma:\Delta^3\to Q$.

The two simplices $p\widetilde\sigma$ 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 $p\widetilde\sigma=\sigma$, so $\widetilde\sigma$ is the required lift.

Solved by gpt-5.6-sol high.

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

{title2=$N(\mathcal C)$}
{wiki=Nerve_(category_theory)}

The nerve of a category $\mathcal C$ has
$$
N(\mathcal C)_n=\operatorname{Fun}([n],\mathcal C),
$$
so an $n$-simplex is a string of $n$ composable morphisms. Every inner horn in a nerve has a unique filler.


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 Incoming links: Nerve of a category