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

The simplex category $\Delta$ has objects $[n]=\{0<1<\cdots <n\}$ for $n\ge 0$ and order-preserving maps as morphisms. A simplicial set is a functor $X:{\Delta }^{\mathrm{op}}\to \mathbf{Set}$.

The standard simplex is the representable simplicial setFor $0\le i\le n$, the simplicial horn ${\Lambda }_i^n\subseteq {\Delta }^n$ is the union of the images of all coface maps ${\Delta }^{n-1}\to {\Delta }^n$ except the face opposite vertex $i$.

Solved by gpt-5.6-sol high.

A quasicategory is a simplicial set having the right lifting property against every inner horn inclusionA Kan complex has the right lifting property against every simplicial horn inclusion, including the two outer horns $i=0,n$.

Solved by gpt-5.6-sol high.

The Yoneda lemma and the definition of the nerve of a category giveThus such a map is precisely a diagram of objects and composable morphismsin $\mathcal{C}$. The remaining edges and higher faces record the composites forced by this string.

Solved by gpt-5.6-sol high.

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.

Solved by gpt-5.6-sol high.

The contravariant simplicial mapping space functor takes the given pushout to the stated strict pullback. Since $X$ is a Kan complex, all four mapping spaces are Kan complexes. Moreover, the monomorphism $A\hookrightarrow B$ induces a Kan fibrationIndeed, a lifting problem against a horn is adjoint to a lifting problem for $X$ against the pushout-product of $A\hookrightarrow B$ with that horn inclusion; this pushout-product is an anodyne monomorphism, and $X$ fills it.

A strict pullback of fibrant simplicial sets along a fibration computes the homotopy pullback. The displayed pullback square is therefore also a homotopy pullback square.

Solved by gpt-5.6-sol high.