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

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.