The <simplex category> $\Delta$ has objects $[n]=\{0<1<\cdots<n\}$ for $n\geq0$ 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 set
$$
\Delta^n_m=\operatorname{Hom}_\Delta([m],[n]).
$$
For $0\leq i\leq n$, 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.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-165/1/i/solution