Simplicial Sets

A simplicial set $X$ is a sequence of sets, $X=\{X_0, X_1,\dots, X_n,\dots\}$ , together with maps

$\begin{aligned}d_i&: X_n\to X_{n-1}\\ s_i&: X_n\to X_{n+1} \end{aligned}$

for each $0\leq i\leq n$ . These maps are required to satisfy the simplicial identities

$\begin{aligned} d_id_j&=d_{j-1}d_i\ \ \ \text{for}\ i<j\\ d_is_i&= \begin{cases}s_{j-1}d_i&\text{for}\ i<j\\ \text{id}&\text{for}\ i=j, j+1\\ s_jd_{i-1}&\text{for}\ i>j+1 \end{cases}\\ s_is_j&=s_{j+1}s_i\ \ \ \text{for}\ i\leq j \end{aligned}$

We can use deleting-doubling for remembering simplicial identities:

$\begin{aligned} d_i&: (x_0,\dots,x_n)\mapsto (x_0,\dots,x_{i-1},x_{i+1},\dots,x_n)\\ s_i&: (x_0,\dots,x_n)\mapsto (x_0,\dots,x_{i-1},x_i,x_i,x_{i+1},\dots,x_n) \end{aligned}$