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}

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