\([n]=\{0\to 1\to 2\to \cdots \to n\}\) is the standard \(n\)-simplex.
\(\delta^i:[n-1]\to [n], n\ge i\) is the \(i\)-th face map, which skips \(i\).
\(\sigma^i:[n+1]\to [n], n\ge i\) is the \(i\)-th degeneracy map, which duplicates \(i\).
The category of simplicial sets is the category of presheaves on the simplex category \(\Delta = \{[0], [1], [2], \ldots\}\).