possible to visualise $S[2]$ as a copy of $[2]$ with a 2-cell going towards the bottom: $$\xymatrix{ &1\ar[dr]&\\
0\ar[rr]\ar[ur]_{\hspace{.5cm}\Downarrow}&&2}$$ The next case $n = 3$ is even more interesting. $S[3](i,j)$ will be (i) empty if $j<i$, (ii) isomorphic to $\Delta[0] $ if $i = j$ or $i = j-1$, (iii) isomorphic to $\Delta[1]$ by the same reasoning as we just saw for $j = i + 2$ and that leaves $S[3](0,3)$. This is a square, $\Delta[1]^2$, as follows: $$\xymatrix@+25pt{(02)(23)\ar[r]^{((02)(23))}&(03)\\
(01)(12)(23)\ar[u]^{((01)(12))((23))}_{\hspace{.5cm} a}\ar[ur]^{diag}\ar[r]_{((01))((12)(23))}&(01)(13)\ar[u]_{((01)(13))}^{b\hspace{6mm}}
}$$ where the diagonal $diag = ((01)(12)(23))$, $a = (((01)(12))((23)))$ and $b = (((01))((12)(23)))$. (It is instructive to check that this is correct, firstly because I may have slipped up (!) as well as seeing the mechanism in action. Removing the innermost brackets is $d_0$, and so on.)

The case of $S[4]$ is worth doing. I will not draw the diagrams here although aspects of it have implications later, but suggest this as an exercise. As might be expected $S[4](0,4)$ is a cube.

**Remark**

The history of this construction is interesting. A variant of it, but with topologically enriched categories as the end result, is in the work of Boardman and Vogt, [@boardmanvogt] and also in Vogt’s paper, [@vogt73]. 