on. Face and degeneracy maps remove or insert brackets, but care must be taken when removing innermost brackets as the compositions that can then take place can result in chains with identities which then need removing, see [@cordier82], that is why the comandic description is so much simpler, as it manages all that itself.

To understand $S(\mathbb{A})$ in general it pays to examine the simplest few cases. The key cases are when $\mathbb{A} = [n]$, the ordinal $\{0< \ldots <n\}$ considered as a category in the usual way. The cases $n=0$ and $n=1$ give no surprises. $S[0]$ has one object 0 and $S[0](0,0$ is isomorphic to $\Delta[0]$, as the only simplices are degenerate copies of the identity. $S[1]$ likewise has a trivial simplicial structure, being just the category $[1]$ considered as an $\mathcal{S}$-category. Things do get more interesting at $n = 2$. The key here is the identification of $S[2](0,2)$. There are two non-degenerate strings or paths that lead from 0 to 2, so $S[2](0,2)$ will have two vertices. The bracketted string $((01)(12))$ on removing inner brackets gives $(02)$ and outer brackets, $(01)(12)$ so represents a 1-simplex $$\xymatrix@+15pt{(01)(12)\ar[r]^{\quad (01)(12))}&(02)}$$ Other simplicial homs are all $\Delta[0]$ or empty. It thus is 