Warning 3.3.3.21. In the statement of Proposition 3.3.3.16, the hypothesis that $X$ is braced cannot be omitted. For example, let $X$ be the simplicial set $\Delta ^{2} \coprod _{ \Delta ^{1} } \Delta ^0$ obtained from the standard $2$-simplex by collapsing a single edge, which we depict informally by the diagram
\[ \xymatrix@R =50pt@C=50pt{ & & \bullet \ar [ddddrr] & & \\ & & & & \\ & & & & \\ & & & & \\ \bullet \ar@ {=}[uuuurr] \ar [rrrr] & & & & \bullet .} \]
Then the subdivision of $X$ is the $2$-dimensional simplicial set depicted in the diagram
\[ \xymatrix@R =50pt@C=50pt{ & & \bullet \ar@ {=}[ddl] \ar [ddr] \ar [ddd] & & \\ & & & & \\ & \bullet \ar [dr] & & \bullet \ar [dl] & \\ & & \bullet & & \\ \bullet \ar [urr] \ar [rr] \ar@ {=}[uur] & & \bullet \ar [u] & & \bullet . \ar [ll] \ar [ull] \ar [uul] } \]
This simplicial set cannot arise as the nerve of a category, because it contains a nondegenerate $2$-simplex $\sigma $ for which $d^{2}_2(\sigma )$ is degenerate.