Exercise 3.3.1.2. Let $\operatorname{\mathcal{C}}$ be a category. Show that the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is braced if and only if $\operatorname{\mathcal{C}}$ satisfies the following condition:
- $(\ast )$
For every pair of morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow X$ in $\operatorname{\mathcal{C}}$ satisfying $g \circ f = \operatorname{id}_{X}$, we have $X = Y$ and $f = g = \operatorname{id}_{X}$.
In particular, for any partially ordered set $Q$, the nerve $\operatorname{N}_{\bullet }(Q)$ is braced.