ScalingStacks

Proposition 3.2. The nerve of $[n]$ is $\Delta[n]$. For categories $C$ and $D$ there are natural isomorphisms

$$\nerve(C\times D)\approx\nerve C\times\nerve D\qquad\text{and}\qquad\nerve(D^{C})\approx\nerve(D)^{\nerve(C)}.$$

The functor $\nerve\colon{\operatorname{\mathcal{C}at}}\rightarrow{\operatorname{\mathcal{S}}}$ is a full embedding of categories. Furthermore, if $C$ is a groupoid then $\nerve(C)$ is a Kan complex.