ScalingStacks

A special case of the classification diagram is the discrete nerve. Let $C_{0}\subset C$ denote the subcategory of $C$ consisting of all its objects and only identity maps between them, and let $\discnerve C=N(C,C_{0})$. Note that $\nerve C=\diag(\discnerve C)$, and that $\discnerve([n])=F(n)$.

It is not hard to see that the functor $\discnerve\colon{\operatorname{\mathcal{C}at}}\rightarrow s{\operatorname{\mathcal{S}}}$ embeds the category of small categories as a full subcategory of simplicial spaces. The discrete nerve functor is awkward from our point of view, since equivalent categories can have non-weakly equivalent discrete nerves.