ScalingStacks

Recall that every discrete simplicial space is Reedy fibrant. A discrete simplicial space $W$ is a Segal space if and only if the maps in (4.3) are isomorphisms. Thus, $W$ is a discrete Segal space if and only if it is isomorphic to the discrete nerve of some small category. In particular, the objects $F(k)$ are Segal spaces.

If $C$ is a category, then its classifying diagram $NC$ (as defined in (3.5)) is a Segal space, by (3.6) and (3.9).