ScalingStacks

Remark 5.6. Recall (3.5) in which we defined an embedding $N\colon{\operatorname{\mathcal{C}at}}\rightarrow s{\operatorname{\mathcal{S}}}$ via the classifying diagram construction (which by (4.4) in fact lands in the subcategory of Segal spaces). By (5.2) we see that $\ho NC\approx C$. It is possible to show that the functor $N$ admits a left adjoint $L\colon s{\operatorname{\mathcal{S}}}\rightarrow{\operatorname{\mathcal{C}at}}$, and that $L(W)\approx\ho W$ whenever $W$ is a Segal space.