Classification diagrams of simplicial categories
Kensuke Arakawa
TL;DR
The paper proves that the classification diagram of a relative simplicial category $(\mathcal{C},\mathcal{W})$ is equivalent to the levelwise nerve by establishing a natural weak equivalence $N_{\mathrm{bi}}^{+}(\mathcal{C},\mathcal{W})\simeq \operatorname{Cls}^{+}(N_{\mathrm{hc}}(\mathcal{C}),\operatorname{mor}\mathcal{W}_{0})$ in the marked CSS model structure. It first treats the unmarked case via a Reedy-fibrant Segal-space replacement and a DK-style fiberwise comparison to $\operatorname{Cls}(N_{\mathrm{hc}}(\mathcal{C}))$, then passes to marked objects using the CSS+ Quillen equivalence, showing that localization is the homotopy colimit of levelwise localizations $N(\mathcal{C}_{n})[N(\mathcal{W}_{n})^{-1}]$. A natural comparison map $B(\mathcal{C})\to N_{\mathrm{hc}}(\mathcal{C})$ is constructed and shown to be a weak equivalence, linking the classical classifying space with the homotopy coherent nerve. The results yield corollaries about preservation of weak equivalences under levelwise localization and interpret localization as a homotopy colimit, with broader impact on understanding localizations of higher categories.
Abstract
We show that the classification diagram of a relative $\infty$-category arising from a relative simplicial category is equivalent to the levelwise nerve. Applications include the comparison of the diagonal of the levelwise nerve and the homotopy coherent nerve, and a result on the levelwise localizations of simplicial categories.