Culf maps and edgewise subdivision

Abstract

We show that, for any simplicial space $X$, the $\infty$-category of culf maps over $X$ is equivalent to the $\infty$-category of right fibrations over $\mathrm{sd}(X)$, the edgewise subdivision of $X$. (When $X$ is a Rezk complete Segal or 2-Segal space, $\mathrm{sd}(X)$ is the twisted arrow category of $X$.) We give two proofs of independent interest; one exploiting comprehensive factorization and the natural transformation from the edgewise subdivision to the nerve of the category of elements, and another exploiting a new factorization system of ambifinal and culf maps, together with the right adjoint to edgewise subdivision. Using this main theorem, we show that the $\infty$-category of decomposition spaces and culf maps is locally an $\infty$-topos.

Appendix coauthored with Jan Steinebrunner. Version 2: 55 pages. Improvements following referee report. To appear in Trans AMS