Culf maps and edgewise subdivision

TL;DR

This paper establishes an equivalence between culf maps over any simplicial space and right fibrations over its edgewise subdivision, revealing new insights into the structure of decomposition spaces and their categorical properties.

Contribution

It proves a novel equivalence between culf maps and right fibrations over edgewise subdivisions, with two independent proofs and implications for decomposition spaces.

Findings

Equivalence between culf maps and right fibrations over sd(X)

Decomposition spaces form a locally an ∞-topos

Two new proofs using factorization systems and natural transformations

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 $\operatorname{sd}(X)$, the edgewise subdivision of $X$. (When $X$ is a Rezk complete Segal or 2-Segal space, $\operatorname{sd}(X)$ 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.