Culf maps and edgewise subdivision
Abstract
We show that, for any simplicial space X X , the ∞ \infty -category of culf maps over X X is equivalent to the ∞ \infty -category of right fibrations over S d ( X ) Sd(X) , the edgewise subdivision of X X . (When X X is a Rezk complete Segal or 2-Segal space, S d ( X ) Sd(X) is the twisted arrow category of X 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.
Fields of Study & Subjects
Homotopy and Cohomology in Algebraic Topology
Advanced Topics in Algebra
Algebraic structures and combinatorial models
Publication details
- Journal / source
- Transactions of the American Mathematical Society
- Volume
- 379
- Issue
- 4
- Pages
- 2955–3007
- Publication date
- 13 January 2026
- Language
- EN