Culf maps and edgewise subdivision
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Culf maps and edgewise subdivision. / Hackney, Philip; Kock, Joachim.
2022.Research output: Working paper › Preprint › Research
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TY - UNPB
T1 - Culf maps and edgewise subdivision
AU - Hackney, Philip
AU - Kock, Joachim
N1 - Appendix coauthored with Jan Steinebrunner. 53 pages
PY - 2022/10/20
Y1 - 2022/10/20
N2 - 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 space or 2-Segal space, this 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.
AB - 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 space or 2-Segal space, this 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.
KW - math.AT
KW - math.CT
KW - 18N50, 55U10, 18N45, 18N60, 18N55, 18A32
M3 - Preprint
BT - Culf maps and edgewise subdivision
ER -
ID: 373038342