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Factorization Systems on ∞-Categories: Un/Straightening and Monadicity

  • Thorger Geiß
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Abstract

We prove two structural results about factorization systems on ∞-categories. Firstly, we classify factorization systems on the total spaces of a class of fibrations of ∞-categories in terms of factorization systems on the base and the fibers. This will be interpreted as an un/straightening equivalence for ∞-categories equipped with a factorization system, using Juran's double ∞-categorical framework. Along the way, we develop the general theory of lifting through faithful functors of n-uple ∞-categories. Secondly, we prove that the forgetful functor from ∞-categories equipped with a factorization system to ∞-categories is a monadic right adjoint.

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Cite

BibTeXRIS
BibTeX
@misc{geiß2026factorization,
  title = {{Factorization Systems on $\infty$-Categories: Un/Straightening and Monadicity}},
  author = {Thorger Geiß},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.29261},
  eprint = {2608.29261},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29261},
}
RIS
TY  - UNPB
TI  - Factorization Systems on ∞-Categories: Un/Straightening and Monadicity
AU  - Thorger Geiß
PY  - 2026
DA  - 2026/08/29
PB  - arXiv
DO  - 10.48550/arxiv.2608.29261
UR  - https://arxiv.org/abs/2608.29261
AB  - We prove two structural results about factorization systems on ∞-categories. Firstly, we classify factorization systems on the total spaces of a class of fibrations of ∞-categories in terms of factorization systems on the base and the fibers. This will be interpreted as an un/straightening equivalence for ∞-categories equipped with a factorization system, using Juran's double ∞-categorical framework. Along the way, we develop the general theory of lifting through faithful functors of n-uple ∞-categories. Secondly, we prove that the forgetful functor from ∞-categories equipped with a factorization system to ∞-categories is a monadic right adjoint.
ER  -