Lax-idempotent monads in homotopy theory

@misc{abellan2026laxidempotent,
  title = {{Lax-idempotent monads in homotopy theory}},
  author = {Fernando Abellán and Thomas Blom},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41904},
  eprint = {2610.41904},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41904},
}

TY  - UNPB
TI  - Lax-idempotent monads in homotopy theory
AU  - Fernando Abellán
AU  - Thomas Blom
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41904
UR  - https://arxiv.org/abs/2610.41904
AB  - We extend the theory of lax-idempotent monads to the setting of (∞,2)-categories. In particular, we prove an analogue of Kock and Zöberlein's characterization of the category of algebras over a lax-idempotent monad. We then apply this theory to many examples that arise naturally in homotopy theory. For example, we show that (∞,1)-toposes are monadic over presentable (∞,1)-categories, with left adjoint given by the symmetric (∞,1)-topos. We also show that ∞-operads are comonadic over symmetric monoidal (∞,1)-categories through the monoidal envelope construction, obtaining along the way a new proof of Barkan-Steinebrunner's result that the envelope functor is a faithful inclusion. We furthermore use this theory to simplify Ramzi's proof that dualizable stable (∞,1)-categories are comonadic over ℵ_1-compactly generated stable (∞,1)-categories.
ER  -