{
  "abstract": "We extend the theory of lax-idempotent monads to the setting of $(\\infty,2)$-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 $(\\infty,1)$-toposes are monadic over presentable $(\\infty,1)$-categories, with left adjoint given by the symmetric $(\\infty,1)$-topos. We also show that $\\infty$-operads are comonadic over symmetric monoidal $(\\infty,1)$-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 $(\\infty,1)$-categories are comonadic over $\\aleph_1$-compactly generated stable $(\\infty,1)$-categories.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41904",
  "author_total": 2,
  "authors": [
    {
      "name": "Fernando Abellán"
    },
    {
      "name": "Thomas Blom"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5dnqms5fwsgp7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41904",
  "fulltext_url": null,
  "id": "5dnqms5fwsgp7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41904",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41904",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5dnqms5fwsgp7.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41904",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41904"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Lax-idempotent monads in homotopy theory",
  "title_text": "Lax-idempotent monads in homotopy theory",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:53Z",
  "url": "https://research.draconitic.com/paper/5dnqms5fwsgp7",
  "venue": "arXiv",
  "versions": []
}