{
  "abstract": "Many kinds of generalised syntax, including internal categories, T-categories or polynomial monads, arise as monads in bicategories of a certain flavour: maps are some kind of span, and composition is some kind of pullback. We explain the pattern behind these similarities: each bicategory is stitched together from two simpler pieces by a distributive law, where the distributivity is governed by pullback. A key novelty is in the kind of distributive law we consider; where Rosebrugh and Wood considered distributive laws between categories, we instead consider pseudo-distributive laws between double categories. A key construction, drawing on work of Weber, obtains such a distributive law from a so-called co-crossed double category: one whose codomain functor is a fibration, and whose cartesian cells are stable under horizontal composition. We show the horizontal Kleisli double category of a monad T is co-crossed just when T is cartesian, and that the composite double category recovers T-spans; that the double category associated to an identity-on-objects functor is co-crossed just when that functor is a calibration in the sense of Bénabou; and that Street's calibrated bicategories yield co-crossed double categories in a similar manner, recovering the double category of polynomials of Gambino--Kock.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41551",
  "author_total": 2,
  "authors": [
    {
      "name": "Mitchell Buckley"
    },
    {
      "name": "Richard Garner"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1affknnv9v56r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41551",
  "fulltext_url": null,
  "id": "1affknnv9v56r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41551",
  "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/2610.41551",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1affknnv9v56r.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41551",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41551"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Composing Double Categories",
  "title_text": "Composing Double Categories",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:34Z",
  "url": "https://research.draconitic.com/paper/1affknnv9v56r",
  "venue": "arXiv",
  "versions": []
}