{
  "abstract": "We examine how the standard proof that left adjoints preserve colimits behaves in the setting of wild categories, a natural setting for synthetic homotopy theory inside homotopy type theory. We show that the proof may fail for adjunctions between wild categories and even produce a wild left adjoint that fails to preserve colimits. Our core contribution, however, is a sufficient condition on the left adjoint for the proof to go through. The condition, which we call 2-coherence, expresses that the naturality structure of the hom-isomorphism commutes with composition of morphisms. We present two useful examples of this condition in action. First, we use it, along with a new version of a known trick for homogeneous types, to show that the suspension functor, as well as a generalization thereof, preserves graph-indexed colimits. Second, we show that every modality, viewed as a functor on coslices of a type universe, is 2-coherent as a left adjoint to the forgetful functor from the subcategory of modal types, thereby proving this subcategory is cocomplete. We have formalized our main results in Agda.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28473",
  "author_total": 1,
  "authors": [
    {
      "name": "Perry Hart"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2vkdb8g67wacm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28473",
  "fulltext_url": null,
  "id": "2vkdb8g67wacm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28473",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LO"
    ],
    [
      "arxiv",
      "math.CT"
    ],
    [
      "openalex",
      "1702|Logic, programming, and type systems"
    ]
  ],
  "openalex": "W7204839617",
  "pdf_url": "https://arxiv.org/pdf/2608.28473",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2vkdb8g67wacm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.28473",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28473"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On Left Adjoints Preserving Colimits in Homotopy Type Theory",
  "title_text": "On Left Adjoints Preserving Colimits in Homotopy Type Theory",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/2vkdb8g67wacm",
  "venue": "arXiv",
  "versions": []
}