{
  "abstract": "We give a new equivalent formulation of Verdier's octahedral axiom. An initial square in a pre-triangulated category determines an obstruction in a quotient of ordinary Hom groups. Its vanishing is equivalent to the existence of a good completion, and $\\mathrm{TR4}$ is equivalent to this vanishing for the squares associated with composable morphisms. As a consequence, if two pre-triangulations have the same underlying additive category and one is triangulated, then the other satisfies $\\mathrm{TR4}$ precisely when their relative inverse Heller comparison admits an exact lift along every short exact sequence in the Freyd category. As applications, we prove that the exotic pre-triangulated category of Díaz Cabrera--Muro is in fact a triangulated category over every algebraically closed field. We construct scalar families for type-$A_{3a-1}$ preprojective algebras over arbitrary fields and for a range of Dynkin preprojective algebras in characteristic two; in each family the zero parameter is the unique triangulated member. The local equations also yield separable descent, including canonical triangulations for non-modular equivariantizations and for the Markman--Mehrotra K3 deformation categories.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08684",
  "author_total": 2,
  "authors": [
    {
      "name": "Ping He"
    },
    {
      "name": "Bo Le"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/80t01h3nhjdqb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08684",
  "fulltext_url": null,
  "id": "80t01h3nhjdqb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08684",
  "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",
      "math.CT"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7212019656",
  "pdf_url": "https://arxiv.org/pdf/2609.08684",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/80t01h3nhjdqb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.08684",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08684"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "From Pre-triangulation to Triangulation: Obstructions and Exact Lifting",
  "title_text": "From Pre-triangulation to Triangulation: Obstructions and Exact Lifting",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/80t01h3nhjdqb",
  "venue": "arXiv",
  "versions": []
}