{
  "abstract": "There are many types of equivalence relations in mathematics, each one enjoying its own defnition, but many of them can also be defined by transformations of morphisms: Green's relations, orbit relations, conjugacy, equality of orbit closures, equivalence of Cauchy sequences, stable isomorphism. We unify them through the comparison category of a comparison datum, read at levels $-1,0,1,2,\\dots$ (comparisons in both directions, connected components, homotopy groups). Its normal form is a full subcategory of a Grothendieck construction $\\int_VH$ cut out by a map $s$; the classical relations are traced back to it by restriction along $s$, enrichment by a cost (limit relations) and filtered colimits (stabilization). If the map $σ$ transporting morphisms into a $2$-category is arbitrary, every relation occurs, so the constraint is the regularity of $σ$. We grade data along seven regularity classes. The rigid and diagonal classes are classified completely (on finite groups, by the double coset relations); completely functorial data contain conjugacy and obey a dichotomy on nil monoids. Whether every relation on a monoid is level $0$ of a completely functorial datum is open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38143",
  "author_total": 1,
  "authors": [
    {
      "name": "Nizar El Idrissi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c1nqfgmsjzzvq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38143",
  "fulltext_url": null,
  "id": "c1nqfgmsjzzvq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38143",
  "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"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38143",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c1nqfgmsjzzvq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38143",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38143"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Comparison categories: the normal form and the regularity of comparison data",
  "title_text": "Comparison categories: the normal form and the regularity of comparison data",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:28Z",
  "url": "https://research.draconitic.com/paper/c1nqfgmsjzzvq",
  "venue": "arXiv",
  "versions": []
}