{
  "abstract": "Let $\\mathcal{C}$ be a small category. We characterize the full initial subcategories of $\\mathcal{C}$ as precisely those whose representables give a free presentation of the constant functor $Δ_{\\mathcal{C}}\\mathbb{K}:\\mathcal{C}\\to\\mathrm{Vect}$. Such a presentation yields a formula for the limit of any functor $\\mathcal{C} \\to \\mathrm{Vect}$ as the kernel of a morphism between products of evaluations. When the linearized endomorphism ring of every object is local, we show that a minimal free presentation, whenever one exists, determines the unique minimal full initial subcategory, up to equivalence. Bass's characterization of perfect rings gives a sufficient condition for the existence of such presentations. In particular, every finite category with no nonidentity endomorphisms, such as a finite poset, contains a unique minimal initial full subcategory. In the poset case, our results recover the minimal initial subposet constructed in [BDL27] and [DL26]. When $\\mathcal{C}$ is the free category on a finite quiver, we give an explicit construction of its minimal initial full subcategories, show that every such subcategory arises from this construction, and provide algorithms with time-complexity bounds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41068",
  "author_total": 3,
  "authors": [
    {
      "name": "Thomas Brüstle"
    },
    {
      "name": "Justin Desrochers"
    },
    {
      "name": "Samuel Leblanc"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c42e4pjq0qbdh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41068",
  "fulltext_url": null,
  "id": "c42e4pjq0qbdh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41068",
  "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"
    ],
    [
      "arxiv",
      "math.RT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41068",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c42e4pjq0qbdh.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41068",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41068"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Initial Subcategories via Free Presentations of the Constant Functor",
  "title_text": "Initial Subcategories via Free Presentations of the Constant Functor",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/c42e4pjq0qbdh",
  "venue": "arXiv",
  "versions": []
}