{
  "abstract": "We study conditions on a small category $\\mathcal{A}$ under which every finitely generated object of its presheaf category is finitely presentable. Example: this holds for a group $\\mathcal{A}$ iff $\\mathcal{A}$ is Noetherian. For ordinals $\\mathcal{A}=α$ this holds iff $α\\leq ω$, whereas this always holds for $\\mathcal{A}=α^{op}$. Various important properties of set functors also apply to endofunctors on locally finitely presentable categories which are graduated. This means that every finitely presentable object $X$ carries a grade (in $\\mathbb{N}$) and grades respect subobjects and strong quotients of $X$. We characterize presheaf categories $\\mathbb{Set}^{\\mathcal{A}^{op}}$ which are graduated for three types of small categories $\\mathcal{A}$. If $\\mathcal{A}$ is a poset, all down sets of elements must be finite. For a group $\\mathcal{A}$, a finite bound on the length of a chain of subgroups must exist. In the case of cartesian categories $\\mathcal{A}$, every object must carry a finite number of sieves. Example: all finitary set functors form a graduated category, here $\\mathcal{A}$ is the dual of finite sets. A closely related concept is a locally finitely presentable category with the descending chain condition (a DCC category): every finitely presentable object has only finite descending chains of subobjects or strong quotients. The presheaf category on a group $G$ is $DCC$ iff $G$ has no infinite chain of subgroups. Example: presheaves on the group $\\mathbb{Z}$ are not $DCC$, but finite generation implies finite presentation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30444",
  "author_total": 2,
  "authors": [
    {
      "name": "Jiri Adamek"
    },
    {
      "name": "Lurdes Sousa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/87dwh6a8qmdtn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30444",
  "fulltext_url": null,
  "id": "87dwh6a8qmdtn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30444",
  "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",
      "1703|Advanced Algebra and Logic"
    ]
  ],
  "openalex": "W7214685149",
  "pdf_url": "https://arxiv.org/pdf/2609.30444",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/87dwh6a8qmdtn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.30444",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30444"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Graduated categories of presheaves",
  "title_text": "Graduated categories of presheaves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/87dwh6a8qmdtn",
  "venue": "arXiv",
  "versions": []
}