{
  "abstract": "Let $\\mathcal{C}$ be a semi-abelian category, $\\mathcal{D}$ a Birkhoff subcategory of $\\mathcal{C}$, and $I$ the associated reflector. For $C\\in \\mathcal{C}$, we establish an isomorphism between $\\mathrm{H}_1(\\mathrm{N}(I(\\check{C}(f,C))))$ (where $\\mathrm{N}$ is the Moore normalization functor and $\\check{C}(f,C)$ is a Cech resolution associated to $C$) and $\\mathrm{H}_1(\\mathrm{N}(I(\\mathbb{G}(C))))$ (where $\\mathbb{G}(C)$ is the canonical $\\mathbb{G}$-simplicial resolution of $C$ for $\\mathbb{G}$ the comonad derived from an adjunction $F\\colon \\mathcal{X}\\to \\mathcal{C}\\dashv U\\colon \\mathcal{C}\\to \\mathcal{X}$ satisfying classical conditions introduced by Quillen). The former corresponds to the Hopf formula for the second homology object associated to $I$, while the latter represents the first \\v Cech derived functor of $I$. Specifically, we show that both can be expressed by a single unified formula. Furthermore, we can also prove that, under certain assumptions, the first \\v Cech derived functor of $I$ is naturally isomorphic to the first non-additive derived functor of $I$ (in the sense of Culot--Renaud--Van~der~Linden).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24720",
  "author_total": 2,
  "authors": [
    {
      "name": "Maxime Culot"
    },
    {
      "name": "Tomas Everaert"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e5qnddh5xayqf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24720",
  "fulltext_url": null,
  "id": "e5qnddh5xayqf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24720",
  "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",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7223056646",
  "pdf_url": "https://arxiv.org/pdf/2610.24720",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e5qnddh5xayqf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24720",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24720"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The first Cech derived functor",
  "title_text": "The first Cech derived functor",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/e5qnddh5xayqf",
  "venue": "arXiv",
  "versions": []
}